From 09acf21bbf4a7e686a26062fb5598d1fbd3fd4f7 Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Thu, 28 Dec 2023 14:29:57 +0200 Subject: [PATCH 01/39] use new k threshold --- R/diagnostics.R | 182 ++++++++++++++++++++++++++++-------------------- 1 file changed, 106 insertions(+), 76 deletions(-) diff --git a/R/diagnostics.R b/R/diagnostics.R index 29a9a1f1..1bf50749 100644 --- a/R/diagnostics.R +++ b/R/diagnostics.R @@ -6,59 +6,64 @@ #' `threshold` value, or plot observation indexes vs. diagnostic estimates. #' The **Details** section below provides a brief overview of the #' diagnostics, but we recommend consulting Vehtari, Gelman, and Gabry (2017) -#' and Vehtari, Simpson, Gelman, Yao, and Gabry (2019) for full details. +#' and Vehtari, Simpson, Gelman, Yao, and Gabry (2022) for full details. #' #' @name pareto-k-diagnostic #' @param x An object created by [loo()] or [psis()]. #' @param threshold For `pareto_k_ids()`, `threshold` is the minimum \eqn{k} -#' value to flag (default is `0.5`). For `mcse_loo()`, if any \eqn{k} -#' estimates are greater than `threshold` the MCSE estimate is returned as -#' `NA` (default is `0.7`). See **Details** for the motivation behind these -#' defaults. +#' value to flag (default is `1 - 1 / log10(S)`). For `mcse_loo()`, if any +#' \eqn{k} estimates are greater than `threshold` the MCSE estimate is +#' returned as `NA` (default is `0.7`). See **Details** for the motivation +#' behind these defaults. #' #' @details #' The reliability and approximate convergence rate of the PSIS-based estimates #' can be assessed using the estimates for the shape parameter \eqn{k} of the #' generalized Pareto distribution: -#' * If \eqn{k < 0.5} then the distribution of raw importance ratios has -#' finite variance and the central limit theorem holds. However, as \eqn{k} -#' approaches \eqn{0.5} the RMSE of plain importance sampling (IS) increases -#' significantly while PSIS has lower RMSE. +#' +#' * If \eqn{k < min(1 - 1 / log10(S), 0.7)}, where \eqn{S} is the +#' (effective) sample size PSIS estimate and the corresponding Monte +#' Carlo standard error estimate are reliable. #' -#' * If \eqn{0.5 \leq k < 1}{0.5 <= k < 1} then the variance of the raw -#' importance ratios is infinite, but the mean exists. TIS and PSIS estimates -#' have finite variance by accepting some bias. The convergence of the -#' estimate is slower with increasing \eqn{k}. -#' If \eqn{k} is between 0.5 and approximately 0.7 then we observe practically -#' useful convergence rates and Monte Carlo error estimates with PSIS (the -#' bias of TIS increases faster than the bias of PSIS). If \eqn{k > 0.7} we -#' observe impractical convergence rates and unreliable Monte Carlo error -#' estimates. +#' * If \eqn{1 - 1 / log10(S) <= k < 0.7} PSIS estimate and the +#' corresponding Monte Carlo standard error estimate are not reliable, +#' but increasing (effective) sample size \eqn{S} above 2200 may help. +#' +#' * If \eqn{0.7 <= k < 1} PSIS estimate and the corresponding Monte +#' Carlo standard error have large bias and are not reliable. Increasing +#' sample size may reduce the uncertainty in \eqn{k} estimate. +#' +#' * If \eqn{k \geq 1}{k >= 1} The target distribution is estimated to +#' have non-finite mean. PSIS estimate and the corresponding Monte +#' Carlo standard error are not well defined. Increasing sample size +#' may reduce the uncertainty in \eqn{k} estimate. #' -#' * If \eqn{k \geq 1}{k >= 1} then neither the variance nor the mean of -#' the raw importance ratios exists. The convergence rate is close to -#' zero and bias can be large with practical sample sizes. +#' \subsection{What if the estimated tail shape parameter \eqn{k} +#' exceeds diagnostic threshold}{ Importance sampling is likely to +#' work less well if the marginal posterior \eqn{p(\theta^s | y)} and +#' LOO posterior \eqn{p(\theta^s | y_{-i})} are very different, which +#' is more likely to happen with a non-robust model and highly +#' influential observations. If the estimated tail shape parameter +#' \eqn{k} exceeds the diagnostic threshold, the user should be +#' warned. (Note: If \eqn{k} is greater than the diagnostic threshold +#' then WAIC is also likely to fail, but WAIC lacks its own +#' diagnostic.) When using PSIS in the context of approximate LOO-CV, +#' we recommend one of the following actions when \eqn{k > 0.7}: #' -#' \subsection{What if the estimated tail shape parameter \eqn{k} exceeds -#' \eqn{0.5}}{ Importance sampling is likely to work less well if the -#' marginal posterior \eqn{p(\theta^s | y)} and LOO posterior -#' \eqn{p(\theta^s | y_{-i})} are very different, which is more likely to -#' happen with a non-robust model and highly influential observations. -#' If the estimated tail shape parameter \eqn{k} exceeds \eqn{0.5}, the user -#' should be warned. (Note: If \eqn{k} is greater than \eqn{0.5} -#' then WAIC is also likely to fail, but WAIC lacks its own diagnostic.) -#' In practice, we have observed good performance for values of \eqn{k} up to 0.7. -#' When using PSIS in the context of approximate LOO-CV, we recommend one of -#' the following actions when \eqn{k > 0.7}: +#' * With some additional computations, it is possible to transform +#' the MCMC draws from the posterior distribution to obtain more +#' reliable importance sampling estimates. This results in a smaller +#' shape parameter \eqn{k}. See [loo_moment_match()] and the +#' vignette *Avoiding model refits in leave-one-out cross-validation +#' with moment matching* for an example of this. #' -#' * With some additional computations, it is possible to transform the MCMC -#' draws from the posterior distribution to obtain more reliable importance -#' sampling estimates. This results in a smaller shape parameter \eqn{k}. -#' See [loo_moment_match()] for an example of this. -#' -#' * Sampling directly from \eqn{p(\theta^s | y_{-i})} for the problematic -#' observations \eqn{i}, or using \eqn{k}-fold cross-validation will generally -#' be more stable. +#' * Sampling from a leave-one-out mixture distribution (see the +#' vignette *Mixture IS leave-one-out cross-validation for +#' high-dimensional Bayesian models*), directly from \eqn{p(\theta^s +#' | y_{-i})} for the problematic observations \eqn{i}, or using +#' \eqn{K}-fold cross-validation (see the vignette *Holdout +#' validation and K-fold cross-validation of Stan programs with the +#' loo package*) will generally be more stable. #' #' * Using a model that is more robust to anomalous observations will #' generally make approximate LOO-CV more stable. @@ -75,9 +80,10 @@ #' obtain the samples from the proposal distribution via MCMC the **loo** #' package also computes estimates for the Monte Carlo error and the effective #' sample size for importance sampling, which are more accurate for PSIS than -#' for IS and TIS (see Vehtari et al (2019) for details). However, the PSIS +#' for IS and TIS (see Vehtari et al (2022) for details). However, the PSIS #' effective sample size estimate will be -#' **over-optimistic when the estimate of \eqn{k} is greater than 0.7**. +#' **over-optimistic when the estimate of \eqn{k} is greater than +#' \eqn{min(1-1/log10(S), 0.7)}**, where \eqn{S} is the sample size. #' } #' #' @seealso @@ -102,13 +108,16 @@ pareto_k_table <- function(x) { n_eff <- rep(NA, length(k)) } - kcut <- k_cut(k) + S <- dim(x)[1] + threshold <- min(ps_khat_threshold(S), 0.7) + kcut <- k_cut(k, threshold) min_n_eff <- min_n_eff_by_k(n_eff, kcut) count <- table(kcut) out <- cbind( Count = count, Proportion = prop.table(count), - "Min. n_eff" = min_n_eff + "Min. n_eff" = min_n_eff, + Threshold = threshold ) structure(out, class = c("pareto_k_table", class(out))) } @@ -117,24 +126,22 @@ pareto_k_table <- function(x) { print.pareto_k_table <- function(x, digits = 1, ...) { count <- x[, "Count"] - if (sum(count[2:4]) == 0) { - cat("\nAll Pareto k estimates are good (k < 0.5).\n") + if (sum(count[2:3]) == 0) { + cat(paste0("\nAll Pareto k estimates are good (k < ", threshold, ").\n")) } else { tab <- cbind( - " " = rep("", 4), - " " = c("(good)", "(ok)", "(bad)", "(very bad)"), + " " = rep("", 3), + " " = c("(good)", "(bad)", "(very bad)"), "Count" = .fr(count, 0), "Pct. " = paste0(.fr(100 * x[, "Proportion"], digits), "%"), - "Min. n_eff" = round(x[, "Min. n_eff"]) + # Print ESS as n_eff terms has been deprecated + "Min. ESS" = round(x[, "Min. n_eff"]) ) tab2 <- rbind(tab) cat("Pareto k diagnostic values:\n") rownames(tab2) <- format(rownames(tab2), justify = "right") print(tab2, quote = FALSE) - if (sum(count[3:4]) == 0) - cat("\nAll Pareto k estimates are ok (k < 0.7).\n") - invisible(x) } } @@ -144,7 +151,7 @@ print.pareto_k_table <- function(x, digits = 1, ...) { #' @return `pareto_k_ids()` returns an integer vector indicating which #' observations have Pareto \eqn{k} estimates above `threshold`. #' -pareto_k_ids <- function(x, threshold = 0.5) { +pareto_k_ids <- function(x, threshold = 0.7) { k <- pareto_k_values(x) which(k > threshold) } @@ -187,7 +194,8 @@ pareto_k_influence_values <- function(x) { psis_n_eff_values <- function(x) { n_eff <- x$diagnostics[["n_eff"]] if (is.null(n_eff)) { - stop("No PSIS n_eff estimates found.", call. = FALSE) + # Print ESS as n_eff terms has been deprecated + stop("No PSIS ESS estimates found.", call. = FALSE) } return(n_eff) } @@ -198,9 +206,11 @@ psis_n_eff_values <- function(x) { #' estimate for PSIS-LOO. MCSE will be NA if any Pareto \eqn{k} values are #' above `threshold`. #' -mcse_loo <- function(x, threshold = 0.7) { +mcse_loo <- function(x) { stopifnot(is.psis_loo(x)) - if (any(pareto_k_values(x) > 0.7, na.rm = TRUE)) { + S <- dim(x)[1] + threshold <- min(ps_khat_threshold(S), 0.7) + if (any(pareto_k_values(x) > threshold, na.rm = TRUE)) { return(NA) } mc_var <- x$pointwise[, "mcse_elpd_loo"]^2 @@ -212,22 +222,22 @@ mcse_loo <- function(x, threshold = 0.7) { #' @export #' @param label_points,... For the `plot()` method, if `label_points` is #' `TRUE` the observation numbers corresponding to any values of \eqn{k} -#' greater than 0.5 will be displayed in the plot. Any arguments specified in -#' `...` will be passed to [graphics::text()] and can be used -#' to control the appearance of the labels. +#' greater than the diagnostic threhold will be displayed in the plot. +#' Any arguments specified in `...` will be passed to [graphics::text()] +#' and can be used to control the appearance of the labels. #' @param diagnostic For the `plot` method, which diagnostic should be #' plotted? The options are `"k"` for Pareto \eqn{k} estimates (the -#' default) or `"n_eff"` for PSIS effective sample size estimates. +#' default) or `"ESS"` (`"n_eff"`) for PSIS effective sample size estimates. #' @param main For the `plot()` method, a title for the plot. #' #' @return The `plot()` method is called for its side effect and does not #' return anything. If `x` is the result of a call to [loo()] #' or [psis()] then `plot(x, diagnostic)` produces a plot of #' the estimates of the Pareto shape parameters (`diagnostic = "k"`) or -#' estimates of the PSIS effective sample sizes (`diagnostic = "n_eff"`). +#' estimates of the PSIS effective sample sizes (`diagnostic = "ESS"`). #' plot.psis_loo <- function(x, - diagnostic = c("k", "n_eff"), + diagnostic = c("k", "ESS", "n_eff"), ..., label_points = FALSE, main = "PSIS diagnostic plot") { @@ -240,15 +250,18 @@ plot.psis_loo <- function(x, "% of Pareto k estimates are Inf/NA/NaN and not plotted.") } - if (diagnostic == "n_eff") { + if (diagnostic == "ESS" | diagnostic == "n_eff") { n_eff <- psis_n_eff_values(x) } else { n_eff <- NULL } + S <- dim(x)[1] + threshold <- min(ps_khat_threshold(S), 0.7) plot_diagnostic( k = k, n_eff = n_eff, + threshold = threshold, ..., label_points = label_points, main = main @@ -262,7 +275,7 @@ plot.loo <- plot.psis_loo #' @export #' @rdname pareto-k-diagnostic -plot.psis <- function(x, diagnostic = c("k", "n_eff"), ..., +plot.psis <- function(x, diagnostic = c("k", "ESS", "n_eff"), ..., label_points = FALSE, main = "PSIS diagnostic plot") { plot.psis_loo(x, diagnostic = diagnostic, ..., @@ -276,6 +289,7 @@ plot.psis <- function(x, diagnostic = c("k", "n_eff"), ..., plot_diagnostic <- function(k, n_eff = NULL, + threshold = 0.7, ..., label_points = FALSE, main = "PSIS diagnostic plot") { @@ -283,7 +297,8 @@ plot_diagnostic <- graphics::plot( x = if (use_n_eff) n_eff else k, xlab = "Data point", - ylab = if (use_n_eff) "PSIS n_eff" else "Pareto shape k", + # Print ESS as n_eff terms has been deprecated + ylab = if (use_n_eff) "PSIS ESS" else "Pareto shape k", type = "n", bty = "l", yaxt = "n", @@ -297,9 +312,9 @@ plot_diagnostic <- if (!use_n_eff) { krange <- range(k, na.rm = TRUE) - breaks <- c(0, 0.5, 0.7, 1) - hex_clrs <- c("#C79999", "#A25050", "#7C0000") - ltys <- c(3, 4, 2, 1) + breaks <- c(0, threshold, 1) + hex_clrs <- c("#C79999", "#7C0000") + ltys <- c(3, 2, 1) for (j in seq_along(breaks)) { val <- breaks[j] if (in_range(val, krange)) @@ -312,20 +327,20 @@ plot_diagnostic <- } } - breaks <- c(-Inf, 0.5, 1) + breaks <- c(-Inf, threshold, 1) hex_clrs <- c("#6497b1", "#005b96", "#03396c") clrs <- ifelse( in_range(k, breaks[1:2]), hex_clrs[1], ifelse(in_range(k, breaks[2:3]), hex_clrs[2], hex_clrs[3]) ) - if (all(k < 0.5) || !label_points) { + if (all(k < threshold) || !label_points) { graphics::points(x = if (use_n_eff) n_eff else k, col = clrs, pch = 3, cex = .6) return(invisible()) } else { - graphics::points(x = if (use_n_eff) n_eff[k < 0.5] else k[k < 0.5], - col = clrs[k < 0.5], pch = 3, cex = .6) + graphics::points(x = if (use_n_eff) n_eff[k < threshold] else k[k < threshold], + col = clrs[k < threshold], pch = 3, cex = .6) sel <- !in_range(k, breaks[1:2]) dots <- list(...) txt_args <- c( @@ -349,13 +364,15 @@ plot_diagnostic <- #' #' @noRd #' @param k Vector of Pareto k estimates. -#' @return A factor variable (the same length as k) with 4 levels. +#' @return A factor variable (the same length as k) with 3 levels. #' -k_cut <- function(k) { +k_cut <- function(k, threshold) { cut( k, - breaks = c(-Inf, 0.5, 0.7, 1, Inf), - labels = c("(-Inf, 0.5]", "(0.5, 0.7]", "(0.7, 1]", "(1, Inf)") + breaks = c(-Inf, threshold, 1, Inf), + labels = c(paste0("(-Inf, ", round(threshold,2), "]"), + paste0("(", round(threshold,2), ", 1]"), + "(1, Inf)") ) } @@ -376,3 +393,16 @@ min_n_eff_by_k <- function(n_eff, kcut) { }) sapply(n_eff_split, min) } + +#' Pareto-smoothing k-hat threshold +#' +#' Given sample size S computes khat threshold for reliable Pareto +#' smoothed estimate (to have small probability of large error). See +#' section 3.2.4, equation (13). +#' @param S sample size +#' @param ... unused +#' @return threshold +#' @noRd +ps_khat_threshold <- function(S, ...) { + 1 - 1 / log10(S) +} From 51249bfce025b06d78cfeb170ecd472c2b86461f Mon Sep 17 00:00:00 2001 From: jgabry Date: Mon, 22 Jan 2024 15:35:52 -0700 Subject: [PATCH 02/39] fix some of the tests (still some failures) --- R/diagnostics.R | 3 +-- tests/testthat/test_print_plot.R | 20 +++++++------------- 2 files changed, 8 insertions(+), 15 deletions(-) diff --git a/R/diagnostics.R b/R/diagnostics.R index 1bf50749..8b9caacf 100644 --- a/R/diagnostics.R +++ b/R/diagnostics.R @@ -125,9 +125,8 @@ pareto_k_table <- function(x) { #' @export print.pareto_k_table <- function(x, digits = 1, ...) { count <- x[, "Count"] - if (sum(count[2:3]) == 0) { - cat(paste0("\nAll Pareto k estimates are good (k < ", threshold, ").\n")) + cat(paste0("\nAll Pareto k estimates are good (k < ", x[1, "Threshold"], ").\n")) } else { tab <- cbind( " " = rep("", 3), diff --git a/tests/testthat/test_print_plot.R b/tests/testthat/test_print_plot.R index 9d355901..c4d439b0 100644 --- a/tests/testthat/test_print_plot.R +++ b/tests/testthat/test_print_plot.R @@ -110,28 +110,22 @@ test_that("pareto_k_table gives correct output", { tab <- pareto_k_table(psis1) expect_output(print(tab), "Pareto k diagnostic values") - expect_identical(colnames(tab), c("Count", "Proportion", "Min. n_eff")) + expect_identical(colnames(tab), c("Count", "Proportion", "Min. n_eff", "Threshold")) expect_equal(sum(tab[, "Count"]), length(k)) expect_equal(sum(tab[, "Proportion"]), 1) - expect_equal(sum(k <= 0.5), tab[1,1]) - expect_equal(sum(k > 0.5 & k <= 0.7), tab[2,1]) - expect_equal(sum(k > 0.7 & k <= 1), tab[3,1]) - expect_equal(sum(k > 1), tab[4,1]) + expect_equal(sum(k <= tab[1, "Threshold"]), tab[1,1]) + expect_equal(sum(k > tab[1, "Threshold"] & k <= 1), tab[2,1]) + expect_equal(sum(k > 1), tab[3,1]) psis1$diagnostics$pareto_k[1:32] <- 0.4 - expect_output(print(pareto_k_table(psis1)), "All Pareto k estimates are good (k < 0.5)", - fixed = TRUE) - - psis1$diagnostics$pareto_k[1:32] <- 0.65 - expect_output(print(pareto_k_table(psis1)), "All Pareto k estimates are ok (k < 0.7)", + expect_output(print(pareto_k_table(psis1)), "All Pareto k estimates are good", fixed = TRUE) # if n_eff is NULL psis1$diagnostics$n_eff <- NULL tab2 <- pareto_k_table(psis1) - expect_output(print(tab2), "") - expect_equal(unname(tab2[, "Min. n_eff"]), rep(NA_real_, 4)) + expect_output(print(tab2), "All Pareto k estimates are good") }) @@ -144,7 +138,7 @@ test_that("psis_n_eff_values extractor works", { expect_identical(psis_n_eff_values(psis1), psis_n_eff_values(loo1)) psis1$diagnostics$n_eff <- NULL - expect_error(psis_n_eff_values(psis1), "No PSIS n_eff estimates found") + expect_error(psis_n_eff_values(psis1), "No PSIS ESS estimates found") }) test_that("mcse_loo extractor gives correct value", { From 97789553c7ae51049d7e01433be336d0e86c7a34 Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Wed, 24 Jan 2024 10:01:43 +0200 Subject: [PATCH 03/39] more updates --- R/diagnostics.R | 46 ++++++++++++++++++++++++++--------------- R/gpdfit.R | 2 +- R/importance_sampling.R | 3 ++- R/loo-glossary.R | 38 +++++++++++++++++++++++++++------- R/loo_moment_matching.R | 31 ++++++++------------------- R/print.R | 4 ++-- R/psis.R | 10 ++++----- 7 files changed, 78 insertions(+), 56 deletions(-) diff --git a/R/diagnostics.R b/R/diagnostics.R index 1bf50749..df9fb57d 100644 --- a/R/diagnostics.R +++ b/R/diagnostics.R @@ -13,31 +13,37 @@ #' @param threshold For `pareto_k_ids()`, `threshold` is the minimum \eqn{k} #' value to flag (default is `1 - 1 / log10(S)`). For `mcse_loo()`, if any #' \eqn{k} estimates are greater than `threshold` the MCSE estimate is -#' returned as `NA` (default is `0.7`). See **Details** for the motivation -#' behind these defaults. +#' returned as `NA` (default is `1 - 1 / log10(S)`). See **Details** for +#' the motivation behind these defaults. #' #' @details #' The reliability and approximate convergence rate of the PSIS-based estimates #' can be assessed using the estimates for the shape parameter \eqn{k} of the #' generalized Pareto distribution: -#' +#' #' * If \eqn{k < min(1 - 1 / log10(S), 0.7)}, where \eqn{S} is the -#' (effective) sample size PSIS estimate and the corresponding Monte -#' Carlo standard error estimate are reliable. +#' sample size PSIS estimate and the corresponding Monte Carlo +#' standard error estimate are reliable. #' #' * If \eqn{1 - 1 / log10(S) <= k < 0.7} PSIS estimate and the #' corresponding Monte Carlo standard error estimate are not reliable, #' but increasing (effective) sample size \eqn{S} above 2200 may help. -#' +#' #' * If \eqn{0.7 <= k < 1} PSIS estimate and the corresponding Monte #' Carlo standard error have large bias and are not reliable. Increasing #' sample size may reduce the uncertainty in \eqn{k} estimate. -#' +#' #' * If \eqn{k \geq 1}{k >= 1} The target distribution is estimated to #' have non-finite mean. PSIS estimate and the corresponding Monte #' Carlo standard error are not well defined. Increasing sample size #' may reduce the uncertainty in \eqn{k} estimate. #' +#' * For simplicity the nominal sample size \eqn{S} is used when +#' computing the sample size specific threshold. This is likely to +#' provide optimistic threshold, but for many purposes this is fine +#' if the MCMC effective sample size is not much smaller than the +#' nominal sample size (e.g. if MCMC-ESS > S/4). +#' #' \subsection{What if the estimated tail shape parameter \eqn{k} #' exceeds diagnostic threshold}{ Importance sampling is likely to #' work less well if the marginal posterior \eqn{p(\theta^s | y)} and @@ -109,25 +115,27 @@ pareto_k_table <- function(x) { } S <- dim(x)[1] - threshold <- min(ps_khat_threshold(S), 0.7) - kcut <- k_cut(k, threshold) + k_threshold <- min(ps_khat_threshold(S), 0.7) + kcut <- k_cut(k, k_threshold) min_n_eff <- min_n_eff_by_k(n_eff, kcut) count <- table(kcut) out <- cbind( Count = count, Proportion = prop.table(count), - "Min. n_eff" = min_n_eff, - Threshold = threshold + "Min. n_eff" = min_n_eff ) + attr(out, "k_threshold") <- k_threshold structure(out, class = c("pareto_k_table", class(out))) } #' @export print.pareto_k_table <- function(x, digits = 1, ...) { count <- x[, "Count"] + k_threshold <- attr(x, "k_threshold") if (sum(count[2:3]) == 0) { - cat(paste0("\nAll Pareto k estimates are good (k < ", threshold, ").\n")) + cat(paste0("\nAll Pareto k estimates are good (k < ", + round(k_threshold,2), ").\n")) } else { tab <- cbind( " " = rep("", 3), @@ -206,11 +214,15 @@ psis_n_eff_values <- function(x) { #' estimate for PSIS-LOO. MCSE will be NA if any Pareto \eqn{k} values are #' above `threshold`. #' -mcse_loo <- function(x) { +mcse_loo <- function(x, threshold = NULL) { stopifnot(is.psis_loo(x)) S <- dim(x)[1] - threshold <- min(ps_khat_threshold(S), 0.7) - if (any(pareto_k_values(x) > threshold, na.rm = TRUE)) { + if (is.null(threshold)) { + k_threshold <- min(ps_khat_threshold(S), 0.7) + } else { + k_threshold <- threshold + } + if (any(pareto_k_values(x) > k_threshold, na.rm = TRUE)) { return(NA) } mc_var <- x$pointwise[, "mcse_elpd_loo"]^2 @@ -256,12 +268,12 @@ plot.psis_loo <- function(x, n_eff <- NULL } S <- dim(x)[1] - threshold <- min(ps_khat_threshold(S), 0.7) + k_threshold <- min(ps_khat_threshold(S), 0.7) plot_diagnostic( k = k, n_eff = n_eff, - threshold = threshold, + threshold = k_threshold, ..., label_points = label_points, main = main diff --git a/R/gpdfit.R b/R/gpdfit.R index b7a14329..7bc7c312 100644 --- a/R/gpdfit.R +++ b/R/gpdfit.R @@ -81,7 +81,7 @@ adjust_k_wip <- function(k, n) { } -#' Inverse CDF of generalized pareto distribution +#' Inverse CDF of generalized Pareto distribution #' (assuming location parameter is 0) #' #' @noRd diff --git a/R/importance_sampling.R b/R/importance_sampling.R index c763fe75..667c8b14 100644 --- a/R/importance_sampling.R +++ b/R/importance_sampling.R @@ -184,6 +184,7 @@ do_importance_sampling <- function(log_ratios, r_eff, cores, method) { assert_importance_sampling_method_is_implemented(method) N <- ncol(log_ratios) S <- nrow(log_ratios) + k_threshold <- min(ps_khat_threshold(S), 0.7) tail_len <- n_pareto(r_eff, S) if (method == "psis") { @@ -223,7 +224,7 @@ do_importance_sampling <- function(log_ratios, r_eff, cores, method) { log_weights <- psis_apply(lw_list, "log_weights", fun_val = numeric(S)) pareto_k <- psis_apply(lw_list, "pareto_k") - throw_pareto_warnings(pareto_k) + throw_pareto_warnings(pareto_k, k_threshold) importance_sampling_object( unnormalized_log_weights = log_weights, diff --git a/R/loo-glossary.R b/R/loo-glossary.R index ab1ba813..abc544f9 100644 --- a/R/loo-glossary.R +++ b/R/loo-glossary.R @@ -68,13 +68,37 @@ #' proposal distribution). The Pareto k diagnostic estimates how far an #' individual leave-one-out distribution is from the full distribution. If #' leaving out an observation changes the posterior too much then importance -#' sampling is not able to give reliable estimate. If `k<0.5`, then the -#' corresponding component of `elpd_loo` is estimated with high accuracy. -#' If `0.50.7`, -#' then importance sampling is not able to provide useful estimate for that -#' component/observation. Pareto k is also useful as a measure of influence of -#' an observation. Highly influential observations have high k values. Very high -#' k values often indicate model misspecification, outliers or mistakes in data +#' sampling is not able to give reliable estimate. Pareto smoothing stabilizes +#' importance sampling and guarantees finite variance estimate with a +#' cost of some bias. +#' +#' The diagnostic threshold for Pareto k depends on sample size +#' \eqn{S}. For simplicity the nominal sample size \eqn{S} is used +#' when computing the sample size specific threshold. This is likely +#' to provide optimistic threshold, but for many purposes this is fine +#' if the MCMC effective sample size is not much smaller than the +#' nominal sample size (e.g. if MCMC-ESS > S/4). +#' +#' * If \eqn{k < min(1 - 1 / log10(S), 0.7)}, where \eqn{S} is the +#' sample size PSIS estimate and the corresponding Monte +#' Carlo standard error estimate are reliable. +#' +#' * If \eqn{1 - 1 / log10(S) <= k < 0.7} PSIS estimate and the +#' corresponding Monte Carlo standard error estimate are not reliable, +#' but increasing (effective) sample size \eqn{S} above 2200 may help. +#' +#' * If \eqn{0.7 <= k < 1} PSIS estimate and the corresponding Monte +#' Carlo standard error have large bias and are not reliable. Increasing +#' sample size may reduce the uncertainty in \eqn{k} estimate. +#' +#' * If \eqn{k \geq 1}{k >= 1} The target distribution is estimated to +#' have non-finite mean. PSIS estimate and the corresponding Monte +#' Carlo standard error are not well defined. Increasing sample size +#' may reduce the uncertainty in \eqn{k} estimate. +#' +#' Pareto k is also useful as a measure of influence of an observation. +#' Highly influential observations have high k values. Very high k values +#' often indicate model misspecification, outliers or mistakes in data #' processing. See Section 6 of Gabry et al. (2019) for an example. #' #' \subsection{Interpreting `p_loo` when Pareto `k` is large}{ diff --git a/R/loo_moment_matching.R b/R/loo_moment_matching.R index ceb9242c..8e860424 100644 --- a/R/loo_moment_matching.R +++ b/R/loo_moment_matching.R @@ -26,7 +26,8 @@ #' reached, there will be a warning, and increasing `max_iters` may improve #' accuracy. #' @param k_threshold Threshold value for Pareto k values above which the moment -#' matching algorithm is used. The default value is 0.5. +#' matching algorithm is used. The default value is `1 - 1 / log10(S)`, +#' where `S` is the sample size. #' @param split Logical; Indicate whether to do the split transformation or not #' at the end of moment matching for each LOO fold. #' @param cov Logical; Indicate whether to match the covariance matrix of the @@ -65,7 +66,7 @@ loo_moment_match <- function(x, ...) { loo_moment_match.default <- function(x, loo, post_draws, log_lik_i, unconstrain_pars, log_prob_upars, log_lik_i_upars, max_iters = 30L, - k_threshold = 0.7, split = TRUE, + k_threshold = NULL, split = TRUE, cov = TRUE, cores = getOption("mc.cores", 1), ...) { @@ -92,6 +93,7 @@ loo_moment_match.default <- function(x, loo, post_draws, log_lik_i, S <- dim(loo)[1] N <- dim(loo)[2] + k_threshold <- min(ps_khat_threshold(S), 0.7) pars <- post_draws(x, ...) # transform the model parameters to unconstrained space upars <- unconstrain_pars(x, pars = pars, ...) @@ -170,10 +172,10 @@ loo_moment_match.default <- function(x, loo, post_draws, log_lik_i, loo$se_looic <- loo$estimates["looic","SE"] # Warn if some Pareto ks are still high - psislw_warnings(loo$diagnostics$pareto_k) + throw_pareto_warnings(loo$diagnostics$pareto_k, k_threshold) # if we don't split, accuracy may be compromised if (!split) { - throw_large_kf_warning(kfs) + throw_large_kf_warning(kfs, k_threshold) } loo @@ -588,9 +590,10 @@ throw_moment_match_max_iters_warning <- function() { ) } -#' Warning message if not using split transformation and accuracy is compromised +#' Warning message if not using split transformation and accuracy is +#' compromised #' @noRd -throw_large_kf_warning <- function(kf, k_threshold = 0.5) { +throw_large_kf_warning <- function(kf, k_threshold) { if (any(kf > k_threshold)) { warning( "The accuracy of self-normalized importance sampling may be bad.\n", @@ -598,21 +601,5 @@ throw_large_kf_warning <- function(kf, k_threshold = 0.5) { call. = FALSE ) } - } -#' warnings about pareto k values ------------------------------------------ -#' @noRd -psislw_warnings <- function(k) { - if (any(k > 0.7)) { - .warn( - "Some Pareto k diagnostic values are too high. ", - .k_help() - ) - } else if (any(k > 0.5)) { - .warn( - "Some Pareto k diagnostic values are slightly high. ", - .k_help() - ) - } -} diff --git a/R/print.R b/R/print.R index 21577e0b..7e064a87 100644 --- a/R/print.R +++ b/R/print.R @@ -39,7 +39,7 @@ print.psis_loo <- function(x, digits = 1, plot_k = FALSE, ...) { print.loo(x, digits = digits, ...) cat("------\n") print_mcse_summary(x, digits = digits) - if (length(pareto_k_ids(x, threshold = 0.5))) { + if (length(pareto_k_ids(x, threshold = 0.7))) { cat("\n") } print(pareto_k_table(x), digits = digits) @@ -65,7 +65,7 @@ print.psis_loo_ap <- function(x, digits = 1, plot_k = FALSE, ...) { cat("------\n") cat("Posterior approximation correction used.\n") print_mcse_summary(x, digits = digits) - if (length(pareto_k_ids(x, threshold = 0.5))) { + if (length(pareto_k_ids(x, threshold = 0.7))) { cat("\n") } print(pareto_k_table(x), digits = digits) diff --git a/R/psis.R b/R/psis.R index 7065217e..642261b7 100644 --- a/R/psis.R +++ b/R/psis.R @@ -293,19 +293,17 @@ enough_tail_samples <- function(tail_len, min_len = 5) { } -#' Throw warnings about pareto k estimates +#' Throw warnings about Pareto k estimates #' #' @noRd -#' @param k A vector of pareto k estimates. +#' @param k A vector of Pareto k estimates. #' @param high The value at which to warn about slighly high estimates. #' @param too_high The value at which to warn about very high estimates. #' @return Nothing, just possibly throws warnings. #' -throw_pareto_warnings <- function(k, high = 0.5, too_high = 0.7) { - if (isTRUE(any(k > too_high))) { +throw_pareto_warnings <- function(k, k_threshold) { + if (isTRUE(any(k > k_threshold))) { .warn("Some Pareto k diagnostic values are too high. ", .k_help()) - } else if (isTRUE(any(k > high))) { - .warn("Some Pareto k diagnostic values are slightly high. ", .k_help()) } } From 26e8a7bd5fcd5c7deb0f26fb0f5c05aeef870e1c Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Wed, 24 Jan 2024 11:11:15 +0200 Subject: [PATCH 04/39] fix some tests --- tests/testthat/test_loo_moment_matching.R | 12 ++++---- tests/testthat/test_print_plot.R | 30 ++++++++----------- .../test_psis_approximate_posterior.R | 2 +- 3 files changed, 20 insertions(+), 24 deletions(-) diff --git a/tests/testthat/test_loo_moment_matching.R b/tests/testthat/test_loo_moment_matching.R index e981a689..10e80139 100644 --- a/tests/testthat/test_loo_moment_matching.R +++ b/tests/testthat/test_loo_moment_matching.R @@ -144,14 +144,14 @@ test_that("loo_moment_match.default warnings work", { expect_warning(loo_moment_match(x, loo_manual, post_draws_test, log_lik_i_test, unconstrain_pars_test, log_prob_upars_test, log_lik_i_upars_test, max_iters = 30L, - k_thres = 100, split = FALSE, - cov = TRUE, cores = 1), "Some Pareto k") + k_thres = 0.5, split = FALSE, + cov = TRUE, cores = 1), "The accuracy of self-normalized importance sampling") - expect_warning(loo_moment_match(x, loo_manual, post_draws_test, log_lik_i_test, + expect_no_warning(loo_moment_match(x, loo_manual, post_draws_test, log_lik_i_test, unconstrain_pars_test, log_prob_upars_test, log_lik_i_upars_test, max_iters = 30L, - k_thres = 0.5, split = FALSE, - cov = TRUE, cores = 1), "The accuracy of self-normalized importance sampling") + k_thres = 100, split = TRUE, + cov = TRUE, cores = 1)) expect_warning(loo_moment_match(x, loo_manual, post_draws_test, log_lik_i_test, unconstrain_pars_test, log_prob_upars_test, @@ -180,7 +180,7 @@ test_that("loo_moment_match.default works", { k_thres = 0.8, split = FALSE, cov = TRUE, cores = 1)) - # diagnostic pareto k decreases but influence pareto k stays the same + # diagnostic Pareto k decreases but influence pareto k stays the same expect_lt(loo_moment_match_object$diagnostics$pareto_k[1], loo_moment_match_object$pointwise[1,"influence_pareto_k"]) expect_equal(loo_moment_match_object$pointwise[,"influence_pareto_k"],loo_manual$pointwise[,"influence_pareto_k"]) expect_equal(loo_moment_match_object$pointwise[,"influence_pareto_k"],loo_manual$diagnostics$pareto_k) diff --git a/tests/testthat/test_print_plot.R b/tests/testthat/test_print_plot.R index 9d355901..a657f8c5 100644 --- a/tests/testthat/test_print_plot.R +++ b/tests/testthat/test_print_plot.R @@ -102,9 +102,9 @@ test_that("pareto_k_ids identifies correct observations", { }) test_that("pareto_k_table gives correct output", { - psis1$diagnostics$pareto_k[1:10] <- runif(10, 0, 0.49) - psis1$diagnostics$pareto_k[11:17] <- runif(7, 0.51, 0.69) - psis1$diagnostics$pareto_k[18:20] <- runif(3, 0.71, 0.99) + threshold <- ps_khat_threshold(dim(psis1)[1]) + psis1$diagnostics$pareto_k[1:10] <- runif(10, 0, threshold) + psis1$diagnostics$pareto_k[11:20] <- runif(10, threshold+0.01, 0.99) psis1$diagnostics$pareto_k[21:32] <- runif(12, 1, 10) k <- pareto_k_values(psis1) tab <- pareto_k_table(psis1) @@ -114,24 +114,20 @@ test_that("pareto_k_table gives correct output", { expect_equal(sum(tab[, "Count"]), length(k)) expect_equal(sum(tab[, "Proportion"]), 1) - expect_equal(sum(k <= 0.5), tab[1,1]) - expect_equal(sum(k > 0.5 & k <= 0.7), tab[2,1]) - expect_equal(sum(k > 0.7 & k <= 1), tab[3,1]) - expect_equal(sum(k > 1), tab[4,1]) - - psis1$diagnostics$pareto_k[1:32] <- 0.4 - expect_output(print(pareto_k_table(psis1)), "All Pareto k estimates are good (k < 0.5)", - fixed = TRUE) - - psis1$diagnostics$pareto_k[1:32] <- 0.65 - expect_output(print(pareto_k_table(psis1)), "All Pareto k estimates are ok (k < 0.7)", - fixed = TRUE) + expect_equal(sum(k <= threshold), tab[1,1]) + expect_equal(sum(k > threshold & k <= 1), tab[2,1]) + expect_equal(sum(k > 1), tab[3,1]) # if n_eff is NULL psis1$diagnostics$n_eff <- NULL tab2 <- pareto_k_table(psis1) expect_output(print(tab2), "") - expect_equal(unname(tab2[, "Min. n_eff"]), rep(NA_real_, 4)) + expect_equal(unname(tab2[, "Min. n_eff"]), rep(NA_real_, 3)) + + psis1$diagnostics$pareto_k[1:32] <- 0.4 + expect_output(print(pareto_k_table(psis1)), + paste0("All Pareto k estimates are good (k < ", round(threshold,2), ")"), + fixed = TRUE) }) @@ -144,7 +140,7 @@ test_that("psis_n_eff_values extractor works", { expect_identical(psis_n_eff_values(psis1), psis_n_eff_values(loo1)) psis1$diagnostics$n_eff <- NULL - expect_error(psis_n_eff_values(psis1), "No PSIS n_eff estimates found") + expect_error(psis_n_eff_values(psis1), "No PSIS ESS estimates found") }) test_that("mcse_loo extractor gives correct value", { diff --git a/tests/testthat/test_psis_approximate_posterior.R b/tests/testthat/test_psis_approximate_posterior.R index 83afd240..0529d0f4 100644 --- a/tests/testthat/test_psis_approximate_posterior.R +++ b/tests/testthat/test_psis_approximate_posterior.R @@ -47,7 +47,7 @@ test_that("Laplace approximation, normal model", { log_p <- test_data_psis_approximate_posterior$laplace_normal$log_p log_g <- test_data_psis_approximate_posterior$laplace_normal$log_q ll <- test_data_psis_approximate_posterior$laplace_normal$log_liks - expect_warning( + expect_no_warning( psis_lap <- psis_approximate_posterior(log_p = log_p, log_g = log_g, cores = 1, save_psis = FALSE) ) From 5abf6e6450995515e0394e11ded2aa78f1caceee Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Wed, 24 Jan 2024 15:11:18 +0200 Subject: [PATCH 05/39] Apply suggestions from Noa's scode review Co-authored-by: n-kall <33577035+n-kall@users.noreply.github.com> --- R/diagnostics.R | 12 ++++++------ R/loo-glossary.R | 8 ++++---- 2 files changed, 10 insertions(+), 10 deletions(-) diff --git a/R/diagnostics.R b/R/diagnostics.R index df9fb57d..1e3ea1a0 100644 --- a/R/diagnostics.R +++ b/R/diagnostics.R @@ -22,18 +22,18 @@ #' generalized Pareto distribution: #' #' * If \eqn{k < min(1 - 1 / log10(S), 0.7)}, where \eqn{S} is the -#' sample size PSIS estimate and the corresponding Monte Carlo +#' sample size, PSIS estimate and the corresponding Monte Carlo #' standard error estimate are reliable. #' -#' * If \eqn{1 - 1 / log10(S) <= k < 0.7} PSIS estimate and the +#' * If \eqn{1 - 1 / log10(S) <= k < 0.7}, PSIS estimate and the #' corresponding Monte Carlo standard error estimate are not reliable, #' but increasing (effective) sample size \eqn{S} above 2200 may help. #' -#' * If \eqn{0.7 <= k < 1} PSIS estimate and the corresponding Monte +#' * If \eqn{0.7 <= k < 1}, PSIS estimate and the corresponding Monte #' Carlo standard error have large bias and are not reliable. Increasing #' sample size may reduce the uncertainty in \eqn{k} estimate. #' -#' * If \eqn{k \geq 1}{k >= 1} The target distribution is estimated to +#' * If \eqn{k \geq 1}{k >= 1}, the target distribution is estimated to #' have non-finite mean. PSIS estimate and the corresponding Monte #' Carlo standard error are not well defined. Increasing sample size #' may reduce the uncertainty in \eqn{k} estimate. @@ -45,7 +45,7 @@ #' nominal sample size (e.g. if MCMC-ESS > S/4). #' #' \subsection{What if the estimated tail shape parameter \eqn{k} -#' exceeds diagnostic threshold}{ Importance sampling is likely to +#' exceeds the diagnostic threshold?}{ Importance sampling is likely to #' work less well if the marginal posterior \eqn{p(\theta^s | y)} and #' LOO posterior \eqn{p(\theta^s | y_{-i})} are very different, which #' is more likely to happen with a non-robust model and highly @@ -234,7 +234,7 @@ mcse_loo <- function(x, threshold = NULL) { #' @export #' @param label_points,... For the `plot()` method, if `label_points` is #' `TRUE` the observation numbers corresponding to any values of \eqn{k} -#' greater than the diagnostic threhold will be displayed in the plot. +#' greater than the diagnostic threshold will be displayed in the plot. #' Any arguments specified in `...` will be passed to [graphics::text()] #' and can be used to control the appearance of the labels. #' @param diagnostic For the `plot` method, which diagnostic should be diff --git a/R/loo-glossary.R b/R/loo-glossary.R index abc544f9..70e071f5 100644 --- a/R/loo-glossary.R +++ b/R/loo-glossary.R @@ -72,7 +72,7 @@ #' importance sampling and guarantees finite variance estimate with a #' cost of some bias. #' -#' The diagnostic threshold for Pareto k depends on sample size +#' The diagnostic threshold for Pareto \eqn{k} depends on sample size #' \eqn{S}. For simplicity the nominal sample size \eqn{S} is used #' when computing the sample size specific threshold. This is likely #' to provide optimistic threshold, but for many purposes this is fine @@ -83,15 +83,15 @@ #' sample size PSIS estimate and the corresponding Monte #' Carlo standard error estimate are reliable. #' -#' * If \eqn{1 - 1 / log10(S) <= k < 0.7} PSIS estimate and the +#' * If \eqn{1 - 1 / log10(S) <= k < 0.7}, PSIS estimate and the #' corresponding Monte Carlo standard error estimate are not reliable, #' but increasing (effective) sample size \eqn{S} above 2200 may help. #' -#' * If \eqn{0.7 <= k < 1} PSIS estimate and the corresponding Monte +#' * If \eqn{0.7 <= k < 1}, PSIS estimate and the corresponding Monte #' Carlo standard error have large bias and are not reliable. Increasing #' sample size may reduce the uncertainty in \eqn{k} estimate. #' -#' * If \eqn{k \geq 1}{k >= 1} The target distribution is estimated to +#' * If \eqn{k \geq 1}{k >= 1}, the target distribution is estimated to #' have non-finite mean. PSIS estimate and the corresponding Monte #' Carlo standard error are not well defined. Increasing sample size #' may reduce the uncertainty in \eqn{k} estimate. From 61c0725897687786a28b2036b42c789c311004d0 Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Wed, 24 Jan 2024 15:12:13 +0200 Subject: [PATCH 06/39] explain 2200 --- R/diagnostics.R | 5 ++++- 1 file changed, 4 insertions(+), 1 deletion(-) diff --git a/R/diagnostics.R b/R/diagnostics.R index 1e3ea1a0..4913c34a 100644 --- a/R/diagnostics.R +++ b/R/diagnostics.R @@ -27,7 +27,10 @@ #' #' * If \eqn{1 - 1 / log10(S) <= k < 0.7}, PSIS estimate and the #' corresponding Monte Carlo standard error estimate are not reliable, -#' but increasing (effective) sample size \eqn{S} above 2200 may help. +#' but increasing (effective) sample size \eqn{S} above 2200 may help +#' (this will increase the sample size specific threshold +#' \eqn{(1-1/log10(2200)>0.7} and then the bias specific threshold +#' 0.7 dominates). #' #' * If \eqn{0.7 <= k < 1}, PSIS estimate and the corresponding Monte #' Carlo standard error have large bias and are not reliable. Increasing From 6565f4537d7f94fb8a000b82477b8cbc4fefc32c Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Fri, 26 Jan 2024 16:01:53 +0200 Subject: [PATCH 07/39] fix threshold in print --- R/print.R | 8 ++++++-- 1 file changed, 6 insertions(+), 2 deletions(-) diff --git a/R/print.R b/R/print.R index 7e064a87..a2247580 100644 --- a/R/print.R +++ b/R/print.R @@ -39,7 +39,9 @@ print.psis_loo <- function(x, digits = 1, plot_k = FALSE, ...) { print.loo(x, digits = digits, ...) cat("------\n") print_mcse_summary(x, digits = digits) - if (length(pareto_k_ids(x, threshold = 0.7))) { + S <- dim(x)[1] + k_threshold <- ps_khat_threshold(S) + if (length(pareto_k_ids(x, threshold = k_threshold))) { cat("\n") } print(pareto_k_table(x), digits = digits) @@ -65,7 +67,9 @@ print.psis_loo_ap <- function(x, digits = 1, plot_k = FALSE, ...) { cat("------\n") cat("Posterior approximation correction used.\n") print_mcse_summary(x, digits = digits) - if (length(pareto_k_ids(x, threshold = 0.7))) { + S <- dim(x)[1] + k_threshold <- ps_khat_threshold(S) + if (length(pareto_k_ids(x, threshold = k_threshold))) { cat("\n") } print(pareto_k_table(x), digits = digits) From 12373daef14f90164d6a87ca743cd3d05520fb02 Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Fri, 26 Jan 2024 16:31:25 +0200 Subject: [PATCH 08/39] simplify use of ps_khat_threshold --- R/diagnostics.R | 15 ++++++++++----- R/importance_sampling.R | 2 +- R/loo_moment_matching.R | 2 +- 3 files changed, 12 insertions(+), 7 deletions(-) diff --git a/R/diagnostics.R b/R/diagnostics.R index 4913c34a..29801468 100644 --- a/R/diagnostics.R +++ b/R/diagnostics.R @@ -118,7 +118,7 @@ pareto_k_table <- function(x) { } S <- dim(x)[1] - k_threshold <- min(ps_khat_threshold(S), 0.7) + k_threshold <- ps_khat_threshold(S) kcut <- k_cut(k, k_threshold) min_n_eff <- min_n_eff_by_k(n_eff, kcut) count <- table(kcut) @@ -221,7 +221,7 @@ mcse_loo <- function(x, threshold = NULL) { stopifnot(is.psis_loo(x)) S <- dim(x)[1] if (is.null(threshold)) { - k_threshold <- min(ps_khat_threshold(S), 0.7) + k_threshold <- ps_khat_threshold(S) } else { k_threshold <- threshold } @@ -271,7 +271,7 @@ plot.psis_loo <- function(x, n_eff <- NULL } S <- dim(x)[1] - k_threshold <- min(ps_khat_threshold(S), 0.7) + k_threshold <- ps_khat_threshold(S) plot_diagnostic( k = k, @@ -413,11 +413,16 @@ min_n_eff_by_k <- function(n_eff, kcut) { #' #' Given sample size S computes khat threshold for reliable Pareto #' smoothed estimate (to have small probability of large error). See -#' section 3.2.4, equation (13). +#' section 3.2.4, equation (13). Sample sizes 100, 320, 1000, 2200, +#' 10000 correspond to thresholds 0.5, 0.6, 0.67, 0.7, 0.75. Although +#' with bigger sample size S we can achieve estimates with small +#' probability of large error, it is difficult to get accurate MCSE +#' estimates as the bias starts to dominate when k > 0.7 (see Section 3.2.3). +#' Thus the sample size dependend k-ht threshold is capped at 0.7. #' @param S sample size #' @param ... unused #' @return threshold #' @noRd ps_khat_threshold <- function(S, ...) { - 1 - 1 / log10(S) + min(1 - 1 / log10(S), 0.7) } diff --git a/R/importance_sampling.R b/R/importance_sampling.R index 667c8b14..a30087ca 100644 --- a/R/importance_sampling.R +++ b/R/importance_sampling.R @@ -184,7 +184,7 @@ do_importance_sampling <- function(log_ratios, r_eff, cores, method) { assert_importance_sampling_method_is_implemented(method) N <- ncol(log_ratios) S <- nrow(log_ratios) - k_threshold <- min(ps_khat_threshold(S), 0.7) + k_threshold <- ps_khat_threshold(S) tail_len <- n_pareto(r_eff, S) if (method == "psis") { diff --git a/R/loo_moment_matching.R b/R/loo_moment_matching.R index 8e860424..e3e83ac6 100644 --- a/R/loo_moment_matching.R +++ b/R/loo_moment_matching.R @@ -93,7 +93,7 @@ loo_moment_match.default <- function(x, loo, post_draws, log_lik_i, S <- dim(loo)[1] N <- dim(loo)[2] - k_threshold <- min(ps_khat_threshold(S), 0.7) + k_threshold <- ps_khat_threshold(S) pars <- post_draws(x, ...) # transform the model parameters to unconstrained space upars <- unconstrain_pars(x, pars = pars, ...) From 7ffd67c52dd6a93dd139e8fdedff4d19b7856489 Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Fri, 26 Jan 2024 17:32:59 +0200 Subject: [PATCH 09/39] fix threshold argument handling --- R/loo_moment_matching.R | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/R/loo_moment_matching.R b/R/loo_moment_matching.R index e3e83ac6..db1340a9 100644 --- a/R/loo_moment_matching.R +++ b/R/loo_moment_matching.R @@ -93,7 +93,9 @@ loo_moment_match.default <- function(x, loo, post_draws, log_lik_i, S <- dim(loo)[1] N <- dim(loo)[2] - k_threshold <- ps_khat_threshold(S) + if (is.null(k_threshold)) { + k_threshold <- ps_khat_threshold(S) + } pars <- post_draws(x, ...) # transform the model parameters to unconstrained space upars <- unconstrain_pars(x, pars = pars, ...) From eaef505cbcce9b952a15cfa7d3e5b1d517a12f45 Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Fri, 26 Jan 2024 17:33:13 +0200 Subject: [PATCH 10/39] fix tests --- tests/testthat/test_loo_subsampling.R | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/tests/testthat/test_loo_subsampling.R b/tests/testthat/test_loo_subsampling.R index ad3e7c5d..03667e1e 100644 --- a/tests/testthat/test_loo_subsampling.R +++ b/tests/testthat/test_loo_subsampling.R @@ -936,23 +936,23 @@ test_that("Test the vignette", { expect_s3_class(looss_2, c("psis_loo_ss", "psis_loo", "loo")) set.seed(4711) - expect_warning(aploo_1 <- loo_approximate_posterior(llfun_logistic, draws = parameter_draws_laplace, data = stan_df, log_p = log_p, log_g = log_g)) + expect_no_warning(aploo_1 <- loo_approximate_posterior(llfun_logistic, draws = parameter_draws_laplace, data = stan_df, log_p = log_p, log_g = log_g)) expect_output(print(aploo_1), "Computed from 2000 by 3020 log-likelihood matrix") expect_output(print(aploo_1), "elpd_loo -1968.4 15.6") expect_output(print(aploo_1), "p_loo 3.2 0.2") expect_output(print(aploo_1), "Posterior approximation correction used.") - expect_output(print(aploo_1), "\\(-Inf, 0.5\\] \\(good\\) 2989 99.0") - expect_output(print(aploo_1), "\\(0.5, 0.7\\] \\(ok\\) 31 1.0") + expect_output(print(aploo_1), "All Pareto k estimates are good") + expect_equal(length(pareto_k_ids(aploo_1,threshold=0.5)), 31) expect_s3_class(aploo_1, c("psis_loo_ap", "psis_loo", "loo")) set.seed(4711) - expect_warning(looapss_1 <- loo_subsample(llfun_logistic, draws = parameter_draws_laplace, data = stan_df, log_p = log_p, log_g = log_g, observations = 100)) + expect_no_warning(looapss_1 <- loo_subsample(llfun_logistic, draws = parameter_draws_laplace, data = stan_df, log_p = log_p, log_g = log_g, observations = 100)) expect_output(print(looapss_1), "Computed from 2000 by 100 subsampled log-likelihood") expect_output(print(looapss_1), "values from 3020 total observations.") expect_output(print(looapss_1), "elpd_loo -1968.2 15.6 0.4") expect_output(print(looapss_1), "p_loo 2.9 0.1 0.5") - expect_output(print(looapss_1), "\\(-Inf, 0.5\\] \\(good\\) 97 97.0") - expect_output(print(looapss_1), "\\(0.5, 0.7\\] \\(ok\\) 3 3.0") + expect_output(print(looapss_1), "All Pareto k estimates are good") + expect_equal(length(pareto_k_ids(looapss_1,threshold=0.5)), 3) # Loo compare set.seed(4711) From 955b4d58abf72355f921e553caaa61ea09a3ddbd Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Fri, 26 Jan 2024 17:37:42 +0200 Subject: [PATCH 11/39] Apply suggestions from code review by jgabry Co-authored-by: Jonah Gabry --- R/diagnostics.R | 26 +++++++++++++------------- 1 file changed, 13 insertions(+), 13 deletions(-) diff --git a/R/diagnostics.R b/R/diagnostics.R index 29801468..1542d4e1 100644 --- a/R/diagnostics.R +++ b/R/diagnostics.R @@ -22,28 +22,28 @@ #' generalized Pareto distribution: #' #' * If \eqn{k < min(1 - 1 / log10(S), 0.7)}, where \eqn{S} is the -#' sample size, PSIS estimate and the corresponding Monte Carlo +#' sample size, the PSIS estimate and the corresponding Monte Carlo #' standard error estimate are reliable. #' -#' * If \eqn{1 - 1 / log10(S) <= k < 0.7}, PSIS estimate and the +#' * If \eqn{1 - 1 / log10(S) <= k < 0.7}, the PSIS estimate and the #' corresponding Monte Carlo standard error estimate are not reliable, -#' but increasing (effective) sample size \eqn{S} above 2200 may help -#' (this will increase the sample size specific threshold -#' \eqn{(1-1/log10(2200)>0.7} and then the bias specific threshold +#' but increasing the (effective) sample size \eqn{S} above 2200 may help +#' (this will increase the sample size specific threshold, +#' \eqn{1-1/log10(2200)>0.7}, and then the bias specific threshold #' 0.7 dominates). #' -#' * If \eqn{0.7 <= k < 1}, PSIS estimate and the corresponding Monte +#' * If \eqn{0.7 <= k < 1}, the PSIS estimate and the corresponding Monte #' Carlo standard error have large bias and are not reliable. Increasing -#' sample size may reduce the uncertainty in \eqn{k} estimate. +#' the sample size may reduce the uncertainty in the \eqn{k} estimate. #' #' * If \eqn{k \geq 1}{k >= 1}, the target distribution is estimated to -#' have non-finite mean. PSIS estimate and the corresponding Monte -#' Carlo standard error are not well defined. Increasing sample size -#' may reduce the uncertainty in \eqn{k} estimate. +#' have non-finite mean. The PSIS estimate and the corresponding Monte +#' Carlo standard error are not well defined. Increasing the sample size +#' may reduce the uncertainty in the \eqn{k} estimate. #' #' * For simplicity the nominal sample size \eqn{S} is used when #' computing the sample size specific threshold. This is likely to -#' provide optimistic threshold, but for many purposes this is fine +#' provide an optimistic threshold, but for many purposes this is fine #' if the MCMC effective sample size is not much smaller than the #' nominal sample size (e.g. if MCMC-ESS > S/4). #' @@ -242,7 +242,7 @@ mcse_loo <- function(x, threshold = NULL) { #' and can be used to control the appearance of the labels. #' @param diagnostic For the `plot` method, which diagnostic should be #' plotted? The options are `"k"` for Pareto \eqn{k} estimates (the -#' default) or `"ESS"` (`"n_eff"`) for PSIS effective sample size estimates. +#' default), or `"ESS"` or `"n_eff"` for PSIS effective sample size estimates. #' @param main For the `plot()` method, a title for the plot. #' #' @return The `plot()` method is called for its side effect and does not @@ -265,7 +265,7 @@ plot.psis_loo <- function(x, "% of Pareto k estimates are Inf/NA/NaN and not plotted.") } - if (diagnostic == "ESS" | diagnostic == "n_eff") { + if (diagnostic == "ESS" || diagnostic == "n_eff") { n_eff <- psis_n_eff_values(x) } else { n_eff <- NULL From b36bc139ccfaab1bc9cf86264d1576dfc7123d84 Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Sun, 28 Jan 2024 15:49:52 +0200 Subject: [PATCH 12/39] reduce r_eff warnings --- R/effective_sample_sizes.R | 2 -- R/importance_sampling.R | 2 +- R/loo.R | 8 ++++---- R/loo_approximate_posterior.R | 1 - R/psis.R | 7 +++++-- 5 files changed, 10 insertions(+), 10 deletions(-) diff --git a/R/effective_sample_sizes.R b/R/effective_sample_sizes.R index 8dd78da7..1c2b6ea4 100644 --- a/R/effective_sample_sizes.R +++ b/R/effective_sample_sizes.R @@ -173,7 +173,6 @@ psis_n_eff <- function(w, ...) { psis_n_eff.default <- function(w, r_eff = NULL, ...) { ss <- sum(w^2) if (is.null(r_eff)) { - warning("PSIS n_eff not adjusted based on MCMC n_eff.", call. = FALSE) return(1 / ss) } stopifnot(length(r_eff) == 1) @@ -182,7 +181,6 @@ psis_n_eff.default <- function(w, r_eff = NULL, ...) { psis_n_eff.matrix <- function(w, r_eff = NULL, ...) { ss <- colSums(w^2) if (is.null(r_eff)) { - warning("PSIS n_eff not adjusted based on MCMC n_eff.", call. = FALSE) return(1 / ss) } if (length(r_eff) != length(ss)) diff --git a/R/importance_sampling.R b/R/importance_sampling.R index a30087ca..583493f6 100644 --- a/R/importance_sampling.R +++ b/R/importance_sampling.R @@ -128,7 +128,7 @@ implemented_is_methods <- function() c("psis", "tis", "sis") #' but unnormalized. #' @param pareto_k Vector of GPD k estimates. #' @param tail_len Vector of tail lengths used to fit GPD. -#' @param r_eff Vector of relative MCMC n_eff for `exp(log lik)` +#' @param r_eff Vector of relative MCMC ESS (n_eff) for `exp(log lik)` #' @template is_method #' @return A list of class `"psis"` with structure described in the main doc at #' the top of this file. diff --git a/R/loo.R b/R/loo.R index fab18e62..850623ca 100644 --- a/R/loo.R +++ b/R/loo.R @@ -13,10 +13,10 @@ #' @param r_eff Vector of relative effective sample size estimates for the #' likelihood (`exp(log_lik)`) of each observation. This is related to #' the relative efficiency of estimating the normalizing term in -#' self-normalizing importance sampling when using posterior draws obtained +#' self-normalized importance sampling when using posterior draws obtained #' with MCMC. If MCMC draws are used and `r_eff` is not provided then #' the reported PSIS effective sample sizes and Monte Carlo error estimates -#' will be over-optimistic. If the posterior draws are independent then +#' can be over-optimistic. If the posterior draws are independent then #' `r_eff=1` and can be omitted. The warning message thrown when `r_eff` is #' not specified can be disabled by setting `r_eff` to `NA`. See the #' [relative_eff()] helper functions for computing `r_eff`. @@ -522,8 +522,8 @@ mcse_elpd <- function(ll, lw, E_elpd, r_eff, n_samples = 1000) { throw_loo_r_eff_warning <- function() { warning( "Relative effective sample sizes ('r_eff' argument) not specified.\n", - "For models fit with MCMC, the reported PSIS effective sample sizes and \n", - "MCSE estimates will be over-optimistic.", + "For models fit with MCMC, the reported PSIS ESS and \n", + "MCSE estimates can be over-optimistic.", call. = FALSE ) } diff --git a/R/loo_approximate_posterior.R b/R/loo_approximate_posterior.R index 72307bb7..4b67c32f 100644 --- a/R/loo_approximate_posterior.R +++ b/R/loo_approximate_posterior.R @@ -176,7 +176,6 @@ loo_approximate_posterior.function <- save_psis = FALSE, is_method) { - if (!is.null(r_eff)) warning("r_eff not implemented for aploo.") if (is_method != "psis") stop(is_method, " not implemented for aploo.") d_i <- data[i, , drop = FALSE] ll_i <- llfun(data_i = d_i, draws = draws, ...) diff --git a/R/psis.R b/R/psis.R index 642261b7..99727cf0 100644 --- a/R/psis.R +++ b/R/psis.R @@ -273,11 +273,14 @@ psis_smooth_tail <- function(x, cutoff) { #' 20% of the total number of weights. #' #' @noRd -#' @param r_eff A N-vector of relative MCMC effective sample sizes of `exp(log-lik matrix)`. +#' @param r_eff A N-vector of relative MCMC effective sample sizes of `exp(log-lik matrix)`. If NULL, relative efficiency of 1 is used. #' @param S The (integer) size of posterior sample. #' @return An N-vector of tail lengths. #' n_pareto <- function(r_eff, S) { + if is.null(r_eff) { + r_eff <- 1 + } ceiling(pmin(0.2 * S, 3 * sqrt(S / r_eff))) } @@ -383,7 +386,7 @@ called_from_loo <- function() { throw_psis_r_eff_warning <- function() { warning( "Relative effective sample sizes ('r_eff' argument) not specified. ", - "PSIS n_eff will not be adjusted based on MCMC n_eff.", + "PSIS ESS (n_eff) will not be adjusted based on MCMC ESS (n_eff).", call. = FALSE ) } From a378f7fcf6c108199860eb76a8531e2b968be4d8 Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Sun, 28 Jan 2024 16:35:38 +0200 Subject: [PATCH 13/39] doc and refs updates --- R/diagnostics.R | 51 ++++++++++++++---------- R/loo-glossary.R | 45 +++++++++++++-------- R/loo-package.R | 4 +- R/loo.R | 2 +- R/loo_compare.R | 3 ++ R/loo_model_weights.R | 2 +- R/psis.R | 2 +- R/waic.R | 2 +- man-roxygen/loo-and-compare-references.R | 10 +++-- man-roxygen/loo-and-psis-references.R | 4 +- man-roxygen/loo-uncertainty-reference.R | 4 ++ 11 files changed, 81 insertions(+), 48 deletions(-) create mode 100644 man-roxygen/loo-uncertainty-reference.R diff --git a/R/diagnostics.R b/R/diagnostics.R index 29801468..1a885f25 100644 --- a/R/diagnostics.R +++ b/R/diagnostics.R @@ -11,41 +11,52 @@ #' @name pareto-k-diagnostic #' @param x An object created by [loo()] or [psis()]. #' @param threshold For `pareto_k_ids()`, `threshold` is the minimum \eqn{k} -#' value to flag (default is `1 - 1 / log10(S)`). For `mcse_loo()`, if any -#' \eqn{k} estimates are greater than `threshold` the MCSE estimate is -#' returned as `NA` (default is `1 - 1 / log10(S)`). See **Details** for -#' the motivation behind these defaults. +#' +#' value to flag (default is a sample size `S` dependend threshold +#' `1 - 1 / log10(S)`). For `mcse_loo()`, if any \eqn{k} estimates are +#' greater than `threshold` the MCSE estimate is returned as `NA` +#' See **Details** for the motivation behind these defaults. #' #' @details -#' The reliability and approximate convergence rate of the PSIS-based estimates -#' can be assessed using the estimates for the shape parameter \eqn{k} of the -#' generalized Pareto distribution: +#' +#' The reliability and approximate convergence rate of the PSIS-based +#' estimates can be assessed using the estimates for the shape +#' parameter \eqn{k} of the generalized Pareto distribution. The +#' diagnostic threshold for Pareto \eqn{k} depends on sample size +#' \eqn{S} (sample size dependent threshold was introduced by Vehtari +#' et al., 2022, and before that fixed thresholds of 0.5 and 0.7 were +#' recommended). For simplicity, `loo` package uses the nominal sample +#' size \eqn{S} when computing the sample size specific +#' threshold. This provides an optimistic threshold if the effective +#' sample size is less than 2200, but if MCMC-ESS > S/2 the difference +#' is usually negligible. Thinning of MCMC draws can be used to +#' improve the ratio ESS/S. #' #' * If \eqn{k < min(1 - 1 / log10(S), 0.7)}, where \eqn{S} is the #' sample size, PSIS estimate and the corresponding Monte Carlo #' standard error estimate are reliable. #' #' * If \eqn{1 - 1 / log10(S) <= k < 0.7}, PSIS estimate and the -#' corresponding Monte Carlo standard error estimate are not reliable, -#' but increasing (effective) sample size \eqn{S} above 2200 may help -#' (this will increase the sample size specific threshold -#' \eqn{(1-1/log10(2200)>0.7} and then the bias specific threshold -#' 0.7 dominates). +#' corresponding Monte Carlo standard error estimate are not +#' reliable, but increasing (effective) sample size \eqn{S} above +#' 2200 may help (this will increase the sample size specific +#' threshold \eqn{(1-1/log10(2200)>0.7} and then the bias specific +#' threshold 0.7 dominates). #' #' * If \eqn{0.7 <= k < 1}, PSIS estimate and the corresponding Monte #' Carlo standard error have large bias and are not reliable. Increasing #' sample size may reduce the uncertainty in \eqn{k} estimate. #' +#' * If \eqn{0.7 <= k < 1}, PSIS estimate and the corresponding Monte +#' Carlo standard error have large bias and are not reliable. Increasing +#' sample size may reduce the variability in \eqn{k} estimate, which +#' may result in lower \eqn{k} estimate, too. +#' #' * If \eqn{k \geq 1}{k >= 1}, the target distribution is estimated to #' have non-finite mean. PSIS estimate and the corresponding Monte #' Carlo standard error are not well defined. Increasing sample size -#' may reduce the uncertainty in \eqn{k} estimate. -#' -#' * For simplicity the nominal sample size \eqn{S} is used when -#' computing the sample size specific threshold. This is likely to -#' provide optimistic threshold, but for many purposes this is fine -#' if the MCMC effective sample size is not much smaller than the -#' nominal sample size (e.g. if MCMC-ESS > S/4). +#' may reduce the variability in \eqn{k} estimate, which +#' may result in lower \eqn{k} estimate, too. #' #' \subsection{What if the estimated tail shape parameter \eqn{k} #' exceeds the diagnostic threshold?}{ Importance sampling is likely to @@ -55,7 +66,7 @@ #' influential observations. If the estimated tail shape parameter #' \eqn{k} exceeds the diagnostic threshold, the user should be #' warned. (Note: If \eqn{k} is greater than the diagnostic threshold -#' then WAIC is also likely to fail, but WAIC lacks its own +#' then WAIC is also likely to fail, but WAIC lacks as accurate #' diagnostic.) When using PSIS in the context of approximate LOO-CV, #' we recommend one of the following actions when \eqn{k > 0.7}: #' diff --git a/R/loo-glossary.R b/R/loo-glossary.R index 70e071f5..e7ae8741 100644 --- a/R/loo-glossary.R +++ b/R/loo-glossary.R @@ -3,6 +3,7 @@ #' @name loo-glossary #' #' @template loo-and-psis-references +#' @template loo-uncertainty-reference #' @template bayesvis-reference #' #' @description @@ -38,7 +39,8 @@ #' estimate is an accurate estimate for the scale, it ignores the skewness. When #' making model comparisons, the SE of the component-wise (pairwise) differences #' should be used instead (see the `se_diff` section below and Eq 24 in -#' VGG2017). +#' VGG2017). Sivula et al. (2022) discuss the conditions when the normal +#' approximation used for SE and `se_diff` is good. #' #' @section Monte Carlo SE of elpd_loo: #' @@ -62,10 +64,10 @@ #' #' @section Pareto k estimates: #' -#' The Pareto `k` estimate is a diagnostic for Pareto smoothed importance +#' The Pareto \eqn{k} estimate is a diagnostic for Pareto smoothed importance #' sampling (PSIS), which is used to compute components of `elpd_loo`. In -#' importance-sampling LOO (the full posterior distribution is used as the -#' proposal distribution). The Pareto k diagnostic estimates how far an +#' importance-sampling LOO the full posterior distribution is used as the +#' proposal distribution. The Pareto k diagnostic estimates how far an #' individual leave-one-out distribution is from the full distribution. If #' leaving out an observation changes the posterior too much then importance #' sampling is not able to give reliable estimate. Pareto smoothing stabilizes @@ -73,33 +75,42 @@ #' cost of some bias. #' #' The diagnostic threshold for Pareto \eqn{k} depends on sample size -#' \eqn{S}. For simplicity the nominal sample size \eqn{S} is used -#' when computing the sample size specific threshold. This is likely -#' to provide optimistic threshold, but for many purposes this is fine -#' if the MCMC effective sample size is not much smaller than the -#' nominal sample size (e.g. if MCMC-ESS > S/4). +#' \eqn{S} (sample size dependent threshold was introduced by Vehtari +#' et al., 2022, and before that fixed thresholds of 0.5 and 0.7 were +#' recommended). For simplicity, `loo` package uses the nominal sample +#' size \eqn{S} when computing the sample size specific +#' threshold. This provides an optimistic threshold if the effective +#' sample size is less than 2200, but even then if ESS/S>1/2 the difference +#' is usually negligible. Thinning of MCMC draws can be used to improve +#' the ratio ESS/S. #' #' * If \eqn{k < min(1 - 1 / log10(S), 0.7)}, where \eqn{S} is the #' sample size PSIS estimate and the corresponding Monte #' Carlo standard error estimate are reliable. #' #' * If \eqn{1 - 1 / log10(S) <= k < 0.7}, PSIS estimate and the -#' corresponding Monte Carlo standard error estimate are not reliable, -#' but increasing (effective) sample size \eqn{S} above 2200 may help. +#' corresponding Monte Carlo standard error estimate are not +#' reliable, but increasing (effective) sample size \eqn{S} above +#' 2200 may help (this will increase the sample size specific +#' threshold \eqn{(1-1/log10(2200)>0.7} and then the bias specific +#' threshold 0.7 dominates). #' #' * If \eqn{0.7 <= k < 1}, PSIS estimate and the corresponding Monte #' Carlo standard error have large bias and are not reliable. Increasing -#' sample size may reduce the uncertainty in \eqn{k} estimate. +#' sample size may reduce the variability in \eqn{k} estimate, which +#' may result in lower \eqn{k} estimate, too. #' #' * If \eqn{k \geq 1}{k >= 1}, the target distribution is estimated to #' have non-finite mean. PSIS estimate and the corresponding Monte #' Carlo standard error are not well defined. Increasing sample size -#' may reduce the uncertainty in \eqn{k} estimate. +#' may reduce the variability in \eqn{k} estimate, which +#' may result in lower \eqn{k} estimate, too. #' -#' Pareto k is also useful as a measure of influence of an observation. -#' Highly influential observations have high k values. Very high k values -#' often indicate model misspecification, outliers or mistakes in data -#' processing. See Section 6 of Gabry et al. (2019) for an example. +#' Pareto \eqn{k} is also useful as a measure of influence of an +#' observation. Highly influential observations have high \eqn{k} +#' values. Very high \eqn{k} values often indicate model +#' misspecification, outliers or mistakes in data processing. See +#' Section 6 of Gabry et al. (2019) for an example. #' #' \subsection{Interpreting `p_loo` when Pareto `k` is large}{ #' If `k > 0.7` then we can also look at the `p_loo` estimate for diff --git a/R/loo-package.R b/R/loo-package.R index a1947b03..92682835 100644 --- a/R/loo-package.R +++ b/R/loo-package.R @@ -13,7 +13,7 @@ #' *Stan Development Team* #' #' This package implements the methods described in Vehtari, Gelman, and -#' Gabry (2017), Vehtari, Simpson, Gelman, Yao, and Gabry (2019), and +#' Gabry (2017), Vehtari, Simpson, Gelman, Yao, and Gabry (2022), and #' Yao et al. (2018). To get started see the **loo** package #' [vignettes](https://mc-stan.org/loo/articles/index.html), the #' [loo()] function for efficient approximate leave-one-out @@ -33,7 +33,7 @@ #' fast and stable computations for approximate LOO-CV laid out in Vehtari, #' Gelman, and Gabry (2017). From existing posterior simulation draws, we #' compute LOO-CV using Pareto smoothed importance sampling (PSIS; Vehtari, -#' Simpson, Gelman, Yao, and Gabry, 2019), a new procedure for stabilizing +#' Simpson, Gelman, Yao, and Gabry, 2022), a new procedure for stabilizing #' and diagnosing importance weights. As a byproduct of our calculations, #' we also obtain approximate standard errors for estimated predictive #' errors and for comparing of predictive errors between two models. diff --git a/R/loo.R b/R/loo.R index 850623ca..9d19df84 100644 --- a/R/loo.R +++ b/R/loo.R @@ -4,7 +4,7 @@ #' CV, efficient approximate leave-one-out (LOO) cross-validation for Bayesian #' models using Pareto smoothed importance sampling ([PSIS][psis()]). This is #' an implementation of the methods described in Vehtari, Gelman, and Gabry -#' (2017) and Vehtari, Simpson, Gelman, Yao, and Gabry (2019). +#' (2017) and Vehtari, Simpson, Gelman, Yao, and Gabry (2022). #' #' @export loo loo.array loo.matrix loo.function #' @param x A log-likelihood array, matrix, or function. The **Methods (by class)** diff --git a/R/loo_compare.R b/R/loo_compare.R index b41801fa..3bd1d765 100644 --- a/R/loo_compare.R +++ b/R/loo_compare.R @@ -41,6 +41,8 @@ #' standard approach of comparing differences of deviances to a Chi-squared #' distribution, a practice derived for Gaussian linear models or #' asymptotically, and which only applies to nested models in any case. +#' Sivula et al. (2022) discuss the conditions when the normal +#' approximation used for SE and `se_diff` is good. #' #' If more than \eqn{11} models are compared, we internally recompute the model #' differences using the median model by ELPD as the baseline model. We then @@ -50,6 +52,7 @@ #' selection process. In that case users are recommended to avoid model #' selection based on LOO-CV, and instead to favor model averaging/stacking or #' projection predictive inference. +#' #' @seealso #' * The [FAQ page](https://mc-stan.org/loo/articles/online-only/faq.html) on #' the __loo__ website for answers to frequently asked questions. diff --git a/R/loo_model_weights.R b/R/loo_model_weights.R index 0cd859cc..ac39c975 100644 --- a/R/loo_model_weights.R +++ b/R/loo_model_weights.R @@ -3,7 +3,7 @@ #' Model averaging via stacking of predictive distributions, pseudo-BMA #' weighting or pseudo-BMA+ weighting with the Bayesian bootstrap. See Yao et #' al. (2018), Vehtari, Gelman, and Gabry (2017), and Vehtari, Simpson, -#' Gelman, Yao, and Gabry (2019) for background. +#' Gelman, Yao, and Gabry (2022) for background. #' #' @export #' @param x A list of `"psis_loo"` objects (objects returned by [loo()]) or diff --git a/R/psis.R b/R/psis.R index 99727cf0..55b5813a 100644 --- a/R/psis.R +++ b/R/psis.R @@ -3,7 +3,7 @@ #' Implementation of Pareto smoothed importance sampling (PSIS), a method for #' stabilizing importance ratios. The version of PSIS implemented here #' corresponds to the algorithm presented in Vehtari, Simpson, Gelman, Yao, -#' and Gabry (2019). +#' and Gabry (2022). #' For PSIS diagnostics see the [pareto-k-diagnostic] page. #' #' @export diff --git a/R/waic.R b/R/waic.R index b95ca65d..c74dc660 100644 --- a/R/waic.R +++ b/R/waic.R @@ -29,7 +29,7 @@ #' @seealso #' * The __loo__ package [vignettes](https://mc-stan.org/loo/articles/) and #' Vehtari, Gelman, and Gabry (2017) and Vehtari, Simpson, Gelman, Yao, -#' and Gabry (2019) for more details on why we prefer `loo()` to `waic()`. +#' and Gabry (2022) for more details on why we prefer `loo()` to `waic()`. #' * [loo_compare()] for comparing models on approximate LOO-CV or WAIC. #' #' @references diff --git a/man-roxygen/loo-and-compare-references.R b/man-roxygen/loo-and-compare-references.R index f8afce31..c0003b4a 100644 --- a/man-roxygen/loo-and-compare-references.R +++ b/man-roxygen/loo-and-compare-references.R @@ -5,10 +5,14 @@ #' ([journal version](https://link.springer.com/article/10.1007/s11222-016-9696-4), #' [preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544)). #' -#' Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). +#' Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). #' Pareto smoothed importance sampling. #' [preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646) #' -#' McLatchie, Y., and Vehtari, A. (2023). -#' Efficient estimation and correction of selection-induced bias with order statistics. +#' Sivula, T, Magnusson, M., Matamoros A. A., and Vehtari, A. (2022). +#' Uncertainty in Bayesian leave-one-out cross-validation based model +#' comparison. [preprint arXiv:2008.10296v3.](https://arxiv.org/abs/2008.10296v3). +#' +#' McLatchie, Y., and Vehtari, A. (2023). Efficient estimation and +#' correction of selection-induced bias with order statistics. #' [preprint arXiv:2309.03742](https://arxiv.org/abs/2309.03742) diff --git a/man-roxygen/loo-and-psis-references.R b/man-roxygen/loo-and-psis-references.R index 6e6de879..24bdb072 100644 --- a/man-roxygen/loo-and-psis-references.R +++ b/man-roxygen/loo-and-psis-references.R @@ -5,7 +5,7 @@ #' ([journal version](https://link.springer.com/article/10.1007/s11222-016-9696-4), #' [preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544)). #' -#' Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). +#' Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). #' Pareto smoothed importance sampling. #' [preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646) -#' +#' diff --git a/man-roxygen/loo-uncertainty-reference.R b/man-roxygen/loo-uncertainty-reference.R new file mode 100644 index 00000000..aca9a9d7 --- /dev/null +++ b/man-roxygen/loo-uncertainty-reference.R @@ -0,0 +1,4 @@ +#' @references Sivula, T, Magnusson, M., Matamoros A. A., and Vehtari, +#' A. (2022). Uncertainty in Bayesian leave-one-out +#' cross-validation based model comparison. [preprint +#' arXiv:2008.10296v3.](https://arxiv.org/abs/2008.10296v3). From 57e9061c8bf08d5972bd4f90fe900536b50a9bf8 Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Sun, 28 Jan 2024 16:46:24 +0200 Subject: [PATCH 14/39] fix tests --- R/psis.R | 2 +- tests/testthat/test_loo_and_waic.R | 6 +++--- tests/testthat/test_psis.R | 4 ++-- 3 files changed, 6 insertions(+), 6 deletions(-) diff --git a/R/psis.R b/R/psis.R index 55b5813a..6c9f7236 100644 --- a/R/psis.R +++ b/R/psis.R @@ -278,7 +278,7 @@ psis_smooth_tail <- function(x, cutoff) { #' @return An N-vector of tail lengths. #' n_pareto <- function(r_eff, S) { - if is.null(r_eff) { + if (is.null(r_eff)) { r_eff <- 1 } ceiling(pmin(0.2 * S, 3 * sqrt(S / r_eff))) diff --git a/tests/testthat/test_loo_and_waic.R b/tests/testthat/test_loo_and_waic.R index 7116011d..e8a4a438 100644 --- a/tests/testthat/test_loo_and_waic.R +++ b/tests/testthat/test_loo_and_waic.R @@ -195,8 +195,8 @@ test_that("save_psis option to loo.function makes correct psis object", { }) test_that("loo throws r_eff warnings", { - expect_warning(loo(-LLarr), "MCSE estimates will be over-optimistic") - expect_warning(loo(-LLmat), "MCSE estimates will be over-optimistic") - expect_warning(loo(llfun, data = data, draws = draws), "MCSE estimates will be over-optimistic") + expect_warning(loo(-LLarr), "MCSE estimates can be over-optimistic") + expect_warning(loo(-LLmat), "MCSE estimates can be over-optimistic") + expect_warning(loo(llfun, data = data, draws = draws), "MCSE estimates can be over-optimistic") }) diff --git a/tests/testthat/test_psis.R b/tests/testthat/test_psis.R index 6163c9e6..b090e62a 100644 --- a/tests/testthat/test_psis.R +++ b/tests/testthat/test_psis.R @@ -123,8 +123,8 @@ test_that("psis_n_eff methods works properly", { psis_n_eff.default(w[, 1], r_eff = 2), psis_n_eff.matrix(w, r_eff = rep(2, ncol(w)))[1] ) - expect_warning(psis_n_eff.default(w[, 1]), "not adjusted based on MCMC n_eff") - expect_warning(psis_n_eff.matrix(w), "not adjusted based on MCMC n_eff") + expect_no_warning(psis_n_eff.default(w[, 1])) + expect_no_warning(psis_n_eff.matrix(w)) }) From 0e03606725338fb19ceee1953f17650a9a5bce8e Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Sun, 28 Jan 2024 18:16:37 +0200 Subject: [PATCH 15/39] fixes suggested by Jonah --- R/diagnostics.R | 7 +++++-- R/psis.R | 3 +-- 2 files changed, 6 insertions(+), 4 deletions(-) diff --git a/R/diagnostics.R b/R/diagnostics.R index 387336fe..b6c9d6a8 100644 --- a/R/diagnostics.R +++ b/R/diagnostics.R @@ -11,7 +11,6 @@ #' @name pareto-k-diagnostic #' @param x An object created by [loo()] or [psis()]. #' @param threshold For `pareto_k_ids()`, `threshold` is the minimum \eqn{k} -#' #' value to flag (default is a sample size `S` dependend threshold #' `1 - 1 / log10(S)`). For `mcse_loo()`, if any \eqn{k} estimates are #' greater than `threshold` the MCSE estimate is returned as `NA` @@ -173,7 +172,11 @@ print.pareto_k_table <- function(x, digits = 1, ...) { #' @return `pareto_k_ids()` returns an integer vector indicating which #' observations have Pareto \eqn{k} estimates above `threshold`. #' -pareto_k_ids <- function(x, threshold = 0.7) { +pareto_k_ids <- function(x, threshold = NULL) { + if (is.null(threshold)) { + S <- dim(x)[1] + threshold <- ps_khat_threshold(S) + } k <- pareto_k_values(x) which(k > threshold) } diff --git a/R/psis.R b/R/psis.R index 6c9f7236..68778084 100644 --- a/R/psis.R +++ b/R/psis.R @@ -300,8 +300,7 @@ enough_tail_samples <- function(tail_len, min_len = 5) { #' #' @noRd #' @param k A vector of Pareto k estimates. -#' @param high The value at which to warn about slighly high estimates. -#' @param too_high The value at which to warn about very high estimates. +#' @param k_threshold The value at which to warn about high Pareto k estimates. #' @return Nothing, just possibly throws warnings. #' throw_pareto_warnings <- function(k, k_threshold) { From a329a135775046457b64baa048b4b21aac4c346f Mon Sep 17 00:00:00 2001 From: jgabry Date: Mon, 29 Jan 2024 14:57:32 -0700 Subject: [PATCH 16/39] regenerate doc --- DESCRIPTION | 2 +- man/compare.Rd | 2 +- man/loo-glossary.Rd | 62 +++++++++++--- man/loo-package.Rd | 37 ++++++++- man/loo.Rd | 8 +- man/loo_compare.Rd | 31 ++++--- man/loo_model_weights.Rd | 4 +- man/loo_moment_match.Rd | 5 +- man/loo_subsample.Rd | 4 +- man/parallel_psis_list.Rd | 4 +- man/pareto-k-diagnostic.Rd | 129 +++++++++++++++++------------- man/psis.Rd | 4 +- man/psis_approximate_posterior.Rd | 2 +- man/psislw.Rd | 2 +- man/sis.Rd | 2 +- man/update.psis_loo_ss.Rd | 4 +- man/waic.Rd | 4 +- 17 files changed, 200 insertions(+), 106 deletions(-) diff --git a/DESCRIPTION b/DESCRIPTION index 6229b79f..9bbe68bc 100644 --- a/DESCRIPTION +++ b/DESCRIPTION @@ -53,5 +53,5 @@ Suggests: VignetteBuilder: knitr Encoding: UTF-8 SystemRequirements: pandoc (>= 1.12.3), pandoc-citeproc -RoxygenNote: 7.2.3 +RoxygenNote: 7.3.0 Roxygen: list(markdown = TRUE) diff --git a/man/compare.Rd b/man/compare.Rd index b0994500..0e8a437f 100644 --- a/man/compare.Rd +++ b/man/compare.Rd @@ -73,7 +73,7 @@ evaluation using leave-one-out cross-validation and WAIC. (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). -Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). +Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} } diff --git a/man/loo-glossary.Rd b/man/loo-glossary.Rd index 041eb11e..6c109374 100644 --- a/man/loo-glossary.Rd +++ b/man/loo-glossary.Rd @@ -39,7 +39,8 @@ the actual SE can even be twice as large. Even for moderate N, when the SE estimate is an accurate estimate for the scale, it ignores the skewness. When making model comparisons, the SE of the component-wise (pairwise) differences should be used instead (see the \code{se_diff} section below and Eq 24 in -VGG2017). +VGG2017). Sivula et al. (2022) discuss the conditions when the normal +approximation used for SE and \code{se_diff} is good. } \section{Monte Carlo SE of elpd_loo}{ @@ -69,20 +70,51 @@ about high Pareto k diagnostic values. \section{Pareto k estimates}{ -The Pareto \code{k} estimate is a diagnostic for Pareto smoothed importance +The Pareto \eqn{k} estimate is a diagnostic for Pareto smoothed importance sampling (PSIS), which is used to compute components of \code{elpd_loo}. In -importance-sampling LOO (the full posterior distribution is used as the -proposal distribution). The Pareto k diagnostic estimates how far an +importance-sampling LOO the full posterior distribution is used as the +proposal distribution. The Pareto k diagnostic estimates how far an individual leave-one-out distribution is from the full distribution. If leaving out an observation changes the posterior too much then importance -sampling is not able to give reliable estimate. If \code{k<0.5}, then the -corresponding component of \code{elpd_loo} is estimated with high accuracy. -If \verb{0.50.7}, -then importance sampling is not able to provide useful estimate for that -component/observation. Pareto k is also useful as a measure of influence of -an observation. Highly influential observations have high k values. Very high -k values often indicate model misspecification, outliers or mistakes in data -processing. See Section 6 of Gabry et al. (2019) for an example. +sampling is not able to give reliable estimate. Pareto smoothing stabilizes +importance sampling and guarantees finite variance estimate with a +cost of some bias. + +The diagnostic threshold for Pareto \eqn{k} depends on sample size +\eqn{S} (sample size dependent threshold was introduced by Vehtari +et al., 2022, and before that fixed thresholds of 0.5 and 0.7 were +recommended). For simplicity, \code{loo} package uses the nominal sample +size \eqn{S} when computing the sample size specific +threshold. This provides an optimistic threshold if the effective +sample size is less than 2200, but even then if ESS/S>1/2 the difference +is usually negligible. Thinning of MCMC draws can be used to improve +the ratio ESS/S. +\itemize{ +\item If \eqn{k < min(1 - 1 / log10(S), 0.7)}, where \eqn{S} is the +sample size PSIS estimate and the corresponding Monte +Carlo standard error estimate are reliable. +\item If \eqn{1 - 1 / log10(S) <= k < 0.7}, PSIS estimate and the +corresponding Monte Carlo standard error estimate are not +reliable, but increasing (effective) sample size \eqn{S} above +2200 may help (this will increase the sample size specific +threshold \eqn{(1-1/log10(2200)>0.7} and then the bias specific +threshold 0.7 dominates). +\item If \eqn{0.7 <= k < 1}, PSIS estimate and the corresponding Monte +Carlo standard error have large bias and are not reliable. Increasing +sample size may reduce the variability in \eqn{k} estimate, which +may result in lower \eqn{k} estimate, too. +\item If \eqn{k \geq 1}{k >= 1}, the target distribution is estimated to +have non-finite mean. PSIS estimate and the corresponding Monte +Carlo standard error are not well defined. Increasing sample size +may reduce the variability in \eqn{k} estimate, which +may result in lower \eqn{k} estimate, too. +} + +Pareto \eqn{k} is also useful as a measure of influence of an +observation. Highly influential observations have high \eqn{k} +values. Very high \eqn{k} values often indicate model +misspecification, outliers or mistakes in data processing. See +Section 6 of Gabry et al. (2019) for an example. \subsection{Interpreting \code{p_loo} when Pareto \code{k} is large}{ If \code{k > 0.7} then we can also look at the \code{p_loo} estimate for @@ -139,10 +171,14 @@ evaluation using leave-one-out cross-validation and WAIC. (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). -Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). +Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} +Sivula, T, Magnusson, M., Matamoros A. A., and Vehtari, +A. (2022). Uncertainty in Bayesian leave-one-out +cross-validation based model comparison. \href{https://arxiv.org/abs/2008.10296v3}{preprint arXiv:2008.10296v3.}. + Gabry, J. , Simpson, D. , Vehtari, A. , Betancourt, M. and Gelman, A. (2019), Visualization in Bayesian workflow. \emph{J. R. Stat. Soc. A}, 182: 389-402. doi:10.1111/rssa.12378 diff --git a/man/loo-package.Rd b/man/loo-package.Rd index d17a5607..b8242e2b 100644 --- a/man/loo-package.Rd +++ b/man/loo-package.Rd @@ -11,7 +11,7 @@ \emph{Stan Development Team} This package implements the methods described in Vehtari, Gelman, and -Gabry (2017), Vehtari, Simpson, Gelman, Yao, and Gabry (2019), and +Gabry (2017), Vehtari, Simpson, Gelman, Yao, and Gabry (2022), and Yao et al. (2018). To get started see the \strong{loo} package \href{https://mc-stan.org/loo/articles/index.html}{vignettes}, the \code{\link[=loo]{loo()}} function for efficient approximate leave-one-out @@ -31,7 +31,7 @@ they involve additional computational steps. This package implements the fast and stable computations for approximate LOO-CV laid out in Vehtari, Gelman, and Gabry (2017). From existing posterior simulation draws, we compute LOO-CV using Pareto smoothed importance sampling (PSIS; Vehtari, -Simpson, Gelman, Yao, and Gabry, 2019), a new procedure for stabilizing +Simpson, Gelman, Yao, and Gabry, 2022), a new procedure for stabilizing and diagnosing importance weights. As a byproduct of our calculations, we also obtain approximate standard errors for estimated predictive errors and for comparing of predictive errors between two models. @@ -47,7 +47,7 @@ evaluation using leave-one-out cross-validation and WAIC. (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). -Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). +Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} @@ -105,3 +105,34 @@ Zhang, J., and Stephens, M. A. (2009). A new and efficient estimation method for the generalized Pareto distribution. \emph{Technometrics} \strong{51}, 316-325. } +\seealso{ +Useful links: +\itemize{ + \item \url{https://mc-stan.org/loo/} + \item \url{https://discourse.mc-stan.org} + \item Report bugs at \url{https://github.com/stan-dev/loo/issues} +} + +} +\author{ +\strong{Maintainer}: Jonah Gabry \email{jsg2201@columbia.edu} + +Authors: +\itemize{ + \item Aki Vehtari \email{Aki.Vehtari@aalto.fi} + \item Mans Magnusson + \item Yuling Yao + \item Paul-Christian Bürkner + \item Topi Paananen + \item Andrew Gelman +} + +Other contributors: +\itemize{ + \item Ben Goodrich [contributor] + \item Juho Piironen [contributor] + \item Bruno Nicenboim [contributor] + \item Leevi Lindgren [contributor] +} + +} diff --git a/man/loo.Rd b/man/loo.Rd index 1698b0ef..cceabdd5 100644 --- a/man/loo.Rd +++ b/man/loo.Rd @@ -63,10 +63,10 @@ each method.} \item{r_eff}{Vector of relative effective sample size estimates for the likelihood (\code{exp(log_lik)}) of each observation. This is related to the relative efficiency of estimating the normalizing term in -self-normalizing importance sampling when using posterior draws obtained +self-normalized importance sampling when using posterior draws obtained with MCMC. If MCMC draws are used and \code{r_eff} is not provided then the reported PSIS effective sample sizes and Monte Carlo error estimates -will be over-optimistic. If the posterior draws are independent then +can be over-optimistic. If the posterior draws are independent then \code{r_eff=1} and can be omitted. The warning message thrown when \code{r_eff} is not specified can be disabled by setting \code{r_eff} to \code{NA}. See the \code{\link[=relative_eff]{relative_eff()}} helper functions for computing \code{r_eff}.} @@ -169,7 +169,7 @@ The \code{loo()} methods for arrays, matrices, and functions compute PSIS-LOO CV, efficient approximate leave-one-out (LOO) cross-validation for Bayesian models using Pareto smoothed importance sampling (\link[=psis]{PSIS}). This is an implementation of the methods described in Vehtari, Gelman, and Gabry -(2017) and Vehtari, Simpson, Gelman, Yao, and Gabry (2019). +(2017) and Vehtari, Simpson, Gelman, Yao, and Gabry (2022). The \code{loo_i()} function enables testing log-likelihood functions for use with the \code{loo.function()} method. @@ -321,7 +321,7 @@ evaluation using leave-one-out cross-validation and WAIC. (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). -Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). +Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} } diff --git a/man/loo_compare.Rd b/man/loo_compare.Rd index 85f876b3..dacee7d9 100644 --- a/man/loo_compare.Rd +++ b/man/loo_compare.Rd @@ -66,16 +66,17 @@ better sense of uncertainty than what is obtained using the current standard approach of comparing differences of deviances to a Chi-squared distribution, a practice derived for Gaussian linear models or asymptotically, and which only applies to nested models in any case. - -If more than \eqn{11} models are compared, then the median model by elpd is -taken as the baseline model, and we recompute (internally) the model -differences to this baseline. We then estimate whether the difference in -predictive performances is potentially due to chance as described by -McLatchie and Vehtari (2023). This will flag a warning if it is deemed that -there is a risk of over-fitting due to the selection process, and users -are recommended to avoid model selection based on LOO-CV, and -instead to favour of model averaging/stacking or projection predictive -inference. +Sivula et al. (2022) discuss the conditions when the normal +approximation used for SE and \code{se_diff} is good. + +If more than \eqn{11} models are compared, we internally recompute the model +differences using the median model by ELPD as the baseline model. We then +estimate whether the differences in predictive performance are potentially +due to chance as described by McLatchie and Vehtari (2023). This will flag +a warning if it is deemed that there is a risk of over-fitting due to the +selection process. In that case users are recommended to avoid model +selection based on LOO-CV, and instead to favor model averaging/stacking or +projection predictive inference. } \examples{ # very artificial example, just for demonstration! @@ -108,12 +109,16 @@ evaluation using leave-one-out cross-validation and WAIC. (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). -Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). +Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} -McLatchie, Y., and Vehtari, A. (2023). -Efficient estimation and correction of selection-induced bias with order statistics. +Sivula, T, Magnusson, M., Matamoros A. A., and Vehtari, A. (2022). +Uncertainty in Bayesian leave-one-out cross-validation based model +comparison. \href{https://arxiv.org/abs/2008.10296v3}{preprint arXiv:2008.10296v3.}. + +McLatchie, Y., and Vehtari, A. (2023). Efficient estimation and +correction of selection-induced bias with order statistics. \href{https://arxiv.org/abs/2309.03742}{preprint arXiv:2309.03742} } \seealso{ diff --git a/man/loo_model_weights.Rd b/man/loo_model_weights.Rd index 1273f61a..99817db9 100644 --- a/man/loo_model_weights.Rd +++ b/man/loo_model_weights.Rd @@ -101,7 +101,7 @@ A numeric vector containing one weight for each model. Model averaging via stacking of predictive distributions, pseudo-BMA weighting or pseudo-BMA+ weighting with the Bayesian bootstrap. See Yao et al. (2018), Vehtari, Gelman, and Gabry (2017), and Vehtari, Simpson, -Gelman, Yao, and Gabry (2019) for background. +Gelman, Yao, and Gabry (2022) for background. } \details{ \code{loo_model_weights()} is a wrapper around the \code{stacking_weights()} and @@ -215,7 +215,7 @@ evaluation using leave-one-out cross-validation and WAIC. (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). -Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). +Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} diff --git a/man/loo_moment_match.Rd b/man/loo_moment_match.Rd index e143e465..e0d07ab4 100644 --- a/man/loo_moment_match.Rd +++ b/man/loo_moment_match.Rd @@ -16,7 +16,7 @@ loo_moment_match(x, ...) log_prob_upars, log_lik_i_upars, max_iters = 30L, - k_threshold = 0.7, + k_threshold = NULL, split = TRUE, cov = TRUE, cores = getOption("mc.cores", 1), @@ -56,7 +56,8 @@ reached, there will be a warning, and increasing \code{max_iters} may improve accuracy.} \item{k_threshold}{Threshold value for Pareto k values above which the moment -matching algorithm is used. The default value is 0.5.} +matching algorithm is used. The default value is \code{1 - 1 / log10(S)}, +where \code{S} is the sample size.} \item{split}{Logical; Indicate whether to do the split transformation or not at the end of moment matching for each LOO fold.} diff --git a/man/loo_subsample.Rd b/man/loo_subsample.Rd index 4a2ce59c..29af1df9 100644 --- a/man/loo_subsample.Rd +++ b/man/loo_subsample.Rd @@ -56,10 +56,10 @@ using MCMC). If not \code{NULL} then they should be specified as described in \item{r_eff}{Vector of relative effective sample size estimates for the likelihood (\code{exp(log_lik)}) of each observation. This is related to the relative efficiency of estimating the normalizing term in -self-normalizing importance sampling when using posterior draws obtained +self-normalized importance sampling when using posterior draws obtained with MCMC. If MCMC draws are used and \code{r_eff} is not provided then the reported PSIS effective sample sizes and Monte Carlo error estimates -will be over-optimistic. If the posterior draws are independent then +can be over-optimistic. If the posterior draws are independent then \code{r_eff=1} and can be omitted. The warning message thrown when \code{r_eff} is not specified can be disabled by setting \code{r_eff} to \code{NA}. See the \code{\link[=relative_eff]{relative_eff()}} helper functions for computing \code{r_eff}.} diff --git a/man/parallel_psis_list.Rd b/man/parallel_psis_list.Rd index 217a1d3d..d8899258 100644 --- a/man/parallel_psis_list.Rd +++ b/man/parallel_psis_list.Rd @@ -45,10 +45,10 @@ below for details on how to specify these arguments.} \item{r_eff}{Vector of relative effective sample size estimates for the likelihood (\code{exp(log_lik)}) of each observation. This is related to the relative efficiency of estimating the normalizing term in -self-normalizing importance sampling when using posterior draws obtained +self-normalized importance sampling when using posterior draws obtained with MCMC. If MCMC draws are used and \code{r_eff} is not provided then the reported PSIS effective sample sizes and Monte Carlo error estimates -will be over-optimistic. If the posterior draws are independent then +can be over-optimistic. If the posterior draws are independent then \code{r_eff=1} and can be omitted. The warning message thrown when \code{r_eff} is not specified can be disabled by setting \code{r_eff} to \code{NA}. See the \code{\link[=relative_eff]{relative_eff()}} helper functions for computing \code{r_eff}.} diff --git a/man/pareto-k-diagnostic.Rd b/man/pareto-k-diagnostic.Rd index c268ffcd..998f7f4b 100644 --- a/man/pareto-k-diagnostic.Rd +++ b/man/pareto-k-diagnostic.Rd @@ -15,7 +15,7 @@ \usage{ pareto_k_table(x) -pareto_k_ids(x, threshold = 0.5) +pareto_k_ids(x, threshold = NULL) pareto_k_values(x) @@ -23,11 +23,11 @@ pareto_k_influence_values(x) psis_n_eff_values(x) -mcse_loo(x, threshold = 0.7) +mcse_loo(x, threshold = NULL) \method{plot}{psis_loo}( x, - diagnostic = c("k", "n_eff"), + diagnostic = c("k", "ESS", "n_eff"), ..., label_points = FALSE, main = "PSIS diagnostic plot" @@ -35,7 +35,7 @@ mcse_loo(x, threshold = 0.7) \method{plot}{psis}( x, - diagnostic = c("k", "n_eff"), + diagnostic = c("k", "ESS", "n_eff"), ..., label_points = FALSE, main = "PSIS diagnostic plot" @@ -45,20 +45,20 @@ mcse_loo(x, threshold = 0.7) \item{x}{An object created by \code{\link[=loo]{loo()}} or \code{\link[=psis]{psis()}}.} \item{threshold}{For \code{pareto_k_ids()}, \code{threshold} is the minimum \eqn{k} -value to flag (default is \code{0.5}). For \code{mcse_loo()}, if any \eqn{k} -estimates are greater than \code{threshold} the MCSE estimate is returned as -\code{NA} (default is \code{0.7}). See \strong{Details} for the motivation behind these -defaults.} +value to flag (default is a sample size \code{S} dependend threshold +\code{1 - 1 / log10(S)}). For \code{mcse_loo()}, if any \eqn{k} estimates are +greater than \code{threshold} the MCSE estimate is returned as \code{NA} +See \strong{Details} for the motivation behind these defaults.} \item{diagnostic}{For the \code{plot} method, which diagnostic should be plotted? The options are \code{"k"} for Pareto \eqn{k} estimates (the -default) or \code{"n_eff"} for PSIS effective sample size estimates.} +default), or \code{"ESS"} or \code{"n_eff"} for PSIS effective sample size estimates.} \item{label_points, ...}{For the \code{plot()} method, if \code{label_points} is \code{TRUE} the observation numbers corresponding to any values of \eqn{k} -greater than 0.5 will be displayed in the plot. Any arguments specified in -\code{...} will be passed to \code{\link[graphics:text]{graphics::text()}} and can be used -to control the appearance of the labels.} +greater than the diagnostic threshold will be displayed in the plot. +Any arguments specified in \code{...} will be passed to \code{\link[graphics:text]{graphics::text()}} +and can be used to control the appearance of the labels.} \item{main}{For the \code{plot()} method, a title for the plot.} } @@ -88,7 +88,7 @@ The \code{plot()} method is called for its side effect and does not return anything. If \code{x} is the result of a call to \code{\link[=loo]{loo()}} or \code{\link[=psis]{psis()}} then \code{plot(x, diagnostic)} produces a plot of the estimates of the Pareto shape parameters (\code{diagnostic = "k"}) or -estimates of the PSIS effective sample sizes (\code{diagnostic = "n_eff"}). +estimates of the PSIS effective sample sizes (\code{diagnostic = "ESS"}). } \description{ Print a diagnostic table summarizing the estimated Pareto shape parameters @@ -97,50 +97,70 @@ the estimated Pareto shape parameter \eqn{k} is larger than some \code{threshold} value, or plot observation indexes vs. diagnostic estimates. The \strong{Details} section below provides a brief overview of the diagnostics, but we recommend consulting Vehtari, Gelman, and Gabry (2017) -and Vehtari, Simpson, Gelman, Yao, and Gabry (2019) for full details. +and Vehtari, Simpson, Gelman, Yao, and Gabry (2022) for full details. } \details{ -The reliability and approximate convergence rate of the PSIS-based estimates -can be assessed using the estimates for the shape parameter \eqn{k} of the -generalized Pareto distribution: +The reliability and approximate convergence rate of the PSIS-based +estimates can be assessed using the estimates for the shape +parameter \eqn{k} of the generalized Pareto distribution. The +diagnostic threshold for Pareto \eqn{k} depends on sample size +\eqn{S} (sample size dependent threshold was introduced by Vehtari +et al., 2022, and before that fixed thresholds of 0.5 and 0.7 were +recommended). For simplicity, \code{loo} package uses the nominal sample +size \eqn{S} when computing the sample size specific +threshold. This provides an optimistic threshold if the effective +sample size is less than 2200, but if MCMC-ESS > S/2 the difference +is usually negligible. Thinning of MCMC draws can be used to +improve the ratio ESS/S. \itemize{ -\item If \eqn{k < 0.5} then the distribution of raw importance ratios has -finite variance and the central limit theorem holds. However, as \eqn{k} -approaches \eqn{0.5} the RMSE of plain importance sampling (IS) increases -significantly while PSIS has lower RMSE. -\item If \eqn{0.5 \leq k < 1}{0.5 <= k < 1} then the variance of the raw -importance ratios is infinite, but the mean exists. TIS and PSIS estimates -have finite variance by accepting some bias. The convergence of the -estimate is slower with increasing \eqn{k}. -If \eqn{k} is between 0.5 and approximately 0.7 then we observe practically -useful convergence rates and Monte Carlo error estimates with PSIS (the -bias of TIS increases faster than the bias of PSIS). If \eqn{k > 0.7} we -observe impractical convergence rates and unreliable Monte Carlo error -estimates. -\item If \eqn{k \geq 1}{k >= 1} then neither the variance nor the mean of -the raw importance ratios exists. The convergence rate is close to -zero and bias can be large with practical sample sizes. +\item If \eqn{k < min(1 - 1 / log10(S), 0.7)}, where \eqn{S} is the +sample size, the PSIS estimate and the corresponding Monte Carlo +standard error estimate are reliable. +\item If \eqn{1 - 1 / log10(S) <= k < 0.7}, PSIS estimate and the +corresponding Monte Carlo standard error estimate are not +reliable, but increasing the (effective) sample size \eqn{S} above +2200 may help (this will increase the sample size specific +threshold \eqn{(1-1/log10(2200)>0.7} and then the bias specific +threshold 0.7 dominates). +\item If \eqn{0.7 <= k < 1}, the PSIS estimate and the corresponding Monte +Carlo standard error have large bias and are not reliable. Increasing +the sample size may reduce the uncertainty in the \eqn{k} estimate. +\item If \eqn{0.7 <= k < 1}, PSIS estimate and the corresponding Monte +Carlo standard error have large bias and are not reliable. Increasing +sample size may reduce the variability in \eqn{k} estimate, which +may result in lower \eqn{k} estimate, too. +\item If \eqn{k \geq 1}{k >= 1}, the target distribution is estimated to +have non-finite mean. The PSIS estimate and the corresponding Monte +Carlo standard error are not well defined. Increasing the sample size +may reduce the variability in \eqn{k} estimate, which +may result in lower \eqn{k} estimate, too. } -\subsection{What if the estimated tail shape parameter \eqn{k} exceeds -\eqn{0.5}}{ Importance sampling is likely to work less well if the -marginal posterior \eqn{p(\theta^s | y)} and LOO posterior -\eqn{p(\theta^s | y_{-i})} are very different, which is more likely to -happen with a non-robust model and highly influential observations. -If the estimated tail shape parameter \eqn{k} exceeds \eqn{0.5}, the user -should be warned. (Note: If \eqn{k} is greater than \eqn{0.5} -then WAIC is also likely to fail, but WAIC lacks its own diagnostic.) -In practice, we have observed good performance for values of \eqn{k} up to 0.7. -When using PSIS in the context of approximate LOO-CV, we recommend one of -the following actions when \eqn{k > 0.7}: +\subsection{What if the estimated tail shape parameter \eqn{k} +exceeds the diagnostic threshold?}{ Importance sampling is likely to +work less well if the marginal posterior \eqn{p(\theta^s | y)} and +LOO posterior \eqn{p(\theta^s | y_{-i})} are very different, which +is more likely to happen with a non-robust model and highly +influential observations. If the estimated tail shape parameter +\eqn{k} exceeds the diagnostic threshold, the user should be +warned. (Note: If \eqn{k} is greater than the diagnostic threshold +then WAIC is also likely to fail, but WAIC lacks as accurate +diagnostic.) When using PSIS in the context of approximate LOO-CV, +we recommend one of the following actions when \eqn{k > 0.7}: \itemize{ -\item With some additional computations, it is possible to transform the MCMC -draws from the posterior distribution to obtain more reliable importance -sampling estimates. This results in a smaller shape parameter \eqn{k}. -See \code{\link[=loo_moment_match]{loo_moment_match()}} for an example of this. -\item Sampling directly from \eqn{p(\theta^s | y_{-i})} for the problematic -observations \eqn{i}, or using \eqn{k}-fold cross-validation will generally -be more stable. +\item With some additional computations, it is possible to transform +the MCMC draws from the posterior distribution to obtain more +reliable importance sampling estimates. This results in a smaller +shape parameter \eqn{k}. See \code{\link[=loo_moment_match]{loo_moment_match()}} and the +vignette \emph{Avoiding model refits in leave-one-out cross-validation +with moment matching} for an example of this. +\item Sampling from a leave-one-out mixture distribution (see the +vignette \emph{Mixture IS leave-one-out cross-validation for +high-dimensional Bayesian models}), directly from \eqn{p(\theta^s + | y_{-i})} for the problematic observations \eqn{i}, or using +\eqn{K}-fold cross-validation (see the vignette \emph{Holdout +validation and K-fold cross-validation of Stan programs with the +loo package}) will generally be more stable. \item Using a model that is more robust to anomalous observations will generally make approximate LOO-CV more stable. } @@ -157,9 +177,10 @@ influence on posterior distribution of the model. These can be obtained with obtain the samples from the proposal distribution via MCMC the \strong{loo} package also computes estimates for the Monte Carlo error and the effective sample size for importance sampling, which are more accurate for PSIS than -for IS and TIS (see Vehtari et al (2019) for details). However, the PSIS +for IS and TIS (see Vehtari et al (2022) for details). However, the PSIS effective sample size estimate will be -\strong{over-optimistic when the estimate of \eqn{k} is greater than 0.7}. +\strong{over-optimistic when the estimate of \eqn{k} is greater than +\eqn{min(1-1/log10(S), 0.7)}}, where \eqn{S} is the sample size. } } \references{ @@ -169,7 +190,7 @@ evaluation using leave-one-out cross-validation and WAIC. (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). -Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). +Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} } diff --git a/man/psis.Rd b/man/psis.Rd index 0b6f01de..ed38f046 100644 --- a/man/psis.Rd +++ b/man/psis.Rd @@ -107,7 +107,7 @@ Method used for importance sampling, here \code{psis}. Implementation of Pareto smoothed importance sampling (PSIS), a method for stabilizing importance ratios. The version of PSIS implemented here corresponds to the algorithm presented in Vehtari, Simpson, Gelman, Yao, -and Gabry (2019). +and Gabry (2022). For PSIS diagnostics see the \link{pareto-k-diagnostic} page. } \section{Methods (by class)}{ @@ -147,7 +147,7 @@ evaluation using leave-one-out cross-validation and WAIC. (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). -Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). +Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} } diff --git a/man/psis_approximate_posterior.Rd b/man/psis_approximate_posterior.Rd index 159e9412..8cf1a03a 100644 --- a/man/psis_approximate_posterior.Rd +++ b/man/psis_approximate_posterior.Rd @@ -64,7 +64,7 @@ evaluation using leave-one-out cross-validation and WAIC. (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). -Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). +Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} } diff --git a/man/psislw.Rd b/man/psislw.Rd index 377f75a4..c340b198 100644 --- a/man/psislw.Rd +++ b/man/psislw.Rd @@ -55,7 +55,7 @@ evaluation using leave-one-out cross-validation and WAIC. (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). -Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). +Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} } diff --git a/man/sis.Rd b/man/sis.Rd index ba66d3e7..981e43df 100644 --- a/man/sis.Rd +++ b/man/sis.Rd @@ -125,7 +125,7 @@ evaluation using leave-one-out cross-validation and WAIC. (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). -Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). +Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} } diff --git a/man/update.psis_loo_ss.Rd b/man/update.psis_loo_ss.Rd index 56950dc2..e8e1b263 100644 --- a/man/update.psis_loo_ss.Rd +++ b/man/update.psis_loo_ss.Rd @@ -41,10 +41,10 @@ previous call to \code{loo_subsample()}. \item{r_eff}{Vector of relative effective sample size estimates for the likelihood (\code{exp(log_lik)}) of each observation. This is related to the relative efficiency of estimating the normalizing term in -self-normalizing importance sampling when using posterior draws obtained +self-normalized importance sampling when using posterior draws obtained with MCMC. If MCMC draws are used and \code{r_eff} is not provided then the reported PSIS effective sample sizes and Monte Carlo error estimates -will be over-optimistic. If the posterior draws are independent then +can be over-optimistic. If the posterior draws are independent then \code{r_eff=1} and can be omitted. The warning message thrown when \code{r_eff} is not specified can be disabled by setting \code{r_eff} to \code{NA}. See the \code{\link[=relative_eff]{relative_eff()}} helper functions for computing \code{r_eff}.} diff --git a/man/waic.Rd b/man/waic.Rd index 1bfc8b66..d4a30476 100644 --- a/man/waic.Rd +++ b/man/waic.Rd @@ -125,7 +125,7 @@ evaluation using leave-one-out cross-validation and WAIC. (\href{https://link.springer.com/article/10.1007/s11222-016-9696-4}{journal version}, \href{https://arxiv.org/abs/1507.04544}{preprint arXiv:1507.04544}). -Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). +Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. \href{https://arxiv.org/abs/1507.02646}{preprint arXiv:1507.02646} } @@ -133,7 +133,7 @@ Pareto smoothed importance sampling. \itemize{ \item The \strong{loo} package \href{https://mc-stan.org/loo/articles/}{vignettes} and Vehtari, Gelman, and Gabry (2017) and Vehtari, Simpson, Gelman, Yao, -and Gabry (2019) for more details on why we prefer \code{loo()} to \code{waic()}. +and Gabry (2022) for more details on why we prefer \code{loo()} to \code{waic()}. \item \code{\link[=loo_compare]{loo_compare()}} for comparing models on approximate LOO-CV or WAIC. } } From 3366ad5e83b474eb337ea365584a6da39d77d0d2 Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Wed, 31 Jan 2024 10:11:37 +0200 Subject: [PATCH 17/39] Apply suggestions from code review by n-kall Co-authored-by: n-kall <33577035+n-kall@users.noreply.github.com> --- R/loo-glossary.R | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/R/loo-glossary.R b/R/loo-glossary.R index e7ae8741..bed6d78c 100644 --- a/R/loo-glossary.R +++ b/R/loo-glossary.R @@ -70,8 +70,8 @@ #' proposal distribution. The Pareto k diagnostic estimates how far an #' individual leave-one-out distribution is from the full distribution. If #' leaving out an observation changes the posterior too much then importance -#' sampling is not able to give reliable estimate. Pareto smoothing stabilizes -#' importance sampling and guarantees finite variance estimate with a +#' sampling is not able to give a reliable estimate. Pareto smoothing stabilizes +#' importance sampling and guarantees a finite variance estimate at the #' cost of some bias. #' #' The diagnostic threshold for Pareto \eqn{k} depends on sample size @@ -80,31 +80,31 @@ #' recommended). For simplicity, `loo` package uses the nominal sample #' size \eqn{S} when computing the sample size specific #' threshold. This provides an optimistic threshold if the effective -#' sample size is less than 2200, but even then if ESS/S>1/2 the difference +#' sample size is less than 2200, but even then if ESS/S > 1/2 the difference #' is usually negligible. Thinning of MCMC draws can be used to improve #' the ratio ESS/S. #' #' * If \eqn{k < min(1 - 1 / log10(S), 0.7)}, where \eqn{S} is the -#' sample size PSIS estimate and the corresponding Monte +#' sample size, PSIS estimate and the corresponding Monte #' Carlo standard error estimate are reliable. #' #' * If \eqn{1 - 1 / log10(S) <= k < 0.7}, PSIS estimate and the #' corresponding Monte Carlo standard error estimate are not #' reliable, but increasing (effective) sample size \eqn{S} above #' 2200 may help (this will increase the sample size specific -#' threshold \eqn{(1-1/log10(2200)>0.7} and then the bias specific +#' threshold \eqn{(1 - 1 / log10(2200) > 0.7} and then the bias specific #' threshold 0.7 dominates). #' #' * If \eqn{0.7 <= k < 1}, PSIS estimate and the corresponding Monte #' Carlo standard error have large bias and are not reliable. Increasing #' sample size may reduce the variability in \eqn{k} estimate, which -#' may result in lower \eqn{k} estimate, too. +#' may result in a lower \eqn{k} estimate, too. #' #' * If \eqn{k \geq 1}{k >= 1}, the target distribution is estimated to #' have non-finite mean. PSIS estimate and the corresponding Monte #' Carlo standard error are not well defined. Increasing sample size #' may reduce the variability in \eqn{k} estimate, which -#' may result in lower \eqn{k} estimate, too. +#' may result in a lower \eqn{k} estimate, too. #' #' Pareto \eqn{k} is also useful as a measure of influence of an #' observation. Highly influential observations have high \eqn{k} From 4013c15a0ca1697c009f0ae667186d09fb7be02a Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Wed, 31 Jan 2024 10:12:11 +0200 Subject: [PATCH 18/39] another doc threshold update --- R/loo-glossary.R | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/R/loo-glossary.R b/R/loo-glossary.R index e7ae8741..80e9a255 100644 --- a/R/loo-glossary.R +++ b/R/loo-glossary.R @@ -113,8 +113,8 @@ #' Section 6 of Gabry et al. (2019) for an example. #' #' \subsection{Interpreting `p_loo` when Pareto `k` is large}{ -#' If `k > 0.7` then we can also look at the `p_loo` estimate for -#' some additional information about the problem: +#' If \eqn{k < min(1 - 1 / log10(S), 0.7)} then we can also look at +#' the `p_loo` estimate for some additional information about the problem: #' #' \itemize{ #' \item If `p_loo << p` (the total number of parameters in the model), From e6fdfda86a45bd4030698af730f395dc40deb7ce Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Wed, 31 Jan 2024 18:11:03 +0200 Subject: [PATCH 19/39] replace r_eff warning with informative message in print --- R/crps.R | 6 +-- R/effective_sample_sizes.R | 4 +- R/importance_sampling.R | 8 ++-- R/loo.R | 39 ++++++++---------- R/loo_approximate_posterior.R | 4 +- R/loo_compare.R | 6 +-- R/loo_predictive_metric.R | 2 +- R/loo_subsample.R | 11 +++-- R/print.R | 28 ++++++++++--- R/psis.R | 35 ++++++++-------- tests/testthat/reference-results/loo.rds | Bin 1826 -> 2088 bytes .../reference-results/moment_match_loo_1.rds | Bin 1721 -> 1745 bytes .../reference-results/moment_match_loo_2.rds | Bin 1723 -> 1742 bytes .../reference-results/moment_match_loo_3.rds | Bin 1723 -> 1746 bytes .../moment_match_var_and_cov.rds | Bin 1900 -> 1931 bytes tests/testthat/reference-results/psis.rds | Bin 238513 -> 238538 bytes tests/testthat/test_crps.R | 9 +--- tests/testthat/test_loo_and_waic.R | 13 +++--- tests/testthat/test_loo_moment_matching.R | 2 +- tests/testthat/test_loo_subsampling.R | 24 +++++++---- tests/testthat/test_model_weighting.R | 2 +- tests/testthat/test_print_plot.R | 5 ++- tests/testthat/test_psis.R | 28 +++++++++---- tests/testthat/test_tisis.R | 32 ++++++++++---- 24 files changed, 150 insertions(+), 108 deletions(-) diff --git a/R/crps.R b/R/crps.R index 062532d6..f805b882 100644 --- a/R/crps.R +++ b/R/crps.R @@ -112,7 +112,7 @@ loo_crps.matrix <- log_lik, ..., permutations = 1, - r_eff = NULL, + r_eff = 1, cores = getOption("mc.cores", 1)) { validate_crps_input(x, x2, y, log_lik) repeats <- replicate(permutations, @@ -154,7 +154,7 @@ loo_scrps.matrix <- log_lik, ..., permutations = 1, - r_eff = NULL, + r_eff = 1, cores = getOption("mc.cores", 1)) { validate_crps_input(x, x2, y, log_lik) repeats <- replicate(permutations, @@ -175,7 +175,7 @@ EXX_compute <- function(x, x2) { } -EXX_loo_compute <- function(x, x2, log_lik, r_eff = NULL, ...) { +EXX_loo_compute <- function(x, x2, log_lik, r_eff = 1, ...) { S <- nrow(x) shuffle <- sample (1:S) x2 <- x2[shuffle,] diff --git a/R/effective_sample_sizes.R b/R/effective_sample_sizes.R index 1c2b6ea4..b66c2207 100644 --- a/R/effective_sample_sizes.R +++ b/R/effective_sample_sizes.R @@ -183,8 +183,8 @@ psis_n_eff.matrix <- function(w, r_eff = NULL, ...) { if (is.null(r_eff)) { return(1 / ss) } - if (length(r_eff) != length(ss)) - stop("r_eff must have length ncol(w).", call. = FALSE) + if (length(r_eff) != length(ss) && length(r_eff) != 1) + stop("r_eff must have length 1 or ncol(w).", call. = FALSE) 1 / ss * r_eff } diff --git a/R/importance_sampling.R b/R/importance_sampling.R index 583493f6..80ec5c8a 100644 --- a/R/importance_sampling.R +++ b/R/importance_sampling.R @@ -19,7 +19,7 @@ importance_sampling <- function(log_ratios, method, ...) { importance_sampling.array <- function(log_ratios, method, ..., - r_eff = NULL, + r_eff = 1, cores = getOption("mc.cores", 1)) { cores <- loo_cores(cores) stopifnot(length(dim(log_ratios)) == 3) @@ -36,7 +36,7 @@ importance_sampling.array <- importance_sampling.matrix <- function(log_ratios, method, ..., - r_eff = NULL, + r_eff = 1, cores = getOption("mc.cores", 1)) { cores <- loo_cores(cores) assert_importance_sampling_method_is_implemented(method) @@ -49,7 +49,7 @@ importance_sampling.matrix <- #' @inheritParams psis #' @export importance_sampling.default <- - function(log_ratios, method, ..., r_eff = NULL) { + function(log_ratios, method, ..., r_eff = 1) { stopifnot(is.null(dim(log_ratios)) || length(dim(log_ratios)) == 1) assert_importance_sampling_method_is_implemented(method) dim(log_ratios) <- c(length(log_ratios), 1) @@ -153,7 +153,7 @@ importance_sampling_object <- out <- structure( list( log_weights = unnormalized_log_weights, - diagnostics = list(pareto_k = pareto_k, n_eff = NULL) + diagnostics = list(pareto_k = pareto_k, n_eff = NULL, r_eff = r_eff) ), # attributes norm_const_log = norm_const_log, diff --git a/R/loo.R b/R/loo.R index 9d19df84..a722cbeb 100644 --- a/R/loo.R +++ b/R/loo.R @@ -16,9 +16,10 @@ #' self-normalized importance sampling when using posterior draws obtained #' with MCMC. If MCMC draws are used and `r_eff` is not provided then #' the reported PSIS effective sample sizes and Monte Carlo error estimates -#' can be over-optimistic. If the posterior draws are independent then -#' `r_eff=1` and can be omitted. The warning message thrown when `r_eff` is -#' not specified can be disabled by setting `r_eff` to `NA`. See the +#' can be over-optimistic. If the posterior draws are (near) independent then +#' `r_eff=1` can be used. `r_eff` has to be a scalar (same value is used +#' for all observations) or a vector with length equal to the number of +#' observations. The default value is 1. #' [relative_eff()] helper functions for computing `r_eff`. #' @param save_psis Should the `"psis"` object created internally by `loo()` be #' saved in the returned object? The `loo()` function calls [psis()] @@ -135,8 +136,8 @@ #' loo_3 <- loo_i(i = 3, llfun = llfun, data = fake_data, draws = fake_posterior, r_eff = NA) #' print(loo_3$pointwise[, "elpd_loo"]) #' -#' # Use loo.function method (setting r_eff=NA since this posterior not obtained via MCMC) -#' loo_with_fn <- loo(llfun, draws = fake_posterior, data = fake_data, r_eff = NA) +#' # Use loo.function method (default r_eff=1 is used as this posterior not obtained via MCMC) +#' loo_with_fn <- loo(llfun, draws = fake_posterior, data = fake_data) #' #' # If we look at the elpd_loo contribution from the 3rd obs it should be the #' # same as what we got above with the loo_i function and i=3: @@ -147,7 +148,7 @@ #' log_lik_matrix <- sapply(1:N, function(i) { #' llfun(data_i = fake_data[i,, drop=FALSE], draws = fake_posterior) #' }) -#' loo_with_mat <- loo(log_lik_matrix, r_eff = NA) +#' loo_with_mat <- loo(log_lik_matrix) #' all.equal(loo_with_mat$estimates, loo_with_fn$estimates) # should be TRUE! #' #' @@ -191,11 +192,10 @@ loo <- function(x, ...) { loo.array <- function(x, ..., - r_eff = NULL, + r_eff = 1, save_psis = FALSE, cores = getOption("mc.cores", 1), is_method = c("psis", "tis", "sis")) { - if (is.null(r_eff)) throw_loo_r_eff_warning() is_method <- match.arg(is_method) psis_out <- importance_sampling.array(log_ratios = -x, r_eff = r_eff, cores = cores, method = is_method) ll <- llarray_to_matrix(x) @@ -216,14 +216,11 @@ loo.array <- loo.matrix <- function(x, ..., - r_eff = NULL, + r_eff = 1, save_psis = FALSE, cores = getOption("mc.cores", 1), is_method = c("psis", "tis", "sis")) { is_method <- match.arg(is_method) - if (is.null(r_eff)) { - throw_loo_r_eff_warning() - } psis_out <- importance_sampling.matrix( log_ratios = -x, @@ -254,7 +251,7 @@ loo.function <- ..., data = NULL, draws = NULL, - r_eff = NULL, + r_eff = 1, save_psis = FALSE, cores = getOption("mc.cores", 1), is_method = c("psis", "tis", "sis")) { @@ -265,11 +262,7 @@ loo.function <- .llfun <- validate_llfun(x) N <- dim(data)[1] - if (is.null(r_eff)) { - throw_loo_r_eff_warning() - } else { - r_eff <- prepare_psis_r_eff(r_eff, len = N) - } + r_eff <- prepare_psis_r_eff(r_eff, len = N) psis_list <- parallel_importance_sampling_list( @@ -294,7 +287,8 @@ loo.function <- diagnostics_list <- lapply(psis_list, "[[", "diagnostics") diagnostics <- list( pareto_k = psis_apply(diagnostics_list, "pareto_k"), - n_eff = psis_apply(diagnostics_list, "n_eff") + n_eff = psis_apply(diagnostics_list, "n_eff"), + r_eff = psis_apply(diagnostics_list, "r_eff") ) } @@ -331,7 +325,7 @@ loo_i <- ..., data = NULL, draws = NULL, - r_eff = NULL, + r_eff = 1, is_method = "psis" ) { stopifnot( @@ -364,7 +358,7 @@ loo_i <- ..., data, draws, - r_eff = NULL, + r_eff = 1, save_psis = FALSE, is_method) { @@ -552,7 +546,8 @@ list2importance_sampling <- function(objects) { log_weights = log_weights, diagnostics = list( pareto_k = psis_apply(diagnostics, item = "pareto_k"), - n_eff = psis_apply(diagnostics, item = "n_eff") + n_eff = psis_apply(diagnostics, item = "n_eff"), + r_eff = psis_apply(diagnostics, item = "r_eff") ) ), norm_const_log = psis_apply(objects, "norm_const_log", fun = "attr"), diff --git a/R/loo_approximate_posterior.R b/R/loo_approximate_posterior.R index 4b67c32f..3a3bce09 100644 --- a/R/loo_approximate_posterior.R +++ b/R/loo_approximate_posterior.R @@ -127,7 +127,7 @@ loo_approximate_posterior.function <- .llfun = .llfun, data = data, draws = draws, - r_eff = NULL, # r_eff is ignored + r_eff = 1, # r_eff is ignored save_psis = save_psis, log_p = log_p, log_g = log_g, @@ -172,7 +172,7 @@ loo_approximate_posterior.function <- draws, log_p, log_g, - r_eff = NULL, + r_eff = 1, save_psis = FALSE, is_method) { diff --git a/R/loo_compare.R b/R/loo_compare.R index 3bd1d765..028fa59b 100644 --- a/R/loo_compare.R +++ b/R/loo_compare.R @@ -61,9 +61,9 @@ #' @examples #' # very artificial example, just for demonstration! #' LL <- example_loglik_array() -#' loo1 <- loo(LL, r_eff = NA) # should be worst model when compared -#' loo2 <- loo(LL + 1, r_eff = NA) # should be second best model when compared -#' loo3 <- loo(LL + 2, r_eff = NA) # should be best model when compared +#' loo1 <- loo(LL) # should be worst model when compared +#' loo2 <- loo(LL + 1) # should be second best model when compared +#' loo3 <- loo(LL + 2) # should be best model when compared #' #' comp <- loo_compare(loo1, loo2, loo3) #' print(comp, digits = 2) diff --git a/R/loo_predictive_metric.R b/R/loo_predictive_metric.R index 009862b7..8ee18bd2 100644 --- a/R/loo_predictive_metric.R +++ b/R/loo_predictive_metric.R @@ -91,7 +91,7 @@ loo_predictive_metric.matrix <- log_lik, ..., metric = c("mae", "rmse", "mse", "acc", "balanced_acc"), - r_eff = NULL, + r_eff = 1, cores = getOption("mc.cores", 1)) { stopifnot( is.numeric(x), diff --git a/R/loo_subsample.R b/R/loo_subsample.R index d02a3ed6..f5bea2ec 100644 --- a/R/loo_subsample.R +++ b/R/loo_subsample.R @@ -103,7 +103,7 @@ loo_subsample.function <- observations = 400, log_p = NULL, log_g = NULL, - r_eff = NULL, + r_eff = 1, save_psis = FALSE, cores = getOption("mc.cores", 1), loo_approximation = "plpd", @@ -124,11 +124,7 @@ loo_subsample.function <- checkmate::assert_null(dim(log_g)) if (is.null(log_p) && is.null(log_g)) { - if (is.null(r_eff)) { - throw_loo_r_eff_warning() - } else { r_eff <- prepare_psis_r_eff(r_eff, len = dim(data)[1]) - } } checkmate::assert_flag(save_psis) cores <- loo_cores(cores) @@ -254,7 +250,7 @@ update.psis_loo_ss <- function(object, ..., data = NULL, draws = NULL, observations = NULL, - r_eff = NULL, + r_eff = 1, cores = getOption("mc.cores", 1), loo_approximation = NULL, loo_approximation_draws = NULL, @@ -976,6 +972,7 @@ rbind.psis_loo_ss <- function(object, x) { object$diagnostics$pareto_k <- c(object$diagnostics$pareto_k, x$diagnostics$pareto_k) object$diagnostics$n_eff <- c(object$diagnostics$n_eff, x$diagnostics$n_eff) + object$diagnostics$r_eff <- c(object$diagnostics$r_eff, x$diagnostics$r_eff) attr(object, "dims")[2] <- nrow(object$pointwise) object } @@ -999,6 +996,7 @@ remove_idx.psis_loo_ss <- function(object, idxs) { object$pointwise <- object$pointwise[-row_map$row_no,,drop = FALSE] object$diagnostics$pareto_k <- object$diagnostics$pareto_k[-row_map$row_no] object$diagnostics$n_eff <- object$diagnostics$n_eff[-row_map$row_no] + object$diagnostics$r_eff <- object$diagnostics$r_eff[-row_map$row_no] attr(object, "dims")[2] <- nrow(object$pointwise) object } @@ -1020,6 +1018,7 @@ order.psis_loo_ss <- function(x, observations) { x$pointwise <- x$pointwise[row_map$row_no_x,,drop = FALSE] x$diagnostics$pareto_k <- x$diagnostics$pareto_k[row_map$row_no_x] x$diagnostics$n_eff <- x$diagnostics$n_eff[row_map$row_no_x] + x$diagnostics$r_eff <- x$diagnostics$r_eff[row_map$row_no_x] x } diff --git a/R/print.R b/R/print.R index a2247580..b739efd7 100644 --- a/R/print.R +++ b/R/print.R @@ -17,6 +17,7 @@ print.loo <- function(x, digits = 1, ...) { cat("\n") print_dims(x) + print_reff_summary(x, digits) if (!("estimates" %in% names(x))) { x <- convert_old_object(x) } @@ -120,7 +121,7 @@ print_dims.importance_sampling <- function(x, ...) { cat( "Computed from", paste(dim(x), collapse = " by "), - "log-weights matrix\n" + "log-weights matrix.\n" ) } @@ -130,7 +131,7 @@ print_dims.psis_loo <- function(x, ...) { cat( "Computed from", paste(dim(x), collapse = " by "), - "log-likelihood matrix\n" + "log-likelihood matrix.\n" ) } @@ -140,7 +141,7 @@ print_dims.importance_sampling_loo <- function(x, ...) { cat( "Computed from", paste(dim(x), collapse = " by "), - "log-likelihood matrix using", class(x)[1], "\n" + "log-likelihood matrix using", class(x)[1], ".\n" ) } @@ -150,7 +151,7 @@ print_dims.waic <- function(x, ...) { cat( "Computed from", paste(dim(x), collapse = " by "), - "log-likelihood matrix\n" + "log-likelihood matrix.\n" ) } @@ -159,7 +160,7 @@ print_dims.waic <- function(x, ...) { print_dims.kfold <- function(x, ...) { K <- attr(x, "K", exact = TRUE) if (!is.null(K)) { - cat("Based on", paste0(K, "-fold"), "cross-validation\n") + cat("Based on", paste0(K, "-fold"), "cross-validation.\n") } } @@ -175,6 +176,22 @@ print_dims.psis_loo_ss <- function(x, ...) { ) } +print_reff_summary <- function(x, digits) { + r_eff <- x$diagnostics$r_eff + if (all(r_eff==1)) { + cat( + "MCSE and ESS estimates assume independent draws (r_eff=1).\n" + ) + } else { + cat(paste0( + "MCSE and ESS estimates assume MCMC draws (r_eff in [", + .fr(min(r_eff), digits), + ", ", + .fr(max(r_eff), digits), + "]).\n" + )) + } +} print_mcse_summary <- function(x, digits) { mcse_val <- mcse_loo(x) @@ -184,7 +201,6 @@ print_mcse_summary <- function(x, digits) { ) } - # print and warning helpers .fr <- function(x, digits) format(round(x, digits), nsmall = digits) .warn <- function(..., call. = FALSE) warning(..., call. = call.) diff --git a/R/psis.R b/R/psis.R index 68778084..67db61da 100644 --- a/R/psis.R +++ b/R/psis.R @@ -19,11 +19,11 @@ #' This is related to the relative efficiency of estimating the normalizing #' term in self-normalizing importance sampling. If `r_eff` is not #' provided then the reported PSIS effective sample sizes and Monte Carlo -#' error estimates will be over-optimistic. See the [relative_eff()] -#' helper function for computing `r_eff`. If using `psis` with -#' draws of the `log_ratios` not obtained from MCMC then the warning -#' message thrown when not specifying `r_eff` can be disabled by -#' setting `r_eff` to `NA`. +#' error estimates can be over-optimistic. If the posterior draws are (near) +#' independent then `r_eff=1` can be used. `r_eff` has to be a scalar (same +#' value is used for all observations) or a vector with length equal to the +#' number of observations. The default value is 1. See the [relative_eff()] +#' helper function for computing `r_eff`. #' #' @return The `psis()` methods return an object of class `"psis"`, #' which is a named list with the following components: @@ -99,7 +99,7 @@ psis <- function(log_ratios, ...) UseMethod("psis") #' psis.array <- function(log_ratios, ..., - r_eff = NULL, + r_eff = 1, cores = getOption("mc.cores", 1)) { importance_sampling.array(log_ratios = log_ratios, ..., r_eff = r_eff, @@ -115,7 +115,7 @@ psis.array <- psis.matrix <- function(log_ratios, ..., - r_eff = NULL, + r_eff = 1, cores = getOption("mc.cores", 1)) { importance_sampling.matrix(log_ratios, ..., @@ -129,7 +129,7 @@ psis.matrix <- #' @template vector #' psis.default <- - function(log_ratios, ..., r_eff = NULL) { + function(log_ratios, ..., r_eff = 1) { importance_sampling.default(log_ratios = log_ratios, ..., r_eff = r_eff, method = "psis") @@ -273,12 +273,13 @@ psis_smooth_tail <- function(x, cutoff) { #' 20% of the total number of weights. #' #' @noRd -#' @param r_eff A N-vector of relative MCMC effective sample sizes of `exp(log-lik matrix)`. If NULL, relative efficiency of 1 is used. +#' @param r_eff A N-vector or scalar of relative MCMC effective sample sizes of +#' `exp(log-lik matrix)`. The default value is 1. #' @param S The (integer) size of posterior sample. #' @return An N-vector of tail lengths. #' n_pareto <- function(r_eff, S) { - if (is.null(r_eff)) { + if (isTRUE(is.null(r_eff) || all(is.na(r_eff)))) { r_eff <- 1 } ceiling(pmin(0.2 * S, 3 * sqrt(S / r_eff))) @@ -347,19 +348,19 @@ throw_tail_length_warnings <- function(tail_lengths) { #' @param len The length `r_eff` should have if not `NULL` or `NA`. #' @return #' * If `r_eff` has length `len` then `r_eff` is returned. -#' * If `r_eff` is `NULL` then a warning is thrown and `rep(1, len)` is returned. -#' * If `r_eff` is `NA` then the warning is skipped and -#' `rep(1, len)` is returned. +#' * If `r_eff` is `NULL` then `rep(1, len)` is returned. +#' * If `r_eff` is `NA` then `rep(1, len)` is returned. +#' * If `r_eff` is a scalar then `rep(r_eff, len)` is returned. +#' * If `r_eff` is not a scalar but the length is not `len` then an error is thrown. #' * If `r_eff` has length `len` but has `NA`s then an error is thrown. #' prepare_psis_r_eff <- function(r_eff, len) { if (isTRUE(is.null(r_eff) || all(is.na(r_eff)))) { - if (!called_from_loo() && is.null(r_eff)) { - throw_psis_r_eff_warning() - } r_eff <- rep(1, len) + } else if (length(r_eff) == 1) { + r_eff <- rep(r_eff, len) } else if (length(r_eff) != len) { - stop("'r_eff' must have one value per observation.", call. = FALSE) + stop("'r_eff' must have one value or one value per observation.", call. = FALSE) } else if (anyNA(r_eff)) { stop("Can't mix NA and not NA values in 'r_eff'.", call. = FALSE) } diff --git a/tests/testthat/reference-results/loo.rds b/tests/testthat/reference-results/loo.rds index 63ea4de268413d48df8eb48a568ef1d1728f6c29..c98cc717a4c2f2bbcbed58438b778f2230f817b7 100644 GIT binary patch literal 2088 zcmV+@2-o)?iwFP!000001MQZ1R8v^R&NYr>;l1I0i7Y%&C<_ZSYV+`q1yz-bjWg(`8S@p9(hyOk#Hi^@ z=2KUL@h@eQ^fxcOyVg09G#i~p$Gk?;Ppq;Wb9V(v3qCX=MPoq{B|RlCEzXfdOK`KT zOhpdx_s*-k)Js}=^OznZJxGRaGUUeSN#cxq%DB7KeR!|!4>){J(j1Ta)(QhjqWs#0 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z_y2kDe@5-!sYomT^KR1rynxO9g)5Lr(BLX9zAi=$M=}C&wt$j=ZTZv z9*3Myp78NKgm3)cb`?u-^A1RC05RU-v7r3lAa*XSJmTIha{_0QIurg(NV_hkT3BT7e67~eKHJ8Np zY0lyE>r|?~oCYz+9h^+R0uVEl*|ko~0mP~lzFpaH9VGj0dqO=mVbin3KJBgXAa>?O zl7EK?Y*PRIBd>53h&5#YX5Y~dVpDhP&t-~&crefC>g;Y1Hw`}Jf3~j>#EMMr91Gh4 zVyngLREKr&K9f)NrQr1qIo-6%#Xv%{!`VDw4#ZxC-)f+@gZR3sm8@-=ASS4l<(Dk? zUsYoGIv#UB)*5NBF--8^V7b|MT<4|9bwv-fsi{b)Nq@+Wyb+ z0O9}pdWZjcZ1q3qe>wP{_y6a0|GD0O&i`Nc^`G Date: Thu, 1 Feb 2024 11:57:21 +0200 Subject: [PATCH 24/39] update glossary and FAQ --- R/loo-glossary.R | 2 +- vignettes/online-only/faq.Rmd | 206 +++++++++++++++++++++------------- vignettes/online-only/faq.bib | 189 ++++++++++++++++++++++++++++++- 3 files changed, 314 insertions(+), 83 deletions(-) diff --git a/R/loo-glossary.R b/R/loo-glossary.R index 789c4d22..b13fd15b 100644 --- a/R/loo-glossary.R +++ b/R/loo-glossary.R @@ -113,7 +113,7 @@ #' Section 6 of Gabry et al. (2019) for an example. #' #' \subsection{Interpreting `p_loo` when Pareto `k` is large}{ -#' If \eqn{k < min(1 - 1 / log10(S), 0.7)} then we can also look at +#' If \eqn{k > 0.7} then we can also look at #' the `p_loo` estimate for some additional information about the problem: #' #' * If `p_loo << p` (the total number of parameters in the model), diff --git a/vignettes/online-only/faq.Rmd b/vignettes/online-only/faq.Rmd index 6ca2490e..1d270e48 100644 --- a/vignettes/online-only/faq.Rmd +++ b/vignettes/online-only/faq.Rmd @@ -44,7 +44,7 @@ More about these cases: 2b) Instead of considering predictions for future, we can consider whether we can generalize from some observations to others. For example, in social science we might make a model explaining poll results with demographical data. To test the model, instead of considering future pollings, we could test whether the model can predict for a new state. If we have observed data from all states in USA, then there are no new states (or it can take unpredictable time before there are new states), but we can simulate a situation where we leave out data from one state and check can we generalize from other states to the left out state. This is sensible approach when we assume that states are exchangeable conditional on the information available (see, e.g., @BDA3 Chapter 5 for exchangeability). The generalization ability from one entity (a person, state, etc) to other similar entity tells us that model has learned something useful. It is very important to think what is the level where the generalization is most interesting. For example, in cognitive science and psychology it would be more interesting to generalize from one person to another than within person data from one trial to another trial for the same person. In cognitive science and psychology studies it is common that the study population is young university students, and in such thus there are limitations what we can say about the generalization to whole human population. In polling data from all US states, the whole population of US states has been observed, but there is limitation how we can generalize to other countries or future years. -2c) In addition of assessing the predictive accuracy and generalizability, it is useful to assess how well calibrated is the uncertainty quantification of the predictive distribution. Cross-validation is useful when we don't trust that the model is well specified, although many bad mis-specifications can be diagnosed also with simpler posterior predictive checking. See, for example, case study [roaches](https://avehtari.github.io/modelselection/roaches.html). +2c) In addition of assessing the predictive accuracy and generalizability, it is useful to assess how well calibrated is the uncertainty quantification of the predictive distribution. Cross-validation is useful when we don't trust that the model is well specified, although many bad mis-specifications can be diagnosed also with simpler posterior predictive checking. See, for example, case study [roaches](https://users.aalto.fi/~ave/modelselection/roaches.html). ## Using cross-validation for many models {#manymodels} @@ -56,31 +56,31 @@ Three basic cases for why to use cross-validation for many models are: More about these cases: -1 ) Use of cross-validation to select the model with best predictive performance is relatively safe if there are small or moderate number of models, and there is a lot of data compared to the model complexity or the best model is clearly best [@Piironen+Vehtari:2017a, @Sivula+etal:2020:loo_uncertainty]. See also Section [How to use cross-validation for model selection?](#modelselection). +1 ) Use of cross-validation to select the model with best predictive performance is relatively safe if there are small or moderate number of models, and there is a lot of data compared to the model complexity or the best model is clearly best [@Piironen+Vehtari:2017a; @Sivula+etal:2020:loo_uncertainty,@McLatchie+Vehtari:2023]. See also Section [How to use cross-validation for model selection?](#modelselection). -2a) Cross-validation is useful especially when there are posterior dependencies between parameters and examining the marginal posterior of a parameter is not very useful to determine whether the component related to that parameter is relevant. This happens, for example, in case of collinear predictors. See, for example, case studies [collinear](https://avehtari.github.io/modelselection/collinear.html), [mesquite](https://avehtari.github.io/modelselection/mesquite.html), and [bodyfat](https://avehtari.github.io/modelselection/bodyfat.html). +2a) Cross-validation is useful especially when there are posterior dependencies between parameters and examining the marginal posterior of a parameter is not very useful to determine whether the component related to that parameter is relevant. This happens, for example, in case of collinear predictors. See, for example, case studies [collinear](https://users.aalto.fi/~ave/modelselection/collinear.html), [mesquite](https://users.aalto.fi/~ave/modelselection/mesquite.html), and [bodyfat](https://users.aalto.fi/~ave/modelselection/bodyfat.html). -2b) Cross-validation is less useful for simple models with no posterior dependencies and assuming that simple model is not mis-specified. In that case the marginal posterior is less variable as it includes the modeling assumptions (which assume to be not mis-specified) while cross-validation uses non-model based approximation of the future data distribution which increases the variability. See, for example, case study [betablockers](https://avehtari.github.io/modelselection/betablockers.html). +2b) Cross-validation is less useful for simple models with no posterior dependencies and assuming that simple model is not mis-specified. In that case the marginal posterior is less variable as it includes the modeling assumptions (which assume to be not mis-specified) while cross-validation uses non-model based approximation of the future data distribution which increases the variability. See, for example, case study [betablockers](https://users.aalto.fi/~ave/modelselection/betablockers.html). 2c) Cross-validation can provide quantitative measure, which should only complement but not replace understanding of qualitative patterns in the data (see, e.g., @Navarro:2019:between). 3 ) See more in [How to use cross-validation for model averaging?](#modelaveraging). -See also the next Section "When not to use cross-validation?", [How is cross-validation related to overfitting?}(#overfitting), and [How to use cross-validation for model selection?](#modelselection). +See also the next Section "When not to use cross-validation?", [How is cross-validation related to overfitting?](#overfitting), and [How to use cross-validation for model selection?](#modelselection). ## When not to use cross-validation? In general there is no need to do any model selection (see more in [How is cross-validation related to overfitting?](#overfitting), and [How to use cross-validation for model selection?](#modelselection)). The best approach is to build a rich model that includes all the uncertainties, do model checking, and possible model adjustments. -Cross-validation cannot answer directly the question "Do the data provide evidence for some effect being non-zero?" Using cross-validation to compare a model with an additional term to a model without that term is a kind of null hypothesis testing. Cross-validation can tell whether that extra term can improve the predictive accuracy. The improvement in the predictive accuracy is a function of signal-to-noise-ratio, the size of the actual effect, and how much the effect is correlating with other included effects. If cross-validation prefers the simpler model, it is not necessarily evidence for an effect being exactly zero, but it is possible that the effect is too small to make a difference, or due to the dependencies it doesn't provide additional information compared to what is already included in the model. Often it makes more sense to just fit the larger model and explore the posterior of the relevant coefficient. Analysing the posterior can however be difficult if there are strong posterior dependencies. +Cross-validation is not answering directly the question "Do the data provide evidence for some effect being non-zero?" Using cross-validation to compare a model with an additional term to a model without that term is a kind of null hypothesis testing. Cross-validation can tell whether that extra term can improve the predictive accuracy. The improvement in the predictive accuracy is a function of signal-to-noise-ratio, the size of the actual effect, and how much the effect is correlating with other included effects. If cross-validation prefers the simpler model, it is not necessarily evidence for an effect being exactly zero, but it is possible that the effect is too small to make a difference, or due to the dependencies it doesn't provide additional information compared to what is already included in the model. Often it makes more sense to just fit the larger model and explore the posterior of the relevant coefficient. Analysing the posterior can however be difficult if there are strong posterior dependencies. -Cross-validation is not good for selecting a model from a large number of models (see [How to use cross-validation for model selection?](#modelselection)) +Using cross-validation to select one model among a large number of models (see [How to use cross-validation for model selection?](#modelselection)) can suffer from the selection induced bias (see, e.g., @McLatchie+Vehtari:2023). # Tutorial material on cross-validation - Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. [online](http://link.springer.com/article/10.1007\%2Fs11222-016-9696-4). - [LOO glossary](https://mc-stan.org/loo/reference/loo-glossary.html) -- [Model selection video lectures](https://avehtari.github.io/modelselection/) and Bayesian Data Analysis lectures [8.2](https://aalto.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=d7849131-0afd-4ae6-ad64-aafb00da36f4), [9.1](https://aalto.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=50b2e73f-af0a-4715-b627-ab0200ca7bbd), [9.2](https://aalto.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=b0299d53-9454-4e33-9086-ab0200db14eeb). +- [Model selection video lectures](https://users.aalto.fi/~ave/modelselection/) and Bayesian Data Analysis lectures [8.2](https://aalto.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=456afda7-0e6d-4903-b0df-b0ab00da8f1e), [9.1](https://aalto.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=a4961b5a-7e42-4603-8aaf-b0b200ca6295), [9.2](https://aalto.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=a4796c79-eab2-436e-b55f-b0b200dac7ce). - Decision theoretical background on Bayesian cross-validation can be found in the article [A survey of Bayesian predictive methods for model assessment, selection and comparison](https://dx.doi.org/10.1214/12-SS102) [@Vehtari+Ojanen:2012]. # What are the parts of cross-validation? {#parts} @@ -89,7 +89,7 @@ It is important to separate 1. the way how the data is divided in cross-validation, e.g. leave-one-out (LOO), leave-one-group-out (LOGO), and leave-future-out (LFO) 2. the utility or loss, e.g. expected log predictive density (ELPD), root mean square error (RMSE), explained variance ($R^2$) -3. the computational method use to compute leave-one-out predictive distributions, e.g. K-fold-CV, Pareto smoothed importance sampling (PSIS), +3. the computational method use to compute leave-one-out predictive distributions, e.g. $K$-fold-CV, Pareto smoothed importance sampling (PSIS), 4. and the estimate obtained by combining these. ## The way how the data is divided in cross-validation @@ -101,13 +101,14 @@ Different partitions of data are held out in different kinds of cross-validation - LFO: leave-future-out cross-validation approach (all future observations). See more in [Can cross-validation be used for time series?](#timeseries). - LOGO: leave-one-group-out cross-validation approach (a group of observations). See more in [Can cross-validation be used for hierarchical / multilevel models?](#hierarchical). -Which unit is systematically left out determines the predictive task that cross-validation assesses model performance on (see more in [When is cross-validation valid?](#valid)). CV, LOO, LFO and LOGO and other cross-validation approaches do not yet specify the utility or loss, or how the computation is made except that it involves estimating cross-validated predictive densities or probabilities. +Which unit is systematically left out determines the predictive task that cross-validation assesses model performance on (see more in [When is cross-validation valid?](#valid)). CV, LOO, LFO and LOGO and other cross-validation approaches do not yet specify the utility or loss, or how the computation is made except that it involves estimating cross-validated predictive densities or probabilities. If the goal is to estimate the predictive performance of a single model used for a specific prediction task, it is often natural to match the actual prediction task and how the data is partitioned in cross-validation, but sometimes better accuracy can be obtained with a different partitioning (possibly slightly increasing bias, but greatly reducing variance). ## The utility or loss First we need to define the utility or loss function which compares predictions to observations. These predictions can be considered to be for future observations, or for other exchangeable entities (see more in [What is cross-validation?](#¤whatis)). Some examples: - LPD or LPPD: Log pointwise predictive density for a new observation. For simplicity the LPD acronym is used also for expected log pointwise predictive probabilities for discrete models. Often a shorter term log score is used. +- LJPD: Log joint pointwise predictive density for several observations (see, e.g., @Cooper+etal:2023). For simplicity the LJPD acronym is used also for expected log joint predictive probabilities for discrete models. - RMSE: Root mean square error. - ACC: Classification accuracy. - $R^2$: Explained variance (see, e.g., @Gelman+etal:2019:BayesR2) @@ -115,7 +116,7 @@ First we need to define the utility or loss function which compares predictions These are examples of utility and loss functions for using the model to predict the future data and then observing that data. Other utility and loss functions could also be used. See more in [Can other utilities or losses be used than log predictive density?](#otherutility), [Scoring rule in Wikipedia](https://en.wikipedia.org/wiki/Scoring_rule), and [Gneiting and Raftery, 2012](https://doi.org/10.1198/016214506000001437). -The value of the loss functions necessarily depends on the data we observe next. We can however try to estimate an _expectation_ of the loss (a summary of average predictive performance over several predictions or expected predictive performance for one prediction) under the assumption that both the covariates and responses we currently have are representative of those we will observe in the future. +The value of the loss functions necessarily depends on the data we observe next. We can however try to estimate an _expectation_ of the loss (a summary of average predictive performance over several predictions or expected predictive performance for one prediction) under the assumption that both the covariates and responses we currently have are representative of those we might observe in the future. - ELPD: The theoretical expected log pointwise predictive density for new observations (or other exchangeable entity) (Eq 1 in @Vehtari+etal:PSIS-LOO:2017). One scenario when we could also actually observe this is if we would get infinite number of future observations from the same data generating mechanism. However, this expected value is valid also when thinking just about one future observation (other exchangeable entity). This can be computed given different data partitions. For simplicity the ELPD acronym is used also for expected log pointwise predictive probabilities for discrete models. @@ -142,14 +143,14 @@ These terms are not yet defining possible computational approximations. ## The computational method used to compute leave-one-out predictive distributions -The choice of partitions to leave out or metric of model performance is independent of the computational method (e.g. PSIS or K-fold-CV). Different computational methods can be used to make the computation faster: +The choice of partitions to leave out or metric of model performance is independent of the computational method (e.g. PSIS or $K$-fold-CV). Different computational methods can be used to make the computation faster: -- K-fold-CV: Each cross-validation fold uses the same inference as is used for the full data. For example, if MCMC is used then MCMC inference needs to be run K times. -- LOO with K-fold-CV: If K=N, where N is the number of observations, then K-fold-CV is LOO. Sometimes this is called exact, naive or brute-force LOO. This can be time consuming as the inference needs to be repeated N times. +- $K$-fold-CV: Each cross-validation fold uses the same inference as is used for the full data. For example, if MCMC is used then MCMC inference needs to be run $K$ times. +- LOO with $K$-fold-CV: If $K=N$, where $N$ is the number of observations, then $K$-fold-CV is LOO. Sometimes this is called exact, naive or brute-force LOO. This can be time consuming as the inference needs to be repeated $N$ times. Sometimes, efficient parallelization can make the wall clock time to be close to the time needed for one model fit [Cooper+etal:2023:parallelCV]. - PSIS-LOO: Pareto smoothed importance sampling leave-one-out cross-validation. Pareto smoothed importance sampling (PSIS, @Vehtari+etal:PSIS-LOO:2017, @Vehtari+etal:PSIS:2019) is used to estimate leave-one-out predictive densities or probabilities. - PSIS: Richard McElreath shortens PSIS-LOO as PSIS in Statistical Rethinking, 2nd ed. -- MM-LOO: Moment matching importance sampling leave-one-out cross-validation [@Paananen+etal:2021:implicit]. Which works better than PSIS-LOO in challenging cases, but is still faster than K-fold-CV with K=N. -- RE-LOO: Run exact LOO (see LOO with K-fold-CV) for those observations for which PSIS diagnostic indicates PSIS-LOO is not accurate (that is, re-fit the model for those leave-one-out cases). +- MM-LOO: Moment matching importance sampling leave-one-out cross-validation [@Paananen+etal:2021:implicit]. Which works better than PSIS-LOO in challenging cases, but is still faster than $K$-fold-CV with K=N. +- RE-LOO: Run exact LOO (see LOO with $K$-fold-CV) for those observations for which PSIS diagnostic indicates PSIS-LOO is not accurate (that is, re-fit the model for those leave-one-out cases). We could write elpd_{psis-loo}, but often drop the specific computational method and report diagnostic information only if that computation may be unreliable. @@ -174,7 +175,7 @@ More about these cases: 2) Overfitting of bigger more complex models is a bigger problem when using less good inference methods. For example, bigger models fitted using maximum likelihood can have much worse predictive performance than simpler models. Overfitting of bigger models is a smaller problem in Bayesian inference because a) integrating over the posterior and b) use of priors. The impact of integration over the posterior is often underestimated compared to the impact of priors. See [a video demonstrating how integration over the posterior is also regularizing and reducing overfit](https://aalto.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=75b9f18f-e379-4557-a5fa-a9f500f11b40). It is still possible to make Bayesian models to overfit by using priors which have much more probability mass for over-complex solutions instead of simple solutions. Combination of (accurate) integration and sensible priors make it common that, for example, adding more predictors doesn't decrease the predictive performance of bigger models even if the number of predictors is much higher than the number of observations (which would be a big problem with maximum likelihood). -3) In (2), it was mentioned that when using maximum likelihood tends to overfit more easily. In Bayesian inference, using approximate integration, for example, using variational inference with normal distribution with diagonal covariance, can overfit more than when using accurate integration. If accurate Bayesian inference is used for each model, but model selection is made using, for example, cross-validation, then the model selection process can overfit badly [@Piironen+Vehtari:2017a]. +3) In (2), it was mentioned that when using maximum likelihood tends to overfit more easily. In Bayesian inference, using approximate integration, for example, using variational inference with normal distribution with diagonal covariance, can overfit more than when using accurate integration. If accurate Bayesian inference is used for each model, but model selection is made using, for example, cross-validation, then the model selection process can overfit [@Piironen+Vehtari:2017a,McLatchie+Vehtari:2023]. # How to use cross-validation for model selection? {#modelselection} @@ -182,26 +183,25 @@ Summary - First avoid model selection by using the model which includes all predictors and includes all uncertain things. Then optimal thing is to integrate over all the uncertainties. When including many components to a model, it is useful to think more carefully about the prior. For example, if there are many predictors, it is useful to use priors that a) state that only some of the effects are big, or b) many effects are big and correlating (it is not possible to have a large number of big independent effects @Tosh+etal:2021:piranha). - If there is explicit utility or loss for observing future predictor values (e.g. medical tests) use decision theory. -- If there is implicit cost for bigger models (e.g. bigger model more difficult to explain or costs of feature measurements are unknown), choose a smaller model which similar predictive performance as the biggest model. If there are only a small number of models, overfitting due to selection process is small. If there are a large number of models, as for example often in variable selection, then the overfitting due to the selection process can be a problem [@Piironen+Vehtari:2017a] and more elaborate approaches, such as projection predictive variable selection is recommended. +- If there is implicit cost for bigger models (e.g. bigger model more difficult to explain or costs of feature measurements are unknown), choose a smaller model which similar predictive performance as the biggest model. If there are only a small number of models, overfitting due to selection process is small [@McLatchie+Vehtari:2023]. If there are a large number of models, as for example often in variable selection, then the overfitting due to the selection process can be a problem [@Piironen+Vehtari:2017a; @McLatchie+Vehtari:2023] and more elaborate approaches, such as projection predictive variable selection is recommended[@McLatchie+etal:2023:projpred_workflow]. - If there is application specific utility or loss function, use that to assess practically relevant difference in predictive performance of two models. - If there is no application specific utility or loss function, use log score, ie elpd. If elpd difference (elpd_diff in `loo` package) is less than 4, the difference is small [@Sivula+etal:2020:loo_uncertainty]). If elpd difference (elpd_diff in loo package) is larger than 4, then compare that difference to standard error of elpd_diff (provided e.g. by `loo` package) [@Sivula+etal:2020:loo_uncertainty]. See also Section [How to interpret in Standard error (SE) of elpd difference (elpd_diff)?](#se_diff). -If there is a large number of models compared, there is possibility of overfitting in model selection. +If there is a large number of models compared, there is possibility of non-negligible overfitting in model selection. - - See video [Model assessment, comparison and selection at Master class in Bayesian statistics, CIRM, Marseille](https://www.youtube.com/watch?v=Re-2yVd0Mqk). - @Vehtari+Ojanen:2012 write: "The model selection induced bias can be taken into account by the double/nested/2-deep cross-validation (e.g. Stone, 1974; Jonathan, Krzanowski and McCarthy, 2000) or making an additional bias correction (Tibshirani and Tibshirani, 2009)." - - @Piironen+Vehtari:2017a write: "Although LOO-CV and WAIC can be used to obtain a nearly unbiased estimate of the predictive ability of a given model, both of these estimates contain a stochastic error term whose variance can be substantial when the dataset is not very large. This variance in the estimate may lead to over-fitting in the selection process causing nonoptimal model selection and inducing bias in the performance estimate for the selected model (e.g., Ambroise and McLachlan 2002; Reunanen 2003; Cawley and Talbot 2010). The overfitting in the selection may be negligible if only a few models are being compared but, as we will demonstrate, may become a problem for a larger number of candidate models, such as in variable selection." + - @Piironen+Vehtari:2017a write: "Although LOO-CV and WAIC can be used to obtain a nearly unbiased estimate of the predictive ability of a given model, both of these estimates contain a stochastic error term whose variance can be substantial when the dataset is not very large. This variance in the estimate may lead to over-fitting in the selection process causing nonoptimal model selection and inducing bias in the performance estimate for the selected model (e.g., @Ambroise+McLachlan:2002; @Reunanen:2003; @Cawley+Talbot:2010). The overfitting in the selection may be negligible if only a few models are being compared but, as we will demonstrate, may become a problem for a larger number of candidate models, such as in variable selection." - Nested CV helps to estimate the overfitting due to the selection but doesn't remove that. - The overfitting is more severe depending on how many degrees of freedom there are in the selection. For example, in predictor selection we can think that we as many indicator variables as there are predictors and then there are combinatorial explosion in possible parameter combinations and overfitting can be severe (as demonstrated by @Piironen+Vehtari:2017a). + - @McLatchie+Vehtari:2023 present a light-weight approach to estimate and correct selection-induced bias, and demonstrate it in exhaustive comparison of modarate number of models and in case of forward search. -Thus if there are a very large number of models to compared, more elaborate approaches are recommended such as projection predictive variable selection [@Piironen+Vehtari:2017a; Piironen+etal:projpred:2020]. +Thus if there are a very large number of models to be compared, either methods that can estimate selection induced bias [@McLatchie+Vehtari:2023] should be used, or more elaborate approaches are recommended such as projection predictive variable selection [@Piironen+Vehtari:2017a; Piironen+etal:projpred:2020; @McLatchie+etal:2023:projpred_workflow]. See more in tutorial videos on using cross-validation for model selection - - [Model assessment, comparison and selection at Master class in Bayesian statistics, CIRM, Marseille](https://www.youtube.com/watch?v=Re-2yVd0Mqk) - Bayesian data analysis lectures -lectures [8.2](https://aalto.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=d7849131-0afd-4ae6-ad64-aafb00da36f4), [9.1](https://aalto.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=50b2e73f-af0a-4715-b627-ab0200ca7bbd), [9.2](https://aalto.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=b0299d53-9454-4e33-9086-ab0200db14eeb, -[9.3](https://aalto.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=4b6eeb48-ae64-4860-a8c3-ab0200e40ad8), and [12.1](https://aalto.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=e998b5dd-bf8e-42da-9f7c-ab1700ca2702). + [8.2](https://aalto.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=456afda7-0e6d-4903-b0df-b0ab00da8f1e), [9.1](https://aalto.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=a4961b5a-7e42-4603-8aaf-b0b200ca6295), [9.2](https://aalto.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=a4796c79-eab2-436e-b55f-b0b200dac7ce). +, and [11.1](https://aalto.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=7ef70bc8-122b-4e86-80fa-b0c000cb5511). # How to use cross-validation for model averaging? {#modelaveraging} @@ -210,23 +210,19 @@ If one of the models in the model selection is not clearly the best, it may be b - CV-weights / LOO-weights idea is derived from assuming elpd_loo can be considered as pseudo marginal likelihood [Geisser+Eddy:1979] and is related also to Akaike-weights [@Burnham+Anderson:2002] - LOO-BB-weights improve LOO-weights by taking into account the uncertainty related to having only a finite sample size to present the future data distribution [@Yao+etal:2018] -- Bayesian stacking optimizes the stacking weights to maximize the elpd_loo of the combined predictive distribution. +- Bayesian stacking optimizes the stacking weights to maximize the elpd_loo of the combined predictive distribution [@Yao+etal:2018] Based on the experiments by @Yao+etal:2018, Bayesian stacking has better performance than LOO-weights and LOO-BB-weights. # When is cross-validation valid? {#valid} -This is about how the data can be divided (see [What are the parts of cross-validation?](#parts)) to estimate expected utility or loss in certain prediction tasks. +This question is often focusing on whether specific data partition (see [What are the parts of cross-validation?](#parts)) matches the predictive task. The folklore says that for valid cross-validation the data partition should match the predictive task. Matching the data partition with the predictive task is assumed to minimize the bias, but there are cases where alternative partition may have negligible bias but much smaller variance (see, e.g., @Cooper+etal:2023). Thus different partitions can be valid even if they don't match the predictive task. -Quite long answer can be found in [a blog post](https://statmodeling.stat.columbia.edu/2018/08/03/loo-cross-validation-approaches-valid/). - -Some of the points from the blog post are extended below. - -LOO and cross-validation in general do not require independence or conditional independence. Exchangeability is sufficient. Even if we are using models with conditional independence structure, we don't require that the true data generating mechanism has such structure, but due to exchangeability and the data collection process we can proceed as if assuming conditional independence. See more in Chapter 5 of BDA3 [@BDA3]. Cross-validation can also be used when the model doesn’t have conditional independence structure. In time series, the observations $y_1,\ldots,y_T$ are not exchangeable as the index has additional information about the similarity in time. If we have model $p(y_t \mid f_t)$, with latent values $f_t$ then pairs $(y_1,f_1),\ldots,(y_T,f_T)$ are exchangeable (see Chapter 5 of BDA3 [@BDA3]) and we can factorize the likelihood trivially. We usually can present time series models with explicit latent values $f_t$ , but sometimes we integrate them analytically out due to computational reasons and then get non-factorized likelihood for exactly the same model. +LOO and cross-validation in general do not require independence or conditional independence. Exchangeability is sufficient. Even if we are using models with conditional independence structure, we don't require that the true data generating mechanism has such structure, but due to exchangeability and the data collection process we can proceed as if assuming conditional independence. See more in Chapter 5 of BDA3 [@BDA3]. Cross-validation can also be used when the model doesn’t have conditional independence structure. In time series, the observations $y_1,\ldots,y_T$ are not exchangeable as the index has additional information about the similarity in time. If we have model $p(y_t \mid f_t)$, with latent values $f_t$ then pairs $(y_1,f_1),\ldots,(y_T,f_T)$ are exchangeable (see Chapter 5 of BDA3 [@BDA3]) and we can factorize the likelihood trivially. We usually can present time series models with explicit latent values $f_t$ , but sometimes we integrate them analytically out due to computational reasons and then get non-factorized likelihood for exactly the same model (see, e.g., @Vehtari+etal:2016:LOO_for_GLVM, Burkner+Gabry+Vehtari:LFO-CV:2020). . If we want to evaluate the goodness of the model part $p(y_t \mid f_t)$, LOO is fine. If we want to evaluate the goodness of the time series model part $p(f_1,\ldots,f_T)$, way may be interested in 1) goodness for predicting missing data in a middle (think about audio restoration of recorded music with missing parts, e.g. due to scratches in the medium) or 2) we may be interested in predicting future (think about stock market or disease transmission models). -If the likelihood is factorizable (and if it’s not we can make it factorizable in some cases, see [Can cross-validation be used for time series?](#timeseries)) then this shows in Stan code as sum of log-likelihood terms. Now it’s possible to define entities which are sums of those individual log likelihood components. If the sums are related to exchangeable parts, we may use terms like leave-one-observation-out (LOO), leave-one-subject-out, leave-one-time-point-out, etc. If we want additionally restrict the information flow, for example, in time series we can add constraint that if $y_t$ is not observed then $y_{t+1},\ldots,y_{T}$ are not observed, we can use leave-future-out (LFO). +If the likelihood is factorizable (and if it’s not we can make it factorizable in some cases, see [Can cross-validation be used for time series?](#timeseries)) then this shows in Stan code as sum of log-likelihood terms. Now it’s possible to define entities which are sums of those individual log likelihood components. If the sums are related to exchangeable parts, we may use terms like leave-one-observation-out (LOO), leave-one-group-out (LOGO), leave-one-subject-out, leave-one-time-point-out, etc. If we want additionally restrict the information flow, for example, in time series we can add constraint that if $y_t$ is not observed then $y_{t+1},\ldots,y_{T}$ are not observed, we can use leave-future-out (LFO). How do we then choose the level of what to leave out in cross-validation? It depends which level of the model is interesting and if many levels are interesting then we can do cross-validation at different levels. @@ -238,28 +234,26 @@ The short answer is "Yes". Hierarchical model is useful, for example, if there a - LOO is valid if the focus is in the conditional observation model. LOO can often already reveal misspecification of the conditional observation model, and as LOO is often easy to compute, it is fine to start with LOO to investigate possible model issues. - LOO is also valid if the prediction task is new individuals in the existing groups. -- If the prediction task is to predict for new groups or we are interested in generalizability for new groups, then leave-one-group-out (LOGO) is a better choice. Computation of LOGO is often more challenging as hierarchical models often have group specific parameters, and removing the all data from one group will change the posterior of those parameters a lot. -- Leave-one-group-out (LOGO) cross-validation usually doesn't work with PSIS approach, unless the group specific parameters are integrated out. This is sometimes possible analytically (@Vehtari+etal:2016:LOO_for_GLVM) or by quadrature (@Merkle+Furr+Rabe_Hesketh:2019, [a Stan code example](https://avehtari.github.io/modelselection/roaches.html#5_Poisson_model_with_%E2%80%9Crandom_effects%E2%80%9D_and_integrated_LOO)) +- If the prediction task is to predict for new groups or we are interested in generalizability for new groups, then leave-one-group-out (LOGO) matches the predictive task more closely and we may assume smaller bias in the estimate. Computation of LOGO is often more challenging as hierarchical models often have group specific parameters, and removing the all data from one group will change the posterior of those parameters a lot. +- Leave-one-group-out (LOGO) cross-validation usually doesn't work with PSIS approach, unless the group specific parameters are integrated out. This is sometimes possible analytically [@Vehtari+etal:2016:LOO_for_GLVM] or by quadrature (@Merkle+Furr+Rabe_Hesketh:2019, [a Stan code example](https://users.aalto.fi/~ave/modelselection/roaches.html#5_Poisson_model_with_%E2%80%9Crandom_effects%E2%80%9D_and_integrated_LOO)) See also - - [A blog post](https://statmodeling.stat.columbia.edu/2018/08/03/loo-cross-validation-approaches-valid/) - - [Model selection videos](https://avehtari.github.io/modelselection/) - - A case study [Cross-validation for hierarchical models](https://avehtari.github.io/modelselection/rats_kcv.html) - - A case study [Roaches cross-validation demo](https://avehtari.github.io/modelselection/roaches.html) with "randome effects" models + - A case study [Cross-validation for hierarchical models](https://users.aalto.fi/~ave/modelselection/rats_kcv.html) + - A case study [Roaches cross-validation demo](https://users.aalto.fi/~ave/modelselection/roaches.html) with "randome effects" models - [Merkel, Furr, and Rabe-Hesketh (2019). Bayesian Comparison of Latent Variable Models: Conditional Versus Marginal Likelihoods. *Psychometrika* 84:802-829.](https://link.springer.com/article/10.1007/s11336-019-09679-0) + - @Cooper+etal:2023 # Can cross-validation be used for time series? {#timeseries} -The short answer is "Yes" (see, e.g. @Burkner+Gabry+Vehtari:LFO-CV:2020). If there is a model $p(y_i \mid f_i,\phi)$ and joint time series prior for $(f_1,...,f_T)$ then $p(y_i \mid f_i,\phi)$ can be considered independent given $f_i$ and $\phi$ and likelihood is factorizable. This is true often and the past values are informative about future values, but conditionally we know $f_i$, the past values are not providing additional information. This should not be confused with that when we don’t know $f_i$ and integrate over the posterior of $(f_1,...,f_T)$, as then $y_i$ are not conditionally independent given $\phi$. Also they are not anymore exchangeable as we have the time ordering telling additional information. $M$-step ahead prediction (see @Burkner+Gabry+Vehtari:LFO-CV:2020) is more about the usual interest in predicting future and evaluating the time series model for $(f_1,...,f_T)$, but leave-one-out cross-validation is valid for assessing conditional part $p(y_i \mid f_i)$. +The short answer is "Yes" (see, e.g. @Burkner+Gabry+Vehtari:LFO-CV:2020 and @Cooper+etal:2023). If there is a model $p(y_i \mid f_i,\phi)$ and joint time series prior for $(f_1,...,f_T)$ then $p(y_i \mid f_i,\phi)$ can be considered independent given $f_i$ and $\phi$ and likelihood is factorizable. This is true often and the past values are informative about future values, but conditionally we know $f_i$, the past values are not providing additional information. This should not be confused with that when we don’t know $f_i$ and integrate over the posterior of $(f_1,...,f_T)$, as then $y_i$ are not conditionally independent given $\phi$. Also they are not anymore exchangeable as we have the time ordering telling additional information. $M$-step ahead prediction (see @Burkner+Gabry+Vehtari:LFO-CV:2020) is more about the usual interest in predicting future and evaluating the time series model for $(f_1,...,f_T)$, but leave-one-out cross-validation is valid for assessing conditional part $p(y_i \mid f_i)$. Even if we are interested in predicting the future, $K$-fold-CV with joint log score can have better model selection performance for time series models than LFO due to smaller variance [@Cooper+etal:2023]. - LOO is valid if the focus is in the conditional observation model. LOO can often already reveal problems in the model, and as LOO is often easy to compute, it is fine to start with LOO to investigate possible model issues. -- If the prediction task is to predict for the future, then leave-future-out (LFO) can be used. Computation of LFO requires a bit more work, but often a fast approximated computation is sufficient (@Burkner+Gabry+Vehtari:LFO-CV:2020). In leave-future-out, it is possible to consider also 1-step-ahead or M-step-ahead predictions depending on the prediction task. +- If the prediction task is to predict for the future, then leave-future-out (LFO) can be used. Computation of LFO requires a bit more work, but often a fast approximated computation is sufficient [@Burkner+Gabry+Vehtari:LFO-CV:2020]. In leave-future-out, it is possible to consider also 1-step-ahead or M-step-ahead predictions depending on the prediction task. +- For model selection, $K$-fold-CV with joint log score is likely to have better performance than LFO [@Cooper+etal:2023]. See also - - [A blog post](https://statmodeling.stat.columbia.edu/2018/08/03/loo-cross-validation-approaches-valid/) - - [Model selection videos](https://avehtari.github.io/modelselection/) - Vignette [Approximate leave-future-out cross-validation for Bayesian time series models](http://mc-stan.org/loo/articles/loo2-lfo.html) # Can cross-validation be used for spatial data? {#spatial} @@ -271,8 +265,6 @@ The short answer is "Yes". This is closely related to [the question about time s See also - - [A blog post](https://statmodeling.stat.columbia.edu/2018/08/03/loo-cross-validation-approaches-valid/) - - [Model selection videos](https://avehtari.github.io/modelselection/) - Vignette [Approximate leave-future-out cross-validation for Bayesian time series models](http://mc-stan.org/loo/articles/loo2-lfo.html) @@ -282,13 +274,16 @@ Short answer is "Yes". @Vehtari+etal:PSIS-LOO:2017 state ``Instead of the log pr Vignette for `loo` package about other utility and loss functions is work in progress, but there are examples elsewhere: - - Sections 4.3 and 4.4 in [Diabetes case study](https://avehtari.github.io/modelselection/diabetes.html), which illustrate how to compute LOO classification accuracy and LOO calibration plots. + - Sections 4.3 and 4.4 in [Diabetes case study](https://users.aalto.fi/~ave/modelselection/diabetes.html), which illustrate how to compute LOO classification accuracy and LOO calibration plots. - Section 2 in [Online appendix for Bayesian $R^2$](https://avehtari.github.io/bayes_R2/bayes_R2.html) shows how to compute LOO-$R^2$, which is just 1-(data variance scaled LOO-MSE). - - projpred case studies such as [collinear](https://avehtari.github.io/modelselection/collinear.html), [diabetes](https://avehtari.github.io/modelselection/diabetes.html), [mesquite](https://avehtari.github.io/modelselection/mesquite.html), [candy](https://avehtari.github.io/modelselection/candy.html), [winequality-red](), [bodyfat](https://avehtari.github.io/modelselection/bodyfat.html), and [projpred](https://mc-stan.org/projpred/articles/quickstart.html) show LOO-RMSE or LOO classification accuracy in addition of ELPD. + - projpred case studies such as [collinear](https://users.aalto.fi/~ave/modelselection/collinear.html), [diabetes](https://users.aalto.fi/~ave/modelselection/diabetes.html), [mesquite](https://users.aalto.fi/~ave/modelselection/mesquite.html), [candy](https://users.aalto.fi/~ave/modelselection/candy.html), [winequality-red](), [bodyfat](https://users.aalto.fi/~ave/modelselection/bodyfat.html), and [projpred](https://mc-stan.org/projpred/articles/quickstart.html) show LOO-RMSE or LOO classification accuracy in addition of ELPD. -`loo` package has functions (http://mc-stan.org/loo/reference/E_loo.html) for computing the necessary expectations. We have plan for adding more on other utilities and loss functions (see a [Github issue](https://github.com/stan-dev/loo/issues/135)). +`loo` package has +- [Functions for computing the necessary LOO expectations](http://mc-stan.org/loo/reference/E_loo.html) +- [Function for computing LOO based mean absolute error (mae), root mean squared error (rmse), mean squared error (mse), classification accuracy (acc), and balanced classification accuracy (balanced_acc)](https://mc-stan.org/loo/reference/loo_predictive_metric.html) +- [Functions for LOO based (scaled) continuously ranked probability score (crps/scrps)](https://mc-stan.org/loo/reference/crps.html) -We recommend log predictive density (log score) for model comparison in general as it measures the goodness of the whole predictive distribution (see also @Vehtari+Ojanen:2012). We also recommend to use application specific utility and loss functions which can provide information whether the predictive accuracy is good enough in practice as compared to application expertise. It is possible that one model is better than others, but still not useful for practice. We are happy to get feedback on other utility and loss functions than log score, RMSE, ACC and $R^2$ that would be even more application specific. +We recommend log predictive density (log score) for model comparison in general as it measures the goodness of the whole predictive distribution including tails (see also @Vehtari+Ojanen:2012). We also recommend to use application specific utility and loss functions which can provide information whether the predictive accuracy is good enough in practice as compared to application expertise. It is possible that one model is better than others, but still not useful for practice. We are happy to get feedback on other utility and loss functions than log score, RMSE, ACC and $R^2$ that would be even more application specific. See also @@ -297,15 +292,15 @@ See also # What is the interpretation of ELPD / elpd_loo / elpd_diff? {#elpd_interpretation} -Log densities and log probabilities can be transformed to densities and probabilities which have intrinsic interpretation, although most are not well calibrated for the values as they are not used to think in densities and probabilities and even less in log densities and log probabilities. +Log densities and log probabilities can be transformed to densities and probabilities which have intrinsic interpretation, although most people are not well calibrated for the values as they are not used to think in densities and probabilities and even less in log densities and log probabilities. The log probabilities are easier. For example, Guido Biele had a problem computing `elpd_loo` with a beta-binomial model for data with 22 categories. Computed individual `elpd_loo` values for observations were around -461. For discrete model with uniform probability for 22 categories log probabilities would be $\log(1/22)\approx -3.1$, and thus there was two orders of magnitude error in log scale. With the fixed code individual `elpd_loo` values were about $-2.3>-3.1$, that is, the model was beating the uniform distribution. The log densities are more difficult as they require knowing possible scaling or transformations of the data. See more in [Can cross-validation be used to compare different observation models / response distributions / likelihoods?](#differentmodels). -Although ELPD is good for model comparison as it measures the goodness of the whole predictive distribution, the difference in ELPD is even more difficult to interpret without some practice, and thus we recommend to use also application specific utility or loss functions. See more in [Can other utility and loss functions be used than log predictive density?](#otherutility). +Although ELPD is good for model comparison as it measures the goodness of the whole predictive distribution, the difference in ELPD is even more difficult to interpret without some practice, and thus we recommend to also use application specific utility or loss functions. See more in [Can other utility and loss functions be used than log predictive density?](#otherutility). -As quick rule: If elpd difference (`elpd_diff` in `loo` package) is less than 4, the difference is small [@Sivula+etal:2020:loo_uncertainty]. If elpd difference (`elpd_diff` in loo package) is larger than 4, then compare that difference to standard error of `elpd_diff` (provided e.g. by `loo` package) [@Sivula+etal:2020:loo_uncertainty]. The value for deciding what is small or large can be based on connection to Pseudo-BMA+-weights [@Yao+etal:2018]. +As quick rule: If elpd difference (`elpd_diff` in `loo` package) is less than 4, the difference is small [@Sivula+etal:2020:loo_uncertainty,McLatchie+etal:2023]. If elpd difference (`elpd_diff` in loo package) is larger than 4, then compare that difference to standard error of `elpd_diff` (provided e.g. by `loo` package) [@Sivula+etal:2020:loo_uncertainty]. The value for deciding what is small or large can be based on connection to Pseudo-BMA+-weights [@Yao+etal:2018,McLatchie+etal:2023]. See also [How to interpret in Standard error (SE) of elpd difference (elpd_diff)?](#se_diff). # Can cross-validation be used to compare different observation models / response distributions / likelihoods? {#differentmodels} @@ -313,8 +308,8 @@ See also [How to interpret in Standard error (SE) of elpd difference (elpd_diff) Short answer is "Yes". First to make the terms more clear, $p(y \mid \theta)$ as a function of $y$ is an observation model and $p(y \mid \theta)$ as a function of $\theta$ is a likelihood. It is better to ask ``Can cross-validation be used to compare different observation models?`` - You can compare models given different discrete observation models and it’s also allowed to have different transformations of $y$ as long as the mapping is bijective (the probabilities will the stay the same). -- You can't compare densities and probabilities directly. Thus you can’t compare model given continuous and discrete observation models, unless you compute probabilities in intervals from the continuous model (also known as discretising continuous model). -- You can compare models given different continuous observation models, but you have exactly the same $y$ (loo functions in `rstanarm` and `brms` check that the hash of $y$ is the same). If y is transformed, then the Jacobian of that transformation needs to be included. There is an example of this in [mesquite case study](https://avehtari.github.io/ROS-Examples/Mesquite/mesquite.html). +- You can't compare densities and probabilities directly. Thus you can’t compare model given continuous and discrete observation models, unless you compute probabilities in intervals from the continuous model (also known as discretising the continuous model). +- You can compare models given different continuous observation models if you have exactly the same $y$ (loo functions in `rstanarm` and `brms` check that the hash of $y$ is the same). If $y$ is transformed, then the Jacobian of that transformation needs to be included. There is an example of this in [mesquite case study](https://avehtari.github.io/ROS-Examples/Mesquite/mesquite.html). - Transformations of variables are briefly discussed in BDA3 p. 21 [@BDA3] and in [Stan Reference Manual Chapter 10](https://mc-stan.org/docs/reference-manual/variable-transforms-chapter.html). @@ -322,9 +317,9 @@ in [Stan Reference Manual Chapter 10](https://mc-stan.org/docs/reference-manual/ See also [Can cross-validation be used to compare different observation models / response distributions / likelihoods?](#differentmodels). -Likelihood is a function with respect to the parameters and, discrete observation model can have continuous likelihood function and continuous observation model can have discrete likelihood function. For example Stan doesn’t allow discrete parameters unless integrated out by summing, and thus in Stan you can mix only discrete and continuous observation models which have continuous likelihood functions. +Likelihood is a function with respect to the parameters and, discrete observation model can have continuous likelihood function and continuous observation model can have discrete likelihood function. For example, Stan doesn’t allow discrete parameters unless integrated out by summing, and thus in Stan you can mix only discrete and continuous observation models which have continuous likelihood functions. -First we need to think which utility or loss functions make sense for different data types. Log score can be used for discrete and continuous. Second we need to be careful with how the continuous data is scaled, as for example in the case of log score, the scaling affects log-densities and then log-probabilities and log-densities of arbitrarily scaled data are not comparable and their contributions would have arbitrary weights in the combined expected utility or loss. +First we need to think which utility or loss functions make sense for different data types. Log score can be used for discrete and continuous. Second we need to be careful with how the continuous data is scaled, as for example in the case of log score, the scaling affects log-densities, and then log-probabilities and log-densities of arbitrarily scaled data are not comparable and their contributions would have arbitrary weights in the combined expected utility or loss. Scaling of the data doesn’t change probabilities in discrete observation model. Scaling of the data does change the probability densities in continuous observation model. People often scale continuous data before modeling, for example, to have standard deviation of 1. The same holds for other transformations, e.g. people might compare Poisson model for discrete counts to normal model for log counts, and then the results are not comparable. When the probabilities don’t change but densities change, then the relative weight of components change. So we need to be careful, either by explicitly discretizing the continuous distribution to probabilities (see "Can cross-validation be used to compare different observation models / response distributions / likelihoods?") or by keeping the scale such that densities correspond directly to sensible discretization. @@ -342,46 +337,81 @@ SE assumes that normal approximation describes well the uncertainty related to t tl;dr When the difference (`elpd_diff`) is larger than 4, the number of observations is larger than 100 and the model is not badly misspecified then normal approximation and SE are quite reliable description of the uncertainty in the difference. Differences smaller than 4 are small and then the models have very similar predictive performance and it doesn't matter if the normal approximation fails or SE is underestimated [@Sivula+etal:2020:loo_uncertainty]. -# What to do if I have many high Pareto $\hat{k}$'s? {#high_khat} +# What to do if I have high Pareto $\hat{k}$'s? {#high_khat} This is about Pareto-$\hat{k}$ (khat) diagnostic for PSIS-LOO. -The Pareto-$\hat{k}$ is a diagnostic for Pareto smoothed importance sampling (PSIS) [@Vehtari+etal:PSIS-LOO:2017], which is used to compute components of `elpd_loo`. In importance-sampling LOO (the full posterior distribution is used as the proposal distribution), the Pareto-$\hat{k}$ diagnostic estimates how far an individual leave-one-out distribution is from the full distribution. If leaving out an observation changes the posterior too much then importance sampling is not able to give reliable estimate. If $\hat{k}<0.5$, then the corresponding component of `elpd_loo` is estimated with high accuracy. If $0.5<\hat{k}<0.7$ the accuracy is lower, but still OK. If $\hat{k}>0.7$, then importance sampling is not able to provide useful estimate for that component/observation. Pareto-$\hat{k}$ is also useful as a measure of influence of an observation. Highly influential observations have high $\hat{k}$ values. Very high $\hat{k}$ values often indicate model misspecification, outliers or mistakes in data processing. See Section 6 of @Gabry+etal:2019:visualization for an example. - -If there are many high $\hat{k}$ values, We can gain additional information by looking at `p_loo` reported, e.g. by `loo` package. `p_loo` is measure of effective number of parameters (see more in [What is the interpretation of p_loo?](#p_loo). +The Pareto-$\hat{k}$ is a diagnostic for Pareto smoothed importance sampling (PSIS) [@Vehtari+etal:PSIS-LOO:2017], which is used to compute components of `elpd_loo`. In importance-sampling LOO (the full posterior distribution is used as the proposal distribution), the Pareto-$\hat{k}$ diagnostic estimates how far an individual leave-one-out distribution is from the full distribution. If leaving out an observation changes the posterior too much then importance sampling is not able to give reliable estimate. Pareto smoothing stabilizes +importance sampling and guarantees a finite variance estimate at the +cost of some bias. + +The diagnostic threshold for Pareto $\hat{k}$ depends on sample size +$S$. Sample size dependent threshold was introduced by +@Vehtari+etal:PSIS:2022, and before that fixed thresholds of 0.5 and +0.7 were recommended. For simplicity, `loo` package uses the nominal +sample size $S$ when computing the sample size specific +threshold. This provides an optimistic threshold if the effective +sample size (ESS) is less than 2200, but even then if $\mathrm{ESS}/S +> 1/2$ the difference is usually negligible. Thinning of MCMC draws +can be used to improve the ratio $\mathrm{ESS}/S. + +- If $\hat{k} < \min(1 - 1 / \log_{10}(S), 0.7)$, where $S$ is the + sample size, PSIS estimate and the corresponding Monte + Carlo standard error estimate are reliable. +- If $1 - 1 / \log_{10}(S) \leq \hat{k} < 0.7$, PSIS estimate and the + corresponding Monte Carlo standard error estimate are not + reliable, but increasing (effective) sample size $S$ above + 2200 may help (this will increase the sample size specific + threshold $(1 - 1 / \log_{10}(2200) > 0.7$ and then the bias specific + threshold 0.7 dominates). +- If $0.7 \leq \hat{k} < 1$, PSIS estimate and the corresponding Monte + Carlo standard error have large bias and are not reliable. Increasing + sample size may reduce the variability in $\hat{k}$ estimate, which + may result in a lower $\hat{k}$ estimate, too. +- If $\hat{k} \geq 1$, the target distribution is estimated to + have non-finite mean. PSIS estimate and the corresponding Monte + Carlo standard error are not well defined. Increasing sample size + may reduce the variability in $\hat{k}$ estimate, which + may result in a lower $\hat{k}$ estimate, too. + +Pareto-$\hat{k}$ is also useful as a measure of influence of an observation. Highly influential observations have high $\hat{k}$ values. Very high $\hat{k}$ values often indicate model misspecification, outliers or mistakes in data processing. See Section 6 of @Gabry+etal:2019:visualization for an example. + +If there are high $\hat{k}$ values, we can gain additional information by looking at `p_loo` reported, e.g. by `loo` package. `p_loo` is measure of effective number of parameters (see more in [What is the interpretation of p_loo?](#p_loo). If $\hat{k} > 0.7$ then we can also look at the p_loo estimate for some additional information about the problem: - If p_loo $\ll p$ (the total number of parameters in the model), then the model is likely to be misspecified. Posterior predictive checks (PPCs) are then likely to also detect the problem. Try using an overdispersed model, or add more structural information (nonlinearity, mixture model, etc.). -- If p_loo $< p$ and the number of parameters $p$ is relatively large compared to the number of observations (e.g., $p>N/5$), it is likely that the model is so flexible or the population prior so weak that it’s difficult to predict the left out observation (even for the true model). This happens, for example, in the simulated 8 schools [@Vehtari+etal:PSIS-LOO:2017], random effect models with a few observations per random effect, and Gaussian processes and spatial models with short correlation lengths. +- If p_loo $< p$ and the number of parameters $p$ is relatively large compared to the number of observations (e.g., $p>N/5$), it is likely that the model is so flexible or the population prior so weak that it’s difficult to predict the left out observation. This can happen even when using the true model, in which case there is now way to improve the model, and we just have to improve the computation. This happens, for example, in the simulated 8 schools [@Vehtari+etal:PSIS-LOO:2017], random effect models with a few observations per random effect, and Gaussian processes and spatial models with short correlation lengths. +- If p_loo $> p$, then the model is likely to be badly misspecified. If the number of parameters $p \ll N$, then PPCs are also likely to detect the problem. See for example the [Roaches case study](https://users.aalto.fi/~ave/modelselection/roaches.html). If $p$ is relatively large compared to the number of observations, say $p>N/5$ (more accurately we should count number of observations influencing each parameter as in hierarchical models some groups may have few observations and other groups many), it is possible that PPCs won't detect the problem. + +Although high $\hat{k}$ values indicate problems with the models or computation, we don't need always fix these problems. If we get high $\hat{k}$ values but the model is clearly bad (based on PPC or much much worse elpd), we can just discard that model without fixing the computation. If we get a small number of high $\hat{k}$ values in the initial part of the workflow, we may proceed without fixing the issues, but if in the final stages we are comparing models that have similar performance, there are some high $\hat{k}$ values, and we want to be minimize probability of wrong conclusions, it's best to fix the problems. There is no threshold for how many high $\hat{k}$ values would be acceptable. -- If p_loo $> p$, then the model is likely to be badly misspecified. If the number of parameters $p \ll N$, then PPCs are also likely to detect the problem. See for example the [Roaches case study](https://avehtari.github.io/modelselection/roaches.html). If $p$ is relatively large compared to the number of observations, say $p>N/5$ (more accurately we should count number of observations influencing each parameter as in hierarchical models some groups may have few observations and other groups many), it is possible that PPCs won't detect the problem. +The number of high Pareto $\hat{k}$'s can be reduced by -For information see +- moment matching by @Paananen+etal:2021:implicit, which is illustrated in vignette [Avoiding model refits in leave-one-out cross-validation with moment matching](https://mc-stan.org/loo/articles/loo2-moment-matching.html) +- mixture importance sampling by @Silva+Zanella:2023, which is illustrated in vignette [Mixture IS leave-one-out cross-validation for high-dimensional Bayesian models](https://mc-stan.org/loo/articles/loo2-mixis.html) + +For more information see - Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. doi:10.1007/s11222-016-9696-4. [Online](http://link.springer.com/article/10.1007\%2Fs11222-016-9696-4). -- Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). +- Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](http://arxiv.org/abs/1507.02646). - Video [Pareto-$\hat{k} as practical pre-asymptotic diagnostic of Monte Carlo estimates](https://www.youtube.com/watch?v=U_EbJMMVdAU&t=278s) (34min) - [Practical pre-asymptotic diagnostic of Monte Carlo estimates in Bayesian inference and machine learning](https://www.youtube.com/watch?v=uIojz7lOz9w&list=PLBqnAso5Dy7PCUJbWHO7z3bdeizDdgOhY&index=2) (50min) -Moment matching LOO can be used to reduce the number of high Pareto $k$'s faster than by refitting all problematic cases. - -- Paananen, T., Piironen, J., Buerkner, P.-C., Vehtari, A. (2020). Implicitly adaptive importance sampling. *Statistics and Computing*, 31, 16. [doi:10.1007/s11222-020-09982-2](https://doi.org/10.1007/s11222-020-09982-2). - # Can I use PSIS-LOO if I have more parameters than observations? {#more_parameters} -Yes, but you are likely to have many high Pareto $k$'s if prior is weak or if there are parameters which see the information only from one observation each (e.g. "random" effect models). See an example in [Section ``Poisson model with “random effects”`` in Roaches cross-validation demo]( https://avehtari.github.io/modelselection/roaches.html#4_poisson_model_with_%E2%80%9Crandom_effects%E2%80%9D vignette). +Yes, but you are likely to have many high Pareto $\hat{k}$'s if prior is weak or if there are parameters which see the information only from one observation each (e.g. varying intercept ("random effect") models). See an example in [Section ``Poisson model with varying intercepts'' in Roaches cross-validation demo](https://users.aalto.fi/~ave/modelselection/roaches.html#5_Poisson_model_with_varying_intercept_and_integrated_LOO). # What is the interpretation of p_loo? {#p_loo} `p_loo` is called the effective number of parameters and can be computed as the difference between `elpd_loo` and the non-cross-validated log posterior predictive density (Equations (4) and (3) in @Vehtari+etal:PSIS-LOO:2017). -It is not needed for `elpd_loo`, but has diagnostic value. +It is not needed to compute `elpd_loo`, but it is obtained easily and it has diagnostic value. It describes how much more difficult it is to predict future data than the observed data. Asymptotically under certain regularity conditions, `p_loo` can be interpreted as the effective number of parameters. In well behaving cases `p_loo` $< N$ and `p_loo` $< p$, where $p$ is the total number of parameters in the model. `p_loo` $> N$ or `p_loo` $> p$ indicates that the model has very weak predictive capability. This can happen even in case of well specified model (as demonstrated in Figure 1 in @Vehtari+etal:PSIS-LOO:2017), but may also indicate a severe model misspecification. See more in "Interpreting p_loo when Pareto-$\hat{k}$ is large" in [LOO Glossary](https://mc-stan.org/loo/reference/loo-glossary.html). # What are the limitations of the cross-validation? {#limitations} -See, for example, [Limitations of “Limitations of Bayesian Leave-one-out Cross-Validation for Model Selection”](https://doi.org/10.1007/s42113-018-0020-6) and [Between the Devil and the Deep Blue Sea: Tensions Between Scientific Judgement and Statistical Model Selection](https://doi.org/10.1007/s42113-018-0019-z). +See for some limitations, for example, [Limitations of “Limitations of Bayesian Leave-one-out Cross-Validation for Model Selection”](https://doi.org/10.1007/s42113-018-0020-6) [@Vehtari+etal:2019:limitations] and [Between the Devil and the Deep Blue Sea: Tensions Between Scientific Judgement and Statistical Model Selection](https://doi.org/10.1007/s42113-018-0019-z) [@Navarro:2019:between]. # How are LOO and WAIC related? {#LOO_WAIC} @@ -393,7 +423,7 @@ In theory, the computational method used in WAIC, which corresponds to a truncat All limitations when LOO is valid or sensible hold also for WAIC (see [When is cross-validation valid?](#valid), [Can cross-validation be used for hierarchical/multilevel models?](#hierarchical), [Can cross-validation be used for time series?](#timeseries)). Thinking in terms of LOO cross-validation, it is easier to move to other cross-validation data division schemes. -@Vehtari+etal:PSIS-LOO:2017 show that PSIS-LOO has usually smaller error in estimating ELPD than WAIC. The exception is the case when p_loo $\ll N$, as then WAIC tends to have slightly smaller error, but in that case both PSIS-LOO and WAIC have very small error and it doesn't matter which computational approximation is used. On the other hand, for flexible models WAIC fails more easily, has significant bias and is less easy to diagnose for failures. WAIC has been included in `loo` package only for comparison purposes and to make it easy to replicate the results in @Vehtari+etal:PSIS-LOO:2017. +@Vehtari+etal:PSIS-LOO:2017 show that PSIS-LOO has usually smaller error in estimating ELPD than WAIC. The exception is the case when p_loo $\ll N$, as then WAIC tends to have slightly smaller error, but in that case both PSIS-LOO and WAIC have very small error and it doesn't matter which computational approximation is used. On the other hand, for flexible models WAIC fails more easily, has significant bias and is less easy to diagnose for failures. WAIC has been included in `loo` package only for comparison purposes and to make it easy to replicate the results by @Vehtari+etal:PSIS-LOO:2017. # How are LOOIC and elpd_loo related? Why LOOIC is -2$*$elpd_loo? {#LOOIC} @@ -403,7 +433,7 @@ The historical -2 was carried on to DIC which still was using point estimates. W If you prefer minimizing losses instead of maximizing utilities, multiply by -1. -The benefit of not having 2, is that then elpd_loo and p_loo are on the same scale and comparing models and using p_loo for diagnostics is easier. +The benefit of not using multiplier 2, is that then elpd_loo and p_loo are on the same scale and comparing models and using p_loo for diagnostics is easier. # What is the relationship between AIC, DIC, WAIC and LOO-CV? {#LOO_and_IC} @@ -422,10 +452,10 @@ Assuming regular and true model, these are asymptotically (with $N \rightarrow \ # What is the relationship between LOO-CV and Bayes factor? {#LOO_and_BF} LOO-CV estimates the predictive performance given $N-1$ observations. Bayes factor can be presented as ratio of predictive performance estimates given $0$ observations. Alternatively Bayes factor can be interpreted as choosing the maximum a posterior model. -- Bayes factor can be sensible when models are well specified and there is lot of data compared to the number of parameters, so that maximum a posteriori estimate is fine and the result is not sensitive to priors -- If there is not a lot of data compared to the number of parameters, Bayes factor can be much more sensitive to prior choice than LOO-CV -- If the models are not very close to the true model, Bayes factor can be more unstable than cross-validation [@Yao+etal:2018; @Oelrich+etal:2020:overconfident] -- Computation of Bayes factor is more challenging. For example, if computed from MCMC sample, usually several orders of magnitude bigger sample sizes are needed for Bayes factor than for LOO-CV +- Bayes factor can be sensible when models are well specified and there is lot of data compared to the number of parameters, so that maximum a posteriori estimate is fine and the result is not sensitive to priors. +- If there is not a lot of data compared to the number of parameters, Bayes factor can be much more sensitive to prior choice than LOO-CV. +- If the models are not very close to the true model, Bayes factor can be more unstable than cross-validation [@Yao+etal:2018; @Oelrich+etal:2020:overconfident]. +- Computation of Bayes factor is more challenging. For example, if computed from MCMC sample, usually several orders of magnitude bigger sample sizes are needed for Bayes factor than for LOO-CV. - If the models are well specified, regular, and there is a lot of data compared to the number of parameters ($n \gg p$), then Bayes factor may have smaller variance than LOO-CV. If the models are nested, instead of Bayes factor, it is also possible to look directly at the posterior of the interesting parameters (see also 2b in [Using cross-validation for many models](#manymodels)). # What is LOO-PIT {#LOO-PIT} @@ -438,7 +468,25 @@ In case of models with a small number of parameters and a large number of observ LOO-PIT (or posterior predictive PIT) values are in finite case only close to uniform and not exactly uniform even if the model would include the true data generating distribution. With small to moderate data sets the difference can be so small that we can't see the difference, but that is why in the above we wrote ``close to uniform'' instead of ``uniform''. The difference can be illustrated with a simple normal model. Assume that the data comes from a normal distribution and consider a model $\mathrm{normal}(\mu, \sigma)$ with classic uninformative priors. The posterior predictive distribution can then be computed analytically and is a Student's $t$ distribution. PIT values from comparison of a Student's $t$ distribution to the normal distributed data are not uniformly distributed, although with increasing data size, the predictive distribution will converge towards the true data generating distribution and the PIT value distribution will converge toward uniform. Thus, in theory, in case of finite data we can see slight deviation from uniformity, but that can be assumed to be small compared to what would be observed in case of bad model misspecification. -At the moment, [`loo` package LOO-PIT functions](https://mc-stan.org/bayesplot/reference/PPC-loo.html) don't yet support PIT values for discrete target distributions, but it is on the todo list of the package maintainers. +At the moment, [`loo` package LOO-PIT functions](https://mc-stan.org/bayesplot/reference/PPC-loo.html) don't yet support PIT values for discrete target distributions, but it will be fixed in first quarter of 2024. + +# How big problem it is that cross-validation is biased? + +Unbiasedness has a special role in statistics, and too often there are dichotomous comments that something is not valid or is inferior because it's not unbiased. However, often the non-zero bias is negligible, and often by modifying the estimator we may even increase bias but reduce the variance a lot providing an overall improved performance. + +In CV the goal is to estimate the predictive performance for unobserved data given the observed data of size $n$. CV has pessimistic bias due to using less than $n$ observation to fit the models. In case of LOO-CV this bias is usually small and negligible. In case of $K$-fold-CV with a small $K$, the bias can be non-negligible, but if the effective number of parameters of the model is much less than $n$, then with $K>10$ the bias is also usually negligible compared to the variance. + +There is a bias correction approach by @Burman:1989 (see also @Fushiki:2011) that reduces CV bias, but even in the cases with non-negligible bias reduction, the variance tends to increase so much that there is no real benefit (see, e.g. @Vehtari+Lampinen:2002b). + +For time series when the task is to predict future (there are other possibilities like missing data imputation) there are specific CV methods such as leave-future-out (LFO) that have lower bias than LOO-CV or $K$-fold-CV [@Burkner+Gabry+Vehtari:LFO-CV:2020]. There are sometimes comments that LOO-CV and $K$-fold-CV would be invalid for time series. Although they tend to have a bigger bias than LFO, they are still valid and can be useful especially in model comparison where bias can cancel out. + +@Cooper+etal:2023 demonstrate how in time series model comparison variance is likely to dominate, it is more important to reduce the variance than bias, and leave-few-observations and use of joint log score is better than use of LFO. The problem with LFO is that the data sets used for fitting models are smaller increasing the variance. + +@Bengio+Grandvalet:2004 proved that there is no unbiased estimate for the variance of CV in general, which has been later used as an argument that there is no hope. Instead of dichotomizing to unbiased or biased, @Sivula+etal:2020:loo_uncertainty consider whether the variance estimates are useful and how to diagnose when the bias is likely to not be negligible (@Sivula+Magnusson+Vehtari:2023 prove also a special case where there actually exists unbiased variance estimate). + +CV tends to have high variance as the sample reuse is not making any modeling assumptions (this holds also for information criteria such as WAIC). Not making modeling assumptions is good when we don't trust our models, but if we trust we can get reduced variance in model comparison, for example, examining directly the posterior or using reference models to filter out noise in the data (see, e.g., @Piironen+etal:projpred:2018 and @Pavone+etal:2020). + +When using CV (information criteria such as WAIC) for model selection, the performance estimate for the *selected model* has additional selection induced bias. In case of small number of models this bias is usually negligible, that is, smaller than the standard deviation of the estimate or smaller than what is practically relevant. In case of negligible bias, we may choose suboptimal model, but the difference to the performance of oracle model is small. In case of a large number models the selection induced bias can be non-negligible, but this bias can be estimated using, for example, nested-CV or ordered statistics approach by @McLatchie+Vehtari:2023. # References {.unnumbered} @@ -446,7 +494,7 @@ At the moment, [`loo` package LOO-PIT functions](https://mc-stan.org/bayesplot/r # Licenses {.unnumbered} -* Text © 2020--2022, Aki Vehtari, licensed under CC-BY-NC 4.0. +* Text © 2020--2024, Aki Vehtari, licensed under CC-BY-NC 4.0. # Acknowledgements {.unnumbered} diff --git a/vignettes/online-only/faq.bib b/vignettes/online-only/faq.bib index 5be8f5f5..604c91b5 100644 --- a/vignettes/online-only/faq.bib +++ b/vignettes/online-only/faq.bib @@ -1,3 +1,23 @@ +@article{Merkle+Furr+Rabe_Hesketh:2019, + title={Bayesian Comparison of Latent Variable Models: Conditional Versus Marginal Likelihoods}, + volume={84}, + number={3}, + journal={Psychometrika}, + author={Merkle, Edgar C. and Furr, Daniel and Rabe-Hesketh, Sophia}, + year={2019}, + pages={802–829} +} + +@article{Vehtari+etal:2016:LOO_for_GLVM, + title={Bayesian leave-one-out cross-validation approximations for Gaussian latent variable models}, + author={Vehtari, Aki and Mononen, Tommi and Tolvanen, Ville and Sivula, Tuomas and Winther, Ole}, + journal={The Journal of Machine Learning Research}, + volume={17}, + number={1}, + pages={3581--3618}, + year={2016} +} + @article{Merkle+Furr+Rabe-Hesketh:2018, title={Bayesian comparison of latent variable models: {C}onditional vs marginal likelihoods}, author={Merkle, Edgar C and Furr, Daniel and Rabe-Hesketh, Sophia}, @@ -214,6 +234,14 @@ @article{Vehtari+etal:PSIS:2019 url={https://arxiv.org/abs/1507.02646v6} } +@article{Vehtari+etal:PSIS:2022, + title={Pareto smoothed importance sampling}, + author={Vehtari, Aki and Simpson, Daniel and Gelman, Andrew and Yao, Yuling and Gabry, Jonah}, + journal={arXiv preprint arXiv:1507.02646}, + year={2022}, + url={https://arxiv.org/abs/1507.02646v6} +} + @article{Piironen+etal:projpred:2018, title={Projective Inference in High-dimensional Problems: Prediction and Feature Selection}, author={Piironen, Juho and Paasiniemi, Markus and Vehtari, Aki}, @@ -243,8 +271,9 @@ @Article{Pavone+etal:2022 title={Using reference models in variable selection}, author={Pavone, Federico and Piironen, Juho and B{\"u}rkner, Paul-Christian and Vehtari, Aki}, journal={Computational Statistics}, -year={2022}, - doi={10.1007/s00180-022-01231-6} +volume = 38, +pges = {349--371}, +year={2022} } @article{Vehtari+etal:PSIS-LOO:2017, @@ -435,7 +464,6 @@ @article{Oelrich+etal:2020:overconfident year={2020} } - @article{Gabry+etal:2019:visualization, author = {Gabry, Jonah and Simpson, Daniel and Vehtari, Aki and Betancourt, Michael and Gelman, Andrew}, title = {Visualization in {Bayesian} workflow}, @@ -445,3 +473,158 @@ @article{Gabry+etal:2019:visualization pages = {389-402}, year = 2019 } + +@Article{Cooper+etal:2023, + author = {Alex Cooper and Dan Simpson and Lauren Kennedy and Catherine Forbes and Aki Vehtari}, + title = {Cross-validatory model selection for Bayesian autoregressions with exogenous regressors}, + journal = {Bayesian Analysis}, + year = {2023}, + note = {Accepted for publication. Preprint arXiv:2301.08276.} +} + +@Article{Burman:1989, + author = {Prabir Burman}, + title = {A Comparative Study of Ordinary Cross-Validation, + $v$-Fold Cross-Validation and the Repeated + Learning-Testing Methods}, + journal = {Biometrika}, + year = {1989}, + volume = {76}, + number = {3}, + pages = {503--514}, + abstract = {Concepts of $\nu$-fold cross-validation and + repeated learning-testing methods have been + introduced here. In many problems, these methods + are computationally much less expensive than + ordinary cross-validation and can be used in its + place. A comparative study of these three methods + has been carried out in detail.} +} + +@article{Bengio+Grandvalet:2004, + title={No unbiased estimator of the variance of $k$-fold cross-validation}, + author={Bengio, Yoshua and Grandvalet, Yves}, + journal={Journal of Machine Learning Research}, + volume={5}, + pages={1089--1105}, + year={2004} +} + +@article{Sivula+Magnusson+Vehtari:2023, + title={Unbiased estimator for the variance of the leave-one-out cross-validation estimator for a Bayesian normal model with fixed variance}, + author={Sivula, Tuomas and Magnusson, M{\aa}ns and Vehtari, Aki}, + journal={Communications in Statistics-Theory and Methods}, + volume={52}, + number={16}, + pages={5877--5899}, + year={2023} +} + +@article{Fushiki:2011, + title={Estimation of prediction error by using K-fold cross-validation}, + author={Fushiki, Tadayoshi}, + journal={Statistics and Computing}, + volume={21}, + pages={137--146}, + year={2011} +} + +@Article{McLatchie+Vehtari:2023, + author = {Yann McLatchie and Aki Vehtari}, + title = {Efficient estimation and correction of selection-induced bias with order statistics}, + journal = {arXiv preprint arXiv:2309.03742}, + year = {2023} +} + +@article{Ambroise+McLachlan:2002, + title = {Selection bias in gene extraction on the basis of microarray gene-expression data}, + volume = {99}, + number = {10}, + journal = {Proceedings of the National Academy of Sciences}, + author = {Ambroise, Christophe and McLachlan, Geoffrey J.}, + year = {2002}, + pages = {6562--6566} +} + +@article{Reunanen:2003, + title = {Overfitting in making comparisons between variable selection methods}, + volume = {3}, + journal = {Journal of Machine Learning Research}, + author = {Reunanen, Juho}, + year = {2003}, + pages = {1371--1382} +} + +@article{Cawley+Talbot:2010, + title = {On Over-Fitting in Model Selection and Subsequent Selection Bias in Performance Evaluation}, + volume = {11}, + journal = {Journal of Machine Learning Research}, + author = {Cawley, Gavin C. and Talbot, Nicola L. C.}, + year = {2010}, + pages = {2079--2107} +} + +@article{McLatchie+etal:2023:projpred_workflow, + title={Robust and efficient projection predictive inference}, + author={McLatchie, Yann and R{\"o}gnvaldsson, S{\"o}lvi and Weber, Frank and Vehtari, Aki}, + journal={arXiv preprint arXiv:2306.15581}, + year={2023} +} + +@article{Zhang+etal:2022:R2D2, + title = {Bayesian Regression Using a Prior on the Model Fit: The {R2-D2} Shrinkage Prior}, + volume = 117, + number = 538, + journal = {Journal of the American Statistical Association}, + author = {Zhang, Yan Dora and Naughton, Brian P. and Bondell, Howard D. and Reich, Brian J.}, + year = 2022, + pages = {862--874} + } + +@article{McLatchie+etal:2023:projpred_workflow, + title = {Robust and Efficient Projection Predictive Inference}, + author = {McLatchie, Yann and R{\"o}gnvaldsson, S{\"o}lvi and Weber, Frank and Vehtari, Aki}, + year = {2023}, + journal = {arXiv:2306.15581}, +} + +@article{Heinze+etal:bodyfat, + title={Variable selection--a review and recommendations for the practicing statistician}, + author={Heinze, Georg and Wallisch, Christine and Dunkler, Daniela}, + journal={Biometrical Journal}, + volume={60}, + number={3}, + pages={431--449}, + year={2018} +} + +@article{Kennedy+etal:2023, + title={Scoring multilevel regression and postratification based population and subpopulation estimates}, + author={Kennedy, Lauren and Vehtari, Aki and Gelman, Andrew}, + journal={arXiv preprint arXiv:2312.06334}, + year={2023} +} + +@article{Cooper+etal:2023:parallelCV, + title={Bayesian cross-validation by parallel {Markov} chain {Monte} {Carlo}}, + author={Cooper, Alex and Vehtari, Aki and Forbes, Catherine and Kennedy, Lauren and Simpson, Dan}, + journal={arXiv preprint arXiv:2310.07002}, + year={2023} +} + +@article{Silva+Zanella:2023, + title={Robust leave-one-out cross-validation for high-dimensional {Bayesian} models}, + author={Silva, Luca Alessandro and Zanella, Giacomo}, + journal={Journal of the American Statistical Association}, + pages={1--13}, + year={2023} +} + +@article{Vehtari+etal:2019:limitations, + title={Limitations of ``{Limitations} of {Bayesian} leave-one-out cross-validation for model selection''}, + author={Vehtari, Aki and Simpson, Daniel P and Yao, Yuling and Gelman, Andrew}, + journal={Computational Brain \& Behavior}, + volume={2}, + pages={22--27}, + year={2019} +} \ No newline at end of file From 448eb3e57b8eb73e1eb8b0fef04a65755191fca1 Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Thu, 1 Feb 2024 12:00:35 +0200 Subject: [PATCH 25/39] fix r_eff default and doc in tis --- R/tis.R | 18 ++++++++++-------- 1 file changed, 10 insertions(+), 8 deletions(-) diff --git a/R/tis.R b/R/tis.R index e427b805..39bf166d 100644 --- a/R/tis.R +++ b/R/tis.R @@ -13,11 +13,13 @@ #' one element per observation. The values provided should be the relative #' effective sample sizes of `1/exp(log_ratios)` (i.e., `1/ratios`). #' This is related to the relative efficiency of estimating the normalizing -#' term in self-normalizing importance sampling. See the [relative_eff()] -#' helper function for computing `r_eff`. If using `psis` with -#' draws of the `log_ratios` not obtained from MCMC then the warning -#' message thrown when not specifying `r_eff` can be disabled by -#' setting `r_eff` to `NA`. +#' term in self-normalizing importance sampling. If `r_eff` is not +#' provided then the reported (T)IS effective sample sizes and Monte Carlo +#' error estimates can be over-optimistic. If the posterior draws are (near) +#' independent then `r_eff=1` can be used. `r_eff` has to be a scalar (same +#' value is used for all observations) or a vector with length equal to the +#' number of observations. The default value is 1. See the [relative_eff()] +#' helper function for computing `r_eff`. #' #' @return The `tis()` methods return an object of class `"tis"`, #' which is a named list with the following components: @@ -88,7 +90,7 @@ tis <- function(log_ratios, ...) UseMethod("tis") #' tis.array <- function(log_ratios, ..., - r_eff = NULL, + r_eff = 1, cores = getOption("mc.cores", 1)) { importance_sampling.array(log_ratios = log_ratios, ..., r_eff = r_eff, @@ -103,7 +105,7 @@ tis.array <- tis.matrix <- function(log_ratios, ..., - r_eff = NULL, + r_eff = 1, cores = getOption("mc.cores", 1)) { importance_sampling.matrix(log_ratios, ..., @@ -117,7 +119,7 @@ tis.matrix <- #' @template vector #' tis.default <- - function(log_ratios, ..., r_eff = NULL) { + function(log_ratios, ..., r_eff = 1) { importance_sampling.default(log_ratios = log_ratios, ..., r_eff = r_eff, method = "tis") } From 52ce79ed1ad0db8b7ea3b29573027e69bd8673ba Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Thu, 1 Feb 2024 12:09:35 +0200 Subject: [PATCH 26/39] print r_eff summary as part of mcse summary --- R/print.R | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/R/print.R b/R/print.R index b739efd7..a6bede1f 100644 --- a/R/print.R +++ b/R/print.R @@ -17,7 +17,6 @@ print.loo <- function(x, digits = 1, ...) { cat("\n") print_dims(x) - print_reff_summary(x, digits) if (!("estimates" %in% names(x))) { x <- convert_old_object(x) } @@ -199,6 +198,7 @@ print_mcse_summary <- function(x, digits) { "Monte Carlo SE of elpd_loo is", paste0(.fr(mcse_val, digits), ".\n") ) + print_reff_summary(x, digits) } # print and warning helpers From bb7d3e24fefae2fc485f398961b42c71f4738340 Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Thu, 1 Feb 2024 12:10:57 +0200 Subject: [PATCH 27/39] Monte Carlo SE -> MCSE --- R/print.R | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/R/print.R b/R/print.R index a6bede1f..b770c082 100644 --- a/R/print.R +++ b/R/print.R @@ -195,7 +195,7 @@ print_reff_summary <- function(x, digits) { print_mcse_summary <- function(x, digits) { mcse_val <- mcse_loo(x) cat( - "Monte Carlo SE of elpd_loo is", + "MCSE of elpd_loo is", paste0(.fr(mcse_val, digits), ".\n") ) print_reff_summary(x, digits) From 4b6fdea2fa3d72d6c5d547968641cd82938dd8e1 Mon Sep 17 00:00:00 2001 From: jgabry Date: Thu, 1 Feb 2024 09:16:38 -0700 Subject: [PATCH 28/39] regenerate doc --- man/loo-glossary.Rd | 2 +- man/tis.Rd | 18 ++++++++++-------- 2 files changed, 11 insertions(+), 9 deletions(-) diff --git a/man/loo-glossary.Rd b/man/loo-glossary.Rd index 084e3c75..abbd1dfa 100644 --- a/man/loo-glossary.Rd +++ b/man/loo-glossary.Rd @@ -117,7 +117,7 @@ misspecification, outliers or mistakes in data processing. See Section 6 of Gabry et al. (2019) for an example. \subsection{Interpreting \code{p_loo} when Pareto \code{k} is large}{ -If \eqn{k < min(1 - 1 / log10(S), 0.7)} then we can also look at +If \eqn{k > 0.7} then we can also look at the \code{p_loo} estimate for some additional information about the problem: \itemize{ \item If \verb{p_loo << p} (the total number of parameters in the model), diff --git a/man/tis.Rd b/man/tis.Rd index a7280b0a..46cf30a8 100644 --- a/man/tis.Rd +++ b/man/tis.Rd @@ -9,11 +9,11 @@ \usage{ tis(log_ratios, ...) -\method{tis}{array}(log_ratios, ..., r_eff = NULL, cores = getOption("mc.cores", 1)) +\method{tis}{array}(log_ratios, ..., r_eff = 1, cores = getOption("mc.cores", 1)) -\method{tis}{matrix}(log_ratios, ..., r_eff = NULL, cores = getOption("mc.cores", 1)) +\method{tis}{matrix}(log_ratios, ..., r_eff = 1, cores = getOption("mc.cores", 1)) -\method{tis}{default}(log_ratios, ..., r_eff = NULL) +\method{tis}{default}(log_ratios, ..., r_eff = 1) } \arguments{ \item{log_ratios}{An array, matrix, or vector of importance ratios on the log @@ -27,11 +27,13 @@ description of how to specify the inputs for each method.} one element per observation. The values provided should be the relative effective sample sizes of \code{1/exp(log_ratios)} (i.e., \code{1/ratios}). This is related to the relative efficiency of estimating the normalizing -term in self-normalizing importance sampling. See the \code{\link[=relative_eff]{relative_eff()}} -helper function for computing \code{r_eff}. If using \code{psis} with -draws of the \code{log_ratios} not obtained from MCMC then the warning -message thrown when not specifying \code{r_eff} can be disabled by -setting \code{r_eff} to \code{NA}.} +term in self-normalizing importance sampling. If \code{r_eff} is not +provided then the reported (T)IS effective sample sizes and Monte Carlo +error estimates can be over-optimistic. If the posterior draws are (near) +independent then \code{r_eff=1} can be used. \code{r_eff} has to be a scalar (same +value is used for all observations) or a vector with length equal to the +number of observations. The default value is 1. See the \code{\link[=relative_eff]{relative_eff()}} +helper function for computing \code{r_eff}.} \item{cores}{The number of cores to use for parallelization. This defaults to the option \code{mc.cores} which can be set for an entire R session by From 90ab04a6dee443fda5d7a2a3365153b631da53f6 Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Fri, 2 Feb 2024 18:26:32 +0200 Subject: [PATCH 29/39] fix print_reff_summary to work well with old objects --- R/print.R | 35 +++++++++++++++++++++++------------ 1 file changed, 23 insertions(+), 12 deletions(-) diff --git a/R/print.R b/R/print.R index b770c082..f05834d6 100644 --- a/R/print.R +++ b/R/print.R @@ -66,6 +66,7 @@ print.psis_loo_ap <- function(x, digits = 1, plot_k = FALSE, ...) { print.loo(x, digits = digits, ...) cat("------\n") cat("Posterior approximation correction used.\n") + attr(x, 'r_eff') <- 1 print_mcse_summary(x, digits = digits) S <- dim(x)[1] k_threshold <- ps_khat_threshold(S) @@ -85,6 +86,7 @@ print.psis_loo_ap <- function(x, digits = 1, plot_k = FALSE, ...) { #' @rdname print.loo print.psis <- function(x, digits = 1, plot_k = FALSE, ...) { print_dims(x) + print_reff_summary(x, digits) print(pareto_k_table(x), digits = digits) cat(.k_help()) if (plot_k) { @@ -177,18 +179,27 @@ print_dims.psis_loo_ss <- function(x, ...) { print_reff_summary <- function(x, digits) { r_eff <- x$diagnostics$r_eff - if (all(r_eff==1)) { - cat( - "MCSE and ESS estimates assume independent draws (r_eff=1).\n" - ) - } else { - cat(paste0( - "MCSE and ESS estimates assume MCMC draws (r_eff in [", - .fr(min(r_eff), digits), - ", ", - .fr(max(r_eff), digits), - "]).\n" - )) + if (is.null(r_eff)) { + if (!is.null(x$psis_object)) { + r_eff <- attr(x$psis_object,'r_eff') + } else { + r_eff <- attr(x,'r_eff') + } + } + if (!is.null(r_eff)) { + if (all(r_eff==1)) { + cat( + "MCSE and ESS estimates assume independent draws (r_eff=1).\n" + ) + } else { + cat(paste0( + "MCSE and ESS estimates assume MCMC draws (r_eff in [", + .fr(min(r_eff), digits), + ", ", + .fr(max(r_eff), digits), + "]).\n" + )) + } } } From cfe1f742422d216413fca935c293c91997d4e165 Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Fri, 2 Feb 2024 20:58:20 +0200 Subject: [PATCH 30/39] 4/9 vignettes fixed --- R/loo_moment_matching.R | 2 +- tests/testthat/test_loo_moment_matching.R | 6 ++++ vignettes/loo2-example.Rmd | 19 ++++++------ vignettes/loo2-large-data.Rmd | 36 +++++++++-------------- vignettes/loo2-lfo.Rmd | 8 ++--- vignettes/loo2-moment-matching.Rmd | 6 ++-- 6 files changed, 37 insertions(+), 40 deletions(-) diff --git a/R/loo_moment_matching.R b/R/loo_moment_matching.R index db1340a9..7de1f06d 100644 --- a/R/loo_moment_matching.R +++ b/R/loo_moment_matching.R @@ -78,7 +78,7 @@ loo_moment_match.default <- function(x, loo, post_draws, log_lik_i, checkmate::assertFunction(log_prob_upars) checkmate::assertFunction(log_lik_i_upars) checkmate::assertNumber(max_iters) - checkmate::assertNumber(k_threshold) + checkmate::assertNumber(k_threshold, null.ok=TRUE) checkmate::assertLogical(split) checkmate::assertLogical(cov) checkmate::assertNumber(cores) diff --git a/tests/testthat/test_loo_moment_matching.R b/tests/testthat/test_loo_moment_matching.R index 7b34a3cd..16a9a426 100644 --- a/tests/testthat/test_loo_moment_matching.R +++ b/tests/testthat/test_loo_moment_matching.R @@ -147,6 +147,12 @@ test_that("loo_moment_match.default warnings work", { k_thres = 0.5, split = FALSE, cov = TRUE, cores = 1), "The accuracy of self-normalized importance sampling") + expect_warning(loo_moment_match(x, loo_manual, post_draws_test, log_lik_i_test, + unconstrain_pars_test, log_prob_upars_test, + log_lik_i_upars_test, max_iters = 30L, + split = FALSE, + cov = TRUE, cores = 1), "The accuracy of self-normalized importance sampling") + expect_no_warning(loo_moment_match(x, loo_manual, post_draws_test, log_lik_i_test, unconstrain_pars_test, log_prob_upars_test, log_lik_i_upars_test, max_iters = 30L, diff --git a/vignettes/loo2-example.Rmd b/vignettes/loo2-example.Rmd index c2b1acc3..749803a9 100644 --- a/vignettes/loo2-example.Rmd +++ b/vignettes/loo2-example.Rmd @@ -30,7 +30,7 @@ encourage readers to refer to the following papers for more details: * Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. Links: [published](https://link.springer.com/article/10.1007/s11222-016-9696-4) | [arXiv preprint](https://arxiv.org/abs/1507.04544). -* Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). +* Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). # Setup @@ -179,22 +179,20 @@ leave-one-out cross-validation marginal posterior predictive checks [Gabry et al (2018)](https://arxiv.org/abs/1709.01449). LOO-PIT values are cumulative probabilities for $y_i$ computed using the LOO marginal predictive distributions $p(y_i|y_{-i})$. For a good model, the distribution of LOO-PIT values should be -uniform. In the following plot the distribution (smoothed density estimate) of -the LOO-PIT values for our model (thick curve) is compared to many -independently generated samples (each the same size as our dataset) from the -standard uniform distribution (thin curves). +uniform. In the following QQ-plot the LOO-PIT values for our model (y-axi) is +compared to standard uniform distribution (x-axis). ```{r ppc_loo_pit_overlay} yrep <- posterior_predict(fit1) -ppc_loo_pit_overlay( +ppc_loo_pit_qq( y = roaches$y, yrep = yrep, lw = weights(loo1$psis_object) ) ``` -The excessive number of values close to 0 indicates that the model is +The excessive number of LOO-PIT values close to 0 indicates that the model is under-dispersed compared to the data, and we should consider a model that allows for greater dispersion. @@ -219,7 +217,8 @@ print(loo2) plot(loo2, label_points = TRUE) ``` -Using the `label_points` argument will label any $k$ values larger than 0.7 with +Using the `label_points` argument will label any $k$ values larger than the +diagnostic threshold with the index of the corresponding data point. These high values are often the result of model misspecification and frequently correspond to data points that would be considered ``outliers'' in the data and surprising according to the @@ -253,7 +252,7 @@ still some degree of model misspecification, but this is much better than the For further model checking we again examine the LOO-PIT values. ```{r ppc_loo_pit_overlay-negbin} yrep <- posterior_predict(fit2) -ppc_loo_pit_overlay(roaches$y, yrep, lw = weights(loo2$psis_object)) +ppc_loo_pit_qq(roaches$y, yrep, lw = weights(loo2$psis_object)) ``` The plot for the negative binomial model looks better than the Poisson plot, but @@ -272,7 +271,7 @@ loo_compare(loo1, loo2) The difference in ELPD is much larger than several times the estimated standard error of the difference again indicating that the negative-binomial model is -expected to have better predictive performance than the Poisson model. However, +xpected to have better predictive performance than the Poisson model. However, according to the LOO-PIT checks there is still some misspecification, and a reasonable guess is that a hurdle or zero-inflated model would be an improvement (we leave that for another case study). diff --git a/vignettes/loo2-large-data.Rmd b/vignettes/loo2-large-data.Rmd index 3d96e202..82f109f3 100644 --- a/vignettes/loo2-large-data.Rmd +++ b/vignettes/loo2-large-data.Rmd @@ -35,7 +35,7 @@ Proceedings of the 23rd International Conference on Artificial Intelligence and * Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. Links: [published](https://link.springer.com/article/10.1007/s11222-016-9696-4) | [arXiv preprint](https://arxiv.org/abs/1507.04544). -* Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544). +* Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544). which provide important background for understanding the methods implemented in the package. @@ -195,8 +195,9 @@ p_loo 3.1 0.1 0.4 looic 3936.9 31.2 0.6 ------ Monte Carlo SE of elpd_loo is 0.0. +MCSE and ESS estimates assume MCMC draws (r_eff in [0.9, 1.0]). -All Pareto k estimates are good (k < 0.5). +All Pareto k estimates are good (k < 0.7). See help('pareto-k-diagnostic') for details. ``` @@ -246,8 +247,9 @@ p_loo 3.2 0.1 0.4 looic 3936.7 31.2 0.5 ------ Monte Carlo SE of elpd_loo is 0.0. +MCSE and ESS estimates assume MCMC draws (r_eff in [0.9, 1.0]). -All Pareto k estimates are good (k < 0.5). +All Pareto k estimates are good (k < 0.7). See help('pareto-k-diagnostic') for details. ``` @@ -290,8 +292,9 @@ p_loo 3.5 0.2 0.5 looic 3937.9 30.7 1.1 ------ Monte Carlo SE of elpd_loo is 0.0. +MCSE and ESS estimates assume MCMC draws (r_eff in [0.9, 1.0]). -All Pareto k estimates are good (k < 0.5). +All Pareto k estimates are good (k < 0.7). See help('pareto-k-diagnostic') for details. ``` @@ -343,15 +346,9 @@ looic 3936.8 31.2 ------ Posterior approximation correction used. Monte Carlo SE of elpd_loo is 0.0. +MCSE and ESS estimates assume independent draws (r_eff=1). -Pareto k diagnostic values: - Count Pct. Min. n_eff -(-Inf, 0.5] (good) 2989 99.0% 1827 - (0.5, 0.7] (ok) 31 1.0% 1996 - (0.7, 1] (bad) 0 0.0% - (1, Inf) (very bad) 0 0.0% - -All Pareto k estimates are ok (k < 0.7). +All Pareto k estimates are good (k < 0.7). See help('pareto-k-diagnostic') for details. ``` @@ -386,15 +383,9 @@ looic 3936.4 31.1 0.8 ------ Posterior approximation correction used. Monte Carlo SE of elpd_loo is 0.0. +MCSE and ESS estimates assume independent draws (r_eff=1). -Pareto k diagnostic values: - Count Pct. Min. n_eff -(-Inf, 0.5] (good) 97 97.0% 1971 - (0.5, 0.7] (ok) 3 3.0% 1997 - (0.7, 1] (bad) 0 0.0% - (1, Inf) (very bad) 0 0.0% - -All Pareto k estimates are ok (k < 0.7). +All Pareto k estimates are good (k < 0.7). See help('pareto-k-diagnostic') for details. ``` @@ -471,8 +462,9 @@ p_loo 2.6 0.1 0.3 looic 3903.9 32.4 0.4 ------ Monte Carlo SE of elpd_loo is 0.0. +MCSE and ESS estimates assume MCMC draws (r_eff in [1.0, 1.1]). -All Pareto k estimates are good (k < 0.5). +All Pareto k estimates are good (k < 0.7). See help('pareto-k-diagnostic') for details. ``` @@ -616,6 +608,6 @@ Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. [online](https://link.springer.com/article/10.1007/s11222-016-9696-4), [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544). -Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto +Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). diff --git a/vignettes/loo2-lfo.Rmd b/vignettes/loo2-lfo.Rmd index 905558e7..09c0e50a 100644 --- a/vignettes/loo2-lfo.Rmd +++ b/vignettes/loo2-lfo.Rmd @@ -54,7 +54,7 @@ leave-one-out cross-validation (LOO-CV). For a data set with $N$ observations, we refit the model $N$ times, each time leaving out one of the $N$ observations and assessing how well the model predicts the left-out observation. LOO-CV is very expensive computationally in most realistic settings, but the Pareto -smoothed importance sampling (PSIS, Vehtari et al, 2017, 2019) algorithm provided by +smoothed importance sampling (PSIS, Vehtari et al, 2017, 2022) algorithm provided by the *loo* package allows for approximating exact LOO-CV with PSIS-LOO-CV. PSIS-LOO-CV requires only a single fit of the full model and comes with diagnostics for assessing the validity of the approximation. @@ -179,7 +179,7 @@ variability of the importance ratios $r_i^{(s)}$ will become too large and importance sampling will fail. We will refer to this particular value of $i$ as $i^\star_1$. To identify the value of $i^\star_1$, we check for which value of $i$ does the estimated shape parameter $k$ of the generalized Pareto -distribution first cross a certain threshold $\tau$ (Vehtari et al, 2019). Only +distribution first cross a certain threshold $\tau$ (Vehtari et al, 2022). Only then do we refit the model using the observations up to $i^\star_1$ and restart the process from there by setting $\theta^{(s)} = \theta^{(s)}_{1:i^\star_1}$ and $i^\star = i^\star_1$ until the next refit. @@ -188,7 +188,7 @@ In some cases we may only need to refit once and in other cases we will find a value $i^\star_2$ that requires a second refitting, maybe an $i^\star_3$ that requires a third refitting, and so on. We refit as many times as is required (only when $k > \tau$) until we arrive at observation $i = N - M$. -For LOO, we recommend to use a threshold of $\tau = 0.7$ (Vehtari et al, 2017, 2019) +For LOO, assuming posterior sample size is 4000 or larger, we recommend to use a threshold of $\tau = 0.7$ (Vehtari et al, 2017, 2022) and it turns out this is a reasonable threshold for LFO as well (Bürkner et al. 2020). ## Autoregressive models @@ -640,7 +640,7 @@ Bürkner P. C., Gabry J., & Vehtari A. (2020). Approximate leave-future-out cros Vehtari A., Gelman A., & Gabry J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. *Statistics and Computing*, 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. [Online](https://link.springer.com/article/10.1007/s11222-016-9696-4). [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544). -Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). +Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646).
diff --git a/vignettes/loo2-moment-matching.Rmd b/vignettes/loo2-moment-matching.Rmd index 2e75ec55..88a3342c 100644 --- a/vignettes/loo2-moment-matching.Rmd +++ b/vignettes/loo2-moment-matching.Rmd @@ -43,7 +43,7 @@ papers * Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. Links: [published](https://link.springer.com/article/10.1007/s11222-016-9696-4) | [arXiv preprint](https://arxiv.org/abs/1507.04544). -* Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). +* Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). @@ -168,7 +168,7 @@ __rstan__. It only requires setting the argument `moment_match` to `TRUE` in the `loo()` function. Optionally, you can also set the argument `k_threshold` which determines the Pareto $k$ threshold, above which moment matching is used. By default, it operates on all observations whose Pareto $k$ value is larger than -0.7. +the sample size ($S$) specific threshold $\min(1 - 1 / \log_{10}(S), 0.7)$ (which is $0.7$ for $S>2200$). ```{r loo_moment_match} # available in rstan >= 2.21 @@ -319,4 +319,4 @@ Implicitly adaptive importance sampling. _Statistics and Computing_, 31, 16. Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. Links: [published](https://link.springer.com/article/10.1007/s11222-016-9696-4) | [arXiv preprint](https://arxiv.org/abs/1507.04544). -Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). +Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). From 122aa102f1e35afb02f2d56605563eb656d9617c Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Fri, 2 Feb 2024 21:35:11 +0200 Subject: [PATCH 31/39] fixedd loo2-mixis vignette --- vignettes/loo2-mixis.Rmd | 40 ++++++++++++++++++++-------------------- 1 file changed, 20 insertions(+), 20 deletions(-) diff --git a/vignettes/loo2-mixis.Rmd b/vignettes/loo2-mixis.Rmd index 47eca538..d2ff7297 100644 --- a/vignettes/loo2-mixis.Rmd +++ b/vignettes/loo2-mixis.Rmd @@ -20,7 +20,7 @@ params: # Introduction -This vignette shows how to perform Bayesian leave-one-out cross-validation (LOO-CV) using the mixture estimators proposed in the paper [Silva and Zanella (2022)](https://arxiv.org/abs/2209.09190). These estimators have shown to be useful in presence of outliers but also, and especially, in high-dimensional settings where the model features many parameters. In these contexts it can happen that a large portion of observations lead to high values of Pareto-$k$ diagnostics and potential instability of PSIS-LOO estimators. +This vignette shows how to perform Bayesian leave-one-out cross-validation (LOO-CV) using the mixture estimators proposed in the paper [Silva and Zanella (2022)](https://arxiv.org/abs/2209.09190). These estimators have shown to be useful in presence of outliers but also, and especially, in high-dimensional settings where the model features many parameters. In these contexts it can happen that a large portion of observations lead to high values of Pareto-$k$ diagnostics and potential instability of PSIS-LOO estimators. For this illustration we consider a high-dimensional Bayesian Logistic regression model applied to the _Voice_ dataset. @@ -44,12 +44,12 @@ This is the Stan code for a logistic regression model with regularized horseshoe # int y[N]; stancode_horseshoe <- " data { - int n; - int p; - array[N] int y; - matrix [n,p] X; + int N; + int P; + array[N] int y; + matrix [N,P] X; real scale_global; - int mixis; + int mixis; } transformed data { real nu_global=1; // degrees of freedom for the half-t priors for tau @@ -59,29 +59,29 @@ transformed data { real slab_df=100; // for the regularized horseshoe } parameters { - vector[p] z; // for non-centered parameterization + vector[P] z; // for non-centered parameterization real tau; // global shrinkage parameter - vector [p] lambda; // local shrinkage parameter + vector [P] lambda; // local shrinkage parameter real caux; } transformed parameters { - vector[p] beta; + vector[P] beta; { - vector[p] lambda_tilde; // 'truncated' local shrinkage parameter + vector[P] lambda_tilde; // 'truncated' local shrinkage parameter real c = slab_scale * sqrt(caux); // slab scale lambda_tilde = sqrt( c^2 * square(lambda) ./ (c^2 + tau^2*square(lambda))); beta = z .* lambda_tilde*tau; } } model { - vector[n] means=X*beta; - vector[n] log_lik; + vector[N] means=X*beta; + vector[N] log_lik; target += std_normal_lpdf(z); target += student_t_lpdf(lambda | nu_local, 0, 1); target += student_t_lpdf(tau | nu_global, 0, scale_global); target += inv_gamma_lpdf(caux | 0.5*slab_df, 0.5*slab_df); - for (index in 1:n) { - log_lik[index]= bernoulli_logit_lpmf(y[index] | means[index]); + for (n in 1:N) { + log_lik[n]= bernoulli_logit_lpmf(y[n] | means[n]); } target += sum(log_lik); if (mixis) { @@ -89,10 +89,10 @@ model { } } generated quantities { - vector[n] means=X*beta; - vector[n] log_lik; - for (index in 1:n) { - log_lik[index] = bernoulli_logit_lpmf(y[index] | means[index]); + vector[N] means=X*beta; + vector[N] log_lik; + for (n in 1:N) { + log_lik[n] = bernoulli_logit_lpmf(y[n] | means[n]); } } " @@ -105,11 +105,11 @@ The _LSVT Voice Rehabilitation Data Set_ (see [link](https://archive.ics.uci.edu data(voice) y <- voice$y X <- voice[2:length(voice)] -n <- dim(X)[1] +p <- dim(X)[1] p <- dim(X)[2] p0 <- 10 scale_global <- 2*p0/(p-p0)/sqrt(n-1) -standata <- list(n = n, p = p, X = as.matrix(X), y = c(y), scale_global = scale_global, mixis = 0) +standata <- list(N = n, P = p, X = as.matrix(X), y = c(y), scale_global = scale_global, mixis = 0) ``` Note that in our prior specification we divide the prior variance by the number of covariates $p$. This is often done in high-dimensional contexts to have a prior variance for the linear predictors $X\beta$ that remains bounded as $p$ increases. From df325d6d63f755523628fa528d0aa96936170cd4 Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Fri, 2 Feb 2024 21:50:19 +0200 Subject: [PATCH 32/39] fixed loo2-non-factorized vignette --- vignettes/loo2-non-factorized.Rmd | 9 ++++----- 1 file changed, 4 insertions(+), 5 deletions(-) diff --git a/vignettes/loo2-non-factorized.Rmd b/vignettes/loo2-non-factorized.Rmd index fb79ea03..78c6b15d 100644 --- a/vignettes/loo2-non-factorized.Rmd +++ b/vignettes/loo2-non-factorized.Rmd @@ -183,7 +183,7 @@ referred to as PSIS-LOO (Vehtari et al, 2017). In order to validate the approximate LOO procedure, and also in order to allow exact computations to be made for a small number of leave-one-out folds for -which the Pareto $k$ diagnostic (Vehtari et al, 2019) indicates an unstable +which the Pareto $k$ diagnostic (Vehtari et al, 2022) indicates an unstable approximation, we need to consider how we might to do _exact_ leave-one-out CV for a non-factorized model. In the case of a Gaussian process that has the marginalization property, we could just drop the one row and column of $C$ @@ -416,9 +416,8 @@ psis_result <- psis(log_ratios) ``` The quality of the PSIS-LOO approximation can be investigated graphically by -plotting the Pareto-k estimate for each observation. Ideally, they should not -exceed $0.5$, but in practice the algorithm turns out to be robust up to values -of $0.7$ (Vehtari et al, 2017, 2019). In the plot below, we see that the fourth +plotting the Pareto-k estimate for each observation. The approximation is robust up to values +of $0.7$ (Vehtari et al, 2017, 2022). In the plot below, we see that the fourth observation is problematic and so may reduce the accuracy of the LOO-CV approximation. @@ -717,4 +716,4 @@ Vehtari A., Mononen T., Tolvanen V., Sivula T., & Winther O. (2016). Bayesian le Vehtari A., Gelman A., & Gabry J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. *Statistics and Computing*, 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. [Online](https://link.springer.com/article/10.1007/s11222-016-9696-4). [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544). -Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). +Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). From 43632a8d329110fabcce311513310c61512f52c8 Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Fri, 2 Feb 2024 21:55:56 +0200 Subject: [PATCH 33/39] fixed loo2-weights vignette --- vignettes/loo2-weights.Rmd | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/vignettes/loo2-weights.Rmd b/vignettes/loo2-weights.Rmd index e3987c12..f07c70c2 100644 --- a/vignettes/loo2-weights.Rmd +++ b/vignettes/loo2-weights.Rmd @@ -366,7 +366,7 @@ Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. [online](https://link.springer.com/article/10.1007/s11222-016-9696-4), [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544). -Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). +Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). Yao, Y., Vehtari, A., Simpson, D., and Gelman, A. (2018). Using stacking to average Bayesian predictive distributions. In Bayesian From 8cd84b0021292c6e77160a8fc4b14026521f4316 Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Fri, 2 Feb 2024 22:08:56 +0200 Subject: [PATCH 34/39] fixed loo2-with-rstan vignette --- vignettes/loo2-with-rstan.Rmd | 10 ++++++---- 1 file changed, 6 insertions(+), 4 deletions(-) diff --git a/vignettes/loo2-with-rstan.Rmd b/vignettes/loo2-with-rstan.Rmd index 3802afdc..8f3d63c4 100644 --- a/vignettes/loo2-with-rstan.Rmd +++ b/vignettes/loo2-with-rstan.Rmd @@ -29,7 +29,7 @@ Some sections from this vignette are excerpted from our papers * Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. _Statistics and Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. Links: [published](https://link.springer.com/article/10.1007/s11222-016-9696-4) | [arXiv preprint](https://arxiv.org/abs/1507.04544). -* Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.02646). +* Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.02646). which provide important background for understanding the methods implemented in the package. @@ -146,8 +146,9 @@ p_loo 3.2 0.1 looic 3937.0 31.2 ------ Monte Carlo SE of elpd_loo is 0.0. +MCSE and ESS estimates assume MCMC draws (r_eff in [0.5, 1.3]). -All Pareto k estimates are good (k < 0.5). +All Pareto k estimates are good (k < 0.7). See help('pareto-k-diagnostic') for details. ``` @@ -187,8 +188,9 @@ p_loo 3.1 0.1 looic 3904.6 32.4 ------ Monte Carlo SE of elpd_loo is 0.0. +MCSE and ESS estimates assume MCMC draws (r_eff in [0.4, 1.2]). -All Pareto k estimates are good (k < 0.5). +All Pareto k estimates are good (k < 0.7). See help('pareto-k-diagnostic') for details. ``` @@ -232,6 +234,6 @@ Computing_. 27(5), 1413--1432. \doi:10.1007/s11222-016-9696-4. [online](https://link.springer.com/article/10.1007/s11222-016-9696-4), [arXiv preprint arXiv:1507.04544](https://arxiv.org/abs/1507.04544). -Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto +Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2022). Pareto smoothed importance sampling. [arXiv preprint arXiv:1507.02646](https://arxiv.org/abs/1507.02646). From 133ea10252d31262630a10006eaeee847e138229 Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Fri, 2 Feb 2024 22:10:30 +0200 Subject: [PATCH 35/39] fixed loo2-lfo vignette --- vignettes/loo2-lfo.Rmd | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/vignettes/loo2-lfo.Rmd b/vignettes/loo2-lfo.Rmd index 09c0e50a..2adb4767 100644 --- a/vignettes/loo2-lfo.Rmd +++ b/vignettes/loo2-lfo.Rmd @@ -220,8 +220,9 @@ interface to Stan to generate a Stan program and fit the model, and also the **bayesplot** and **ggplot2** packages for plotting. ```{r pkgs, cache=FALSE} -library("loo") library("brms") +library("loo") +devtools::load_all('~/proj/loo') library("bayesplot") library("ggplot2") color_scheme_set("brightblue") From 1a818f352820f555cc1451f71ac954cbdd9d2c38 Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Mon, 5 Feb 2024 11:48:51 +0200 Subject: [PATCH 36/39] Remove forgotten testing line in vignettes/loo2-lfo.Rmd Co-authored-by: n-kall <33577035+n-kall@users.noreply.github.com> --- vignettes/loo2-lfo.Rmd | 1 - 1 file changed, 1 deletion(-) diff --git a/vignettes/loo2-lfo.Rmd b/vignettes/loo2-lfo.Rmd index 2adb4767..4751bda4 100644 --- a/vignettes/loo2-lfo.Rmd +++ b/vignettes/loo2-lfo.Rmd @@ -222,7 +222,6 @@ interface to Stan to generate a Stan program and fit the model, and also the ```{r pkgs, cache=FALSE} library("brms") library("loo") -devtools::load_all('~/proj/loo') library("bayesplot") library("ggplot2") color_scheme_set("brightblue") From 8d73b37fa66efc851ec27d352b1d6bfb2b6a353e Mon Sep 17 00:00:00 2001 From: Aki Vehtari Date: Wed, 7 Feb 2024 20:19:39 +0200 Subject: [PATCH 37/39] typo fix --- vignettes/loo2-mixis.Rmd | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/vignettes/loo2-mixis.Rmd b/vignettes/loo2-mixis.Rmd index d2ff7297..4f417c02 100644 --- a/vignettes/loo2-mixis.Rmd +++ b/vignettes/loo2-mixis.Rmd @@ -105,7 +105,7 @@ The _LSVT Voice Rehabilitation Data Set_ (see [link](https://archive.ics.uci.edu data(voice) y <- voice$y X <- voice[2:length(voice)] -p <- dim(X)[1] +n <- dim(X)[1] p <- dim(X)[2] p0 <- 10 scale_global <- 2*p0/(p-p0)/sqrt(n-1) From 5f7864861fbbeb792a665d37b28e0ce9fdd39e8c Mon Sep 17 00:00:00 2001 From: jgabry Date: Wed, 7 Feb 2024 13:31:32 -0700 Subject: [PATCH 38/39] Add Aki's news items to NEWS.md with a few minor edits --- NEWS.md | 40 +++++++++++++++++++++++++++++++++++----- 1 file changed, 35 insertions(+), 5 deletions(-) diff --git a/NEWS.md b/NEWS.md index 16d8a16b..fa2793b1 100644 --- a/NEWS.md +++ b/NEWS.md @@ -1,13 +1,43 @@ # loo 2.6.0.9000 -### New features +### Major changes + +* Use of new sample size specific diagnostic threshold for Pareto `k`. The pre-2022 version +of the [PSIS paper](https://arxiv.org/abs/1507.02646) recommended diagnostic +thresholds of +`k < 0.5 "good"`, `0.5 <= k < 0.7 "ok"`, +`0.7 <= k < 1 "bad"`, `k>=1 "very bad"`. +The 2022 revision of the PSIS paper now recommends +`k < min(1 - 1/log10(S), 0.7) "good"`, `min(1 - 1/log10(S), 0.7) <= k < 1 "bad"`, +`k > 1 "very bad"`, where `S` is the sample size. +There is now one fewer diagnostic threshold (`"ok"` has been removed), and the +most important threshold now depends on the sample size `S`. With sample sizes +`100`, `320`, `1000`, `2200`, `10000` the sample size specific part +`1 - 1/log10(S)` corresponds to thresholds of `0.5`, `0.6`, `0.67`, `0.7`, `0.75`. +Even if the sample size grows, the bias in the PSIS estimate dominates if +`0.7 <= k < 1`, and thus the diagnostic threshold for good is capped at +`0.7` (if `k > 1`, the mean does not exist and bias is not a valid measure). +The new recommended thresholds are based on more careful bias-variance analysis +of PSIS based on truncated Pareto sums theory. For those who use the Stan +default 4000 posterior draws, the `0.7` threshold will be roughly the same, but +there will be fewer warnings as there will be no diagnostic message for `0.5 <= +k < 0.7`. Those who use smaller sample sizes may see diagnostic messages with a +threshold less than `0.7`, and they can simply increase the sample size to about +`2200` to get the threshold to `0.7`. + +* There are no more warnings if the `r_eff` argument is not provided, and the +default is now `r_eff = 1`. The summary print output showing MCSE and ESS now +shows diagnostic information on the range of `r_eff`. The change was made to +reduce unnecessary warnings. The use of `r_eff` does not change the expected +value of `elpd_loo`, `p_loo`, and Pareto `k`, and is needed only to estimate +MCSE and ESS. Thus it is better to show the diagnostic information about `r_eff` +only when MCSE and ESS values are shown. + +### Other changes * `E_loo` now allows `type="sd"`. - - -### Bug fixes - * Fix bug in `E_loo` when `type=variance`. + # loo 2.6.0 From 372cb9297d06d23c4a64f148b5b56cc1d7ab5239 Mon Sep 17 00:00:00 2001 From: jgabry Date: Wed, 7 Feb 2024 13:47:03 -0700 Subject: [PATCH 39/39] diagnostics.R: a few minor doc edits --- R/diagnostics.R | 26 +++++++++++++------------- man/pareto-k-diagnostic.Rd | 20 ++++++++++---------- 2 files changed, 23 insertions(+), 23 deletions(-) diff --git a/R/diagnostics.R b/R/diagnostics.R index b6c9d6a8..33dd66f1 100644 --- a/R/diagnostics.R +++ b/R/diagnostics.R @@ -17,13 +17,13 @@ #' See **Details** for the motivation behind these defaults. #' #' @details -#' +#' #' The reliability and approximate convergence rate of the PSIS-based #' estimates can be assessed using the estimates for the shape #' parameter \eqn{k} of the generalized Pareto distribution. The #' diagnostic threshold for Pareto \eqn{k} depends on sample size #' \eqn{S} (sample size dependent threshold was introduced by Vehtari -#' et al., 2022, and before that fixed thresholds of 0.5 and 0.7 were +#' et al. (2022), and before that fixed thresholds of 0.5 and 0.7 were #' recommended). For simplicity, `loo` package uses the nominal sample #' size \eqn{S} when computing the sample size specific #' threshold. This provides an optimistic threshold if the effective @@ -35,7 +35,7 @@ #' sample size, the PSIS estimate and the corresponding Monte Carlo #' standard error estimate are reliable. #' -#' * If \eqn{1 - 1 / log10(S) <= k < 0.7}, PSIS estimate and the +#' * If \eqn{1 - 1 / log10(S) <= k < 0.7}, the PSIS estimate and the #' corresponding Monte Carlo standard error estimate are not #' reliable, but increasing the (effective) sample size \eqn{S} above #' 2200 may help (this will increase the sample size specific @@ -44,19 +44,19 @@ #' #' * If \eqn{0.7 <= k < 1}, the PSIS estimate and the corresponding Monte #' Carlo standard error have large bias and are not reliable. Increasing -#' the sample size may reduce the uncertainty in the \eqn{k} estimate. +#' the sample size may reduce the uncertainty in the \eqn{k} estimate. #' -#' * If \eqn{0.7 <= k < 1}, PSIS estimate and the corresponding Monte +#' * If \eqn{0.7 <= k < 1}, the PSIS estimate and the corresponding Monte #' Carlo standard error have large bias and are not reliable. Increasing -#' sample size may reduce the variability in \eqn{k} estimate, which +#' the sample size may reduce the variability in \eqn{k} estimate, which #' may result in lower \eqn{k} estimate, too. #' #' * If \eqn{k \geq 1}{k >= 1}, the target distribution is estimated to -#' have non-finite mean. The PSIS estimate and the corresponding Monte +#' have a non-finite mean. The PSIS estimate and the corresponding Monte #' Carlo standard error are not well defined. Increasing the sample size -#' may reduce the variability in \eqn{k} estimate, which -#' may result in lower \eqn{k} estimate, too. -#' +#' may reduce the variability in the \eqn{k} estimate, which +#' may also result in a lower \eqn{k} estimate. +#' #' \subsection{What if the estimated tail shape parameter \eqn{k} #' exceeds the diagnostic threshold?}{ Importance sampling is likely to #' work less well if the marginal posterior \eqn{p(\theta^s | y)} and @@ -67,7 +67,7 @@ #' warned. (Note: If \eqn{k} is greater than the diagnostic threshold #' then WAIC is also likely to fail, but WAIC lacks as accurate #' diagnostic.) When using PSIS in the context of approximate LOO-CV, -#' we recommend one of the following actions when \eqn{k > 0.7}: +#' we recommend one of the following actions: #' #' * With some additional computations, it is possible to transform #' the MCMC draws from the posterior distribution to obtain more @@ -101,8 +101,8 @@ #' sample size for importance sampling, which are more accurate for PSIS than #' for IS and TIS (see Vehtari et al (2022) for details). However, the PSIS #' effective sample size estimate will be -#' **over-optimistic when the estimate of \eqn{k} is greater than -#' \eqn{min(1-1/log10(S), 0.7)}**, where \eqn{S} is the sample size. +#' **over-optimistic when the estimate of \eqn{k} is greater than** +#' \eqn{min(1-1/log10(S), 0.7)}, where \eqn{S} is the sample size. #' } #' #' @seealso diff --git a/man/pareto-k-diagnostic.Rd b/man/pareto-k-diagnostic.Rd index 998f7f4b..4ae28b25 100644 --- a/man/pareto-k-diagnostic.Rd +++ b/man/pareto-k-diagnostic.Rd @@ -105,7 +105,7 @@ estimates can be assessed using the estimates for the shape parameter \eqn{k} of the generalized Pareto distribution. The diagnostic threshold for Pareto \eqn{k} depends on sample size \eqn{S} (sample size dependent threshold was introduced by Vehtari -et al., 2022, and before that fixed thresholds of 0.5 and 0.7 were +et al. (2022), and before that fixed thresholds of 0.5 and 0.7 were recommended). For simplicity, \code{loo} package uses the nominal sample size \eqn{S} when computing the sample size specific threshold. This provides an optimistic threshold if the effective @@ -116,7 +116,7 @@ improve the ratio ESS/S. \item If \eqn{k < min(1 - 1 / log10(S), 0.7)}, where \eqn{S} is the sample size, the PSIS estimate and the corresponding Monte Carlo standard error estimate are reliable. -\item If \eqn{1 - 1 / log10(S) <= k < 0.7}, PSIS estimate and the +\item If \eqn{1 - 1 / log10(S) <= k < 0.7}, the PSIS estimate and the corresponding Monte Carlo standard error estimate are not reliable, but increasing the (effective) sample size \eqn{S} above 2200 may help (this will increase the sample size specific @@ -125,15 +125,15 @@ threshold 0.7 dominates). \item If \eqn{0.7 <= k < 1}, the PSIS estimate and the corresponding Monte Carlo standard error have large bias and are not reliable. Increasing the sample size may reduce the uncertainty in the \eqn{k} estimate. -\item If \eqn{0.7 <= k < 1}, PSIS estimate and the corresponding Monte +\item If \eqn{0.7 <= k < 1}, the PSIS estimate and the corresponding Monte Carlo standard error have large bias and are not reliable. Increasing -sample size may reduce the variability in \eqn{k} estimate, which +the sample size may reduce the variability in \eqn{k} estimate, which may result in lower \eqn{k} estimate, too. \item If \eqn{k \geq 1}{k >= 1}, the target distribution is estimated to -have non-finite mean. The PSIS estimate and the corresponding Monte +have a non-finite mean. The PSIS estimate and the corresponding Monte Carlo standard error are not well defined. Increasing the sample size -may reduce the variability in \eqn{k} estimate, which -may result in lower \eqn{k} estimate, too. +may reduce the variability in the \eqn{k} estimate, which +may also result in a lower \eqn{k} estimate. } \subsection{What if the estimated tail shape parameter \eqn{k} @@ -146,7 +146,7 @@ influential observations. If the estimated tail shape parameter warned. (Note: If \eqn{k} is greater than the diagnostic threshold then WAIC is also likely to fail, but WAIC lacks as accurate diagnostic.) When using PSIS in the context of approximate LOO-CV, -we recommend one of the following actions when \eqn{k > 0.7}: +we recommend one of the following actions: \itemize{ \item With some additional computations, it is possible to transform the MCMC draws from the posterior distribution to obtain more @@ -179,8 +179,8 @@ package also computes estimates for the Monte Carlo error and the effective sample size for importance sampling, which are more accurate for PSIS than for IS and TIS (see Vehtari et al (2022) for details). However, the PSIS effective sample size estimate will be -\strong{over-optimistic when the estimate of \eqn{k} is greater than -\eqn{min(1-1/log10(S), 0.7)}}, where \eqn{S} is the sample size. +\strong{over-optimistic when the estimate of \eqn{k} is greater than} +\eqn{min(1-1/log10(S), 0.7)}, where \eqn{S} is the sample size. } } \references{

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zQ}r`fe!zy6Y3u9j%0NtbSldL{8^qR}7u%yXhtIE5t@#`XV$Rz+nE{0$W-PmNowgH* z)hK?wyx|&1_1pJEcx%DNrzw3pTarNR>MVNNwSN=})5Wqu7w3m*B~#T=h@Iv4!yFmv+x%IkbJE#_20+=bQK zxS6BR;;JZ@o-(KT8yZ_e+L#mC613Wnwao7)?!3KyR)YC0#b)bILTpxGk~kzluXaSNJ#8|78pRzt_-?9rSlU?nd~Z zQU1Sodix#u?}4SddOG|4M Date: Wed, 31 Jan 2024 12:06:01 -0700 Subject: [PATCH 20/39] regenerate doc --- man/crps.Rd | 4 ++-- man/importance_sampling.Rd | 16 ++++++++-------- man/loo-glossary.Rd | 18 +++++++++--------- man/loo.Rd | 29 +++++++++++------------------ man/loo_compare.Rd | 6 +++--- man/loo_predictive_metric.Rd | 2 +- man/loo_subsample.Rd | 9 +++++---- man/parallel_psis_list.Rd | 7 ++++--- man/psis.Rd | 16 ++++++++-------- man/update.psis_loo_ss.Rd | 9 +++++---- 10 files changed, 56 insertions(+), 60 deletions(-) diff --git a/man/crps.Rd b/man/crps.Rd index 30cdfdea..2f8c098a 100644 --- a/man/crps.Rd +++ b/man/crps.Rd @@ -32,7 +32,7 @@ loo_scrps(x, ...) log_lik, ..., permutations = 1, - r_eff = NULL, + r_eff = 1, cores = getOption("mc.cores", 1) ) @@ -47,7 +47,7 @@ loo_scrps(x, ...) log_lik, ..., permutations = 1, - r_eff = NULL, + r_eff = 1, cores = getOption("mc.cores", 1) ) } diff --git a/man/importance_sampling.Rd b/man/importance_sampling.Rd index 06d4e89a..b9cdf75d 100644 --- a/man/importance_sampling.Rd +++ b/man/importance_sampling.Rd @@ -13,7 +13,7 @@ importance_sampling(log_ratios, method, ...) log_ratios, method, ..., - r_eff = NULL, + r_eff = 1, cores = getOption("mc.cores", 1) ) @@ -21,11 +21,11 @@ importance_sampling(log_ratios, method, ...) log_ratios, method, ..., - r_eff = NULL, + r_eff = 1, cores = getOption("mc.cores", 1) ) -\method{importance_sampling}{default}(log_ratios, method, ..., r_eff = NULL) +\method{importance_sampling}{default}(log_ratios, method, ..., r_eff = 1) } \arguments{ \item{log_ratios}{An array, matrix, or vector of importance ratios on the log @@ -50,11 +50,11 @@ effective sample sizes of \code{1/exp(log_ratios)} (i.e., \code{1/ratios}). This is related to the relative efficiency of estimating the normalizing term in self-normalizing importance sampling. If \code{r_eff} is not provided then the reported PSIS effective sample sizes and Monte Carlo -error estimates will be over-optimistic. See the \code{\link[=relative_eff]{relative_eff()}} -helper function for computing \code{r_eff}. If using \code{psis} with -draws of the \code{log_ratios} not obtained from MCMC then the warning -message thrown when not specifying \code{r_eff} can be disabled by -setting \code{r_eff} to \code{NA}.} +error estimates can be over-optimistic. If the posterior draws are (near) +independent then \code{r_eff=1} can be used. \code{r_eff} has to be a scalar (same +value is used for all observations) or a vector with length equal to the +number of observations. The default value is 1. See the \code{\link[=relative_eff]{relative_eff()}} +helper function for computing \code{r_eff}.} \item{cores}{The number of cores to use for parallelization. This defaults to the option \code{mc.cores} which can be set for an entire R session by diff --git a/man/loo-glossary.Rd b/man/loo-glossary.Rd index 6c109374..9f8173b6 100644 --- a/man/loo-glossary.Rd +++ b/man/loo-glossary.Rd @@ -76,8 +76,8 @@ importance-sampling LOO the full posterior distribution is used as the proposal distribution. The Pareto k diagnostic estimates how far an individual leave-one-out distribution is from the full distribution. If leaving out an observation changes the posterior too much then importance -sampling is not able to give reliable estimate. Pareto smoothing stabilizes -importance sampling and guarantees finite variance estimate with a +sampling is not able to give a reliable estimate. Pareto smoothing stabilizes +importance sampling and guarantees a finite variance estimate at the cost of some bias. The diagnostic threshold for Pareto \eqn{k} depends on sample size @@ -86,28 +86,28 @@ et al., 2022, and before that fixed thresholds of 0.5 and 0.7 were recommended). For simplicity, \code{loo} package uses the nominal sample size \eqn{S} when computing the sample size specific threshold. This provides an optimistic threshold if the effective -sample size is less than 2200, but even then if ESS/S>1/2 the difference +sample size is less than 2200, but even then if ESS/S > 1/2 the difference is usually negligible. Thinning of MCMC draws can be used to improve the ratio ESS/S. \itemize{ \item If \eqn{k < min(1 - 1 / log10(S), 0.7)}, where \eqn{S} is the -sample size PSIS estimate and the corresponding Monte +sample size, PSIS estimate and the corresponding Monte Carlo standard error estimate are reliable. \item If \eqn{1 - 1 / log10(S) <= k < 0.7}, PSIS estimate and the corresponding Monte Carlo standard error estimate are not reliable, but increasing (effective) sample size \eqn{S} above 2200 may help (this will increase the sample size specific -threshold \eqn{(1-1/log10(2200)>0.7} and then the bias specific +threshold \eqn{(1 - 1 / log10(2200) > 0.7} and then the bias specific threshold 0.7 dominates). \item If \eqn{0.7 <= k < 1}, PSIS estimate and the corresponding Monte Carlo standard error have large bias and are not reliable. Increasing sample size may reduce the variability in \eqn{k} estimate, which -may result in lower \eqn{k} estimate, too. +may result in a lower \eqn{k} estimate, too. \item If \eqn{k \geq 1}{k >= 1}, the target distribution is estimated to have non-finite mean. PSIS estimate and the corresponding Monte Carlo standard error are not well defined. Increasing sample size may reduce the variability in \eqn{k} estimate, which -may result in lower \eqn{k} estimate, too. +may result in a lower \eqn{k} estimate, too. } Pareto \eqn{k} is also useful as a measure of influence of an @@ -117,8 +117,8 @@ misspecification, outliers or mistakes in data processing. See Section 6 of Gabry et al. (2019) for an example. \subsection{Interpreting \code{p_loo} when Pareto \code{k} is large}{ -If \code{k > 0.7} then we can also look at the \code{p_loo} estimate for -some additional information about the problem: +If \eqn{k < min(1 - 1 / log10(S), 0.7)} then we can also look at +the \code{p_loo} estimate for some additional information about the problem: \itemize{ \item If \verb{p_loo << p} (the total number of parameters in the model), diff --git a/man/loo.Rd b/man/loo.Rd index cceabdd5..c67e10af 100644 --- a/man/loo.Rd +++ b/man/loo.Rd @@ -15,7 +15,7 @@ loo(x, ...) \method{loo}{array}( x, ..., - r_eff = NULL, + r_eff = 1, save_psis = FALSE, cores = getOption("mc.cores", 1), is_method = c("psis", "tis", "sis") @@ -24,7 +24,7 @@ loo(x, ...) \method{loo}{matrix}( x, ..., - r_eff = NULL, + r_eff = 1, save_psis = FALSE, cores = getOption("mc.cores", 1), is_method = c("psis", "tis", "sis") @@ -35,21 +35,13 @@ loo(x, ...) ..., data = NULL, draws = NULL, - r_eff = NULL, + r_eff = 1, save_psis = FALSE, cores = getOption("mc.cores", 1), is_method = c("psis", "tis", "sis") ) -loo_i( - i, - llfun, - ..., - data = NULL, - draws = NULL, - r_eff = NULL, - is_method = "psis" -) +loo_i(i, llfun, ..., data = NULL, draws = NULL, r_eff = 1, is_method = "psis") is.loo(x) @@ -66,9 +58,10 @@ the relative efficiency of estimating the normalizing term in self-normalized importance sampling when using posterior draws obtained with MCMC. If MCMC draws are used and \code{r_eff} is not provided then the reported PSIS effective sample sizes and Monte Carlo error estimates -can be over-optimistic. If the posterior draws are independent then -\code{r_eff=1} and can be omitted. The warning message thrown when \code{r_eff} is -not specified can be disabled by setting \code{r_eff} to \code{NA}. See the +can be over-optimistic. If the posterior draws are (near) independent then +\code{r_eff=1} can be used. \code{r_eff} has to be a scalar (same value is used +for all observations) or a vector with length equal to the number of +observations. The default value is 1. \code{\link[=relative_eff]{relative_eff()}} helper functions for computing \code{r_eff}.} \item{save_psis}{Should the \code{"psis"} object created internally by \code{loo()} be @@ -269,8 +262,8 @@ llfun <- function(data_i, draws) { loo_3 <- loo_i(i = 3, llfun = llfun, data = fake_data, draws = fake_posterior, r_eff = NA) print(loo_3$pointwise[, "elpd_loo"]) -# Use loo.function method (setting r_eff=NA since this posterior not obtained via MCMC) -loo_with_fn <- loo(llfun, draws = fake_posterior, data = fake_data, r_eff = NA) +# Use loo.function method (default r_eff=1 is used as this posterior not obtained via MCMC) +loo_with_fn <- loo(llfun, draws = fake_posterior, data = fake_data) # If we look at the elpd_loo contribution from the 3rd obs it should be the # same as what we got above with the loo_i function and i=3: @@ -281,7 +274,7 @@ print(loo_3$pointwise[, "elpd_loo"]) log_lik_matrix <- sapply(1:N, function(i) { llfun(data_i = fake_data[i,, drop=FALSE], draws = fake_posterior) }) -loo_with_mat <- loo(log_lik_matrix, r_eff = NA) +loo_with_mat <- loo(log_lik_matrix) all.equal(loo_with_mat$estimates, loo_with_fn$estimates) # should be TRUE! diff --git a/man/loo_compare.Rd b/man/loo_compare.Rd index dacee7d9..0fc43392 100644 --- a/man/loo_compare.Rd +++ b/man/loo_compare.Rd @@ -81,9 +81,9 @@ projection predictive inference. \examples{ # very artificial example, just for demonstration! LL <- example_loglik_array() -loo1 <- loo(LL, r_eff = NA) # should be worst model when compared -loo2 <- loo(LL + 1, r_eff = NA) # should be second best model when compared -loo3 <- loo(LL + 2, r_eff = NA) # should be best model when compared +loo1 <- loo(LL) # should be worst model when compared +loo2 <- loo(LL + 1) # should be second best model when compared +loo3 <- loo(LL + 2) # should be best model when compared comp <- loo_compare(loo1, loo2, loo3) print(comp, digits = 2) diff --git a/man/loo_predictive_metric.Rd b/man/loo_predictive_metric.Rd index 7df7c6c4..b82f7e30 100644 --- a/man/loo_predictive_metric.Rd +++ b/man/loo_predictive_metric.Rd @@ -13,7 +13,7 @@ loo_predictive_metric(x, ...) log_lik, ..., metric = c("mae", "rmse", "mse", "acc", "balanced_acc"), - r_eff = NULL, + r_eff = 1, cores = getOption("mc.cores", 1) ) } diff --git a/man/loo_subsample.Rd b/man/loo_subsample.Rd index 29af1df9..c494448b 100644 --- a/man/loo_subsample.Rd +++ b/man/loo_subsample.Rd @@ -15,7 +15,7 @@ loo_subsample(x, ...) observations = 400, log_p = NULL, log_g = NULL, - r_eff = NULL, + r_eff = 1, save_psis = FALSE, cores = getOption("mc.cores", 1), loo_approximation = "plpd", @@ -59,9 +59,10 @@ the relative efficiency of estimating the normalizing term in self-normalized importance sampling when using posterior draws obtained with MCMC. If MCMC draws are used and \code{r_eff} is not provided then the reported PSIS effective sample sizes and Monte Carlo error estimates -can be over-optimistic. If the posterior draws are independent then -\code{r_eff=1} and can be omitted. The warning message thrown when \code{r_eff} is -not specified can be disabled by setting \code{r_eff} to \code{NA}. See the +can be over-optimistic. If the posterior draws are (near) independent then +\code{r_eff=1} can be used. \code{r_eff} has to be a scalar (same value is used +for all observations) or a vector with length equal to the number of +observations. The default value is 1. \code{\link[=relative_eff]{relative_eff()}} helper functions for computing \code{r_eff}.} \item{save_psis}{Should the \code{"psis"} object created internally by diff --git a/man/parallel_psis_list.Rd b/man/parallel_psis_list.Rd index d8899258..5863758a 100644 --- a/man/parallel_psis_list.Rd +++ b/man/parallel_psis_list.Rd @@ -48,9 +48,10 @@ the relative efficiency of estimating the normalizing term in self-normalized importance sampling when using posterior draws obtained with MCMC. If MCMC draws are used and \code{r_eff} is not provided then the reported PSIS effective sample sizes and Monte Carlo error estimates -can be over-optimistic. If the posterior draws are independent then -\code{r_eff=1} and can be omitted. The warning message thrown when \code{r_eff} is -not specified can be disabled by setting \code{r_eff} to \code{NA}. See the +can be over-optimistic. If the posterior draws are (near) independent then +\code{r_eff=1} can be used. \code{r_eff} has to be a scalar (same value is used +for all observations) or a vector with length equal to the number of +observations. The default value is 1. \code{\link[=relative_eff]{relative_eff()}} helper functions for computing \code{r_eff}.} \item{save_psis}{Should the \code{"psis"} object created internally by \code{loo()} be diff --git a/man/psis.Rd b/man/psis.Rd index ed38f046..78c6dfe7 100644 --- a/man/psis.Rd +++ b/man/psis.Rd @@ -12,11 +12,11 @@ \usage{ psis(log_ratios, ...) -\method{psis}{array}(log_ratios, ..., r_eff = NULL, cores = getOption("mc.cores", 1)) +\method{psis}{array}(log_ratios, ..., r_eff = 1, cores = getOption("mc.cores", 1)) -\method{psis}{matrix}(log_ratios, ..., r_eff = NULL, cores = getOption("mc.cores", 1)) +\method{psis}{matrix}(log_ratios, ..., r_eff = 1, cores = getOption("mc.cores", 1)) -\method{psis}{default}(log_ratios, ..., r_eff = NULL) +\method{psis}{default}(log_ratios, ..., r_eff = 1) is.psis(x) @@ -38,11 +38,11 @@ effective sample sizes of \code{1/exp(log_ratios)} (i.e., \code{1/ratios}). This is related to the relative efficiency of estimating the normalizing term in self-normalizing importance sampling. If \code{r_eff} is not provided then the reported PSIS effective sample sizes and Monte Carlo -error estimates will be over-optimistic. See the \code{\link[=relative_eff]{relative_eff()}} -helper function for computing \code{r_eff}. If using \code{psis} with -draws of the \code{log_ratios} not obtained from MCMC then the warning -message thrown when not specifying \code{r_eff} can be disabled by -setting \code{r_eff} to \code{NA}.} +error estimates can be over-optimistic. If the posterior draws are (near) +independent then \code{r_eff=1} can be used. \code{r_eff} has to be a scalar (same +value is used for all observations) or a vector with length equal to the +number of observations. The default value is 1. See the \code{\link[=relative_eff]{relative_eff()}} +helper function for computing \code{r_eff}.} \item{cores}{The number of cores to use for parallelization. This defaults to the option \code{mc.cores} which can be set for an entire R session by diff --git a/man/update.psis_loo_ss.Rd b/man/update.psis_loo_ss.Rd index e8e1b263..c562a5b3 100644 --- a/man/update.psis_loo_ss.Rd +++ b/man/update.psis_loo_ss.Rd @@ -10,7 +10,7 @@ data = NULL, draws = NULL, observations = NULL, - r_eff = NULL, + r_eff = 1, cores = getOption("mc.cores", 1), loo_approximation = NULL, loo_approximation_draws = NULL, @@ -44,9 +44,10 @@ the relative efficiency of estimating the normalizing term in self-normalized importance sampling when using posterior draws obtained with MCMC. If MCMC draws are used and \code{r_eff} is not provided then the reported PSIS effective sample sizes and Monte Carlo error estimates -can be over-optimistic. If the posterior draws are independent then -\code{r_eff=1} and can be omitted. The warning message thrown when \code{r_eff} is -not specified can be disabled by setting \code{r_eff} to \code{NA}. See the +can be over-optimistic. If the posterior draws are (near) independent then +\code{r_eff=1} can be used. \code{r_eff} has to be a scalar (same value is used +for all observations) or a vector with length equal to the number of +observations. The default value is 1. \code{\link[=relative_eff]{relative_eff()}} helper functions for computing \code{r_eff}.} \item{cores}{The number of cores to use for parallelization. This defaults to From 8c0d4490467de9ce2e9046b4d7cc58aef8712e6d Mon Sep 17 00:00:00 2001 From: jgabry Date: Wed, 31 Jan 2024 12:39:04 -0700 Subject: [PATCH 21/39] a few minor doc edits --- R/effective_sample_sizes.R | 3 ++- R/loo-glossary.R | 30 ++++++++++++++---------------- R/loo.R | 15 ++++++++------- man/loo-glossary.Rd | 23 ++++++++++------------- 4 files changed, 34 insertions(+), 37 deletions(-) diff --git a/R/effective_sample_sizes.R b/R/effective_sample_sizes.R index b66c2207..127e4ea4 100644 --- a/R/effective_sample_sizes.R +++ b/R/effective_sample_sizes.R @@ -183,8 +183,9 @@ psis_n_eff.matrix <- function(w, r_eff = NULL, ...) { if (is.null(r_eff)) { return(1 / ss) } - if (length(r_eff) != length(ss) && length(r_eff) != 1) + if (length(r_eff) != length(ss) && length(r_eff) != 1) { stop("r_eff must have length 1 or ncol(w).", call. = FALSE) + } 1 / ss * r_eff } diff --git a/R/loo-glossary.R b/R/loo-glossary.R index 76265caa..789c4d22 100644 --- a/R/loo-glossary.R +++ b/R/loo-glossary.R @@ -85,27 +85,27 @@ #' the ratio ESS/S. #' #' * If \eqn{k < min(1 - 1 / log10(S), 0.7)}, where \eqn{S} is the -#' sample size, PSIS estimate and the corresponding Monte +#' sample size, the PSIS estimate and the corresponding Monte #' Carlo standard error estimate are reliable. #' -#' * If \eqn{1 - 1 / log10(S) <= k < 0.7}, PSIS estimate and the +#' * If \eqn{1 - 1 / log10(S) <= k < 0.7}, the PSIS estimate and the #' corresponding Monte Carlo standard error estimate are not -#' reliable, but increasing (effective) sample size \eqn{S} above +#' reliable, but increasing the (effective) sample size \eqn{S} above #' 2200 may help (this will increase the sample size specific #' threshold \eqn{(1 - 1 / log10(2200) > 0.7} and then the bias specific #' threshold 0.7 dominates). #' -#' * If \eqn{0.7 <= k < 1}, PSIS estimate and the corresponding Monte +#' * If \eqn{0.7 <= k < 1}, the PSIS estimate and the corresponding Monte #' Carlo standard error have large bias and are not reliable. Increasing -#' sample size may reduce the variability in \eqn{k} estimate, which -#' may result in a lower \eqn{k} estimate, too. +#' the sample size may reduce the variability in the \eqn{k} estimate, which +#' may also result in a lower \eqn{k} estimate. #' #' * If \eqn{k \geq 1}{k >= 1}, the target distribution is estimated to -#' have non-finite mean. PSIS estimate and the corresponding Monte -#' Carlo standard error are not well defined. Increasing sample size -#' may reduce the variability in \eqn{k} estimate, which -#' may result in a lower \eqn{k} estimate, too. -#' +#' have non-finite mean. The PSIS estimate and the corresponding Monte +#' Carlo standard error are not well defined. Increasing the sample size +#' may reduce the variability in \eqn{k} estimate, which may also result in +#' a lower \eqn{k} estimate. +#' #' Pareto \eqn{k} is also useful as a measure of influence of an #' observation. Highly influential observations have high \eqn{k} #' values. Very high \eqn{k} values often indicate model @@ -116,14 +116,13 @@ #' If \eqn{k < min(1 - 1 / log10(S), 0.7)} then we can also look at #' the `p_loo` estimate for some additional information about the problem: #' -#' \itemize{ -#' \item If `p_loo << p` (the total number of parameters in the model), +#' * If `p_loo << p` (the total number of parameters in the model), #' then the model is likely to be misspecified. Posterior predictive checks #' (PPCs) are then likely to also detect the problem. Try using an overdispersed #' model, or add more structural information (nonlinearity, mixture model, #' etc.). #' -#' \item If `p_loo < p` and the number of parameters `p` is relatively +#' * If `p_loo < p` and the number of parameters `p` is relatively #' large compared to the number of observations (e.g., `p>N/5`), it is #' likely that the model is so flexible or the population prior so weak that it’s #' difficult to predict the left out observation (even for the true model). @@ -131,7 +130,7 @@ #' effect models with a few observations per random effect, and Gaussian #' processes and spatial models with short correlation lengths. #' -#' \item If `p_loo > p`, then the model is likely to be badly misspecified. +#' * If `p_loo > p`, then the model is likely to be badly misspecified. #' If the number of parameters `p< for an example. @@ -141,7 +140,6 @@ #' may have few observations and other groups many), it is possible that PPCs won't #' detect the problem. #' } -#' } #' #' @section elpd_diff: #' `elpd_diff` is the difference in `elpd_loo` for two models. If more diff --git a/R/loo.R b/R/loo.R index a722cbeb..a321ff08 100644 --- a/R/loo.R +++ b/R/loo.R @@ -19,15 +19,16 @@ #' can be over-optimistic. If the posterior draws are (near) independent then #' `r_eff=1` can be used. `r_eff` has to be a scalar (same value is used #' for all observations) or a vector with length equal to the number of -#' observations. The default value is 1. -#' [relative_eff()] helper functions for computing `r_eff`. -#' @param save_psis Should the `"psis"` object created internally by `loo()` be +#' observations. The default value is 1. See the [relative_eff()] helper +#' functions for help computing `r_eff`. +#' @param save_psis Should the `psis` object created internally by `loo()` be #' saved in the returned object? The `loo()` function calls [psis()] -#' internally but by default discards the (potentially large) `"psis"` object +#' internally but by default discards the (potentially large) `psis` object #' after using it to compute the LOO-CV summaries. Setting `save_psis=TRUE` #' will add a `psis_object` component to the list returned by `loo`. -#' Currently this is only needed if you plan to use the [E_loo()] function to -#' compute weighted expectations after running `loo`. +#' This is useful if you plan to use the [E_loo()] function to compute +#' weighted expectations after running `loo`. Several functions in the +#' \pkg{bayesplot} package also accept `psis` objects. #' @template cores #' @template is_method #' @@ -133,7 +134,7 @@ #' #' # Use the loo_i function to check that llfun works on a single observation #' # before running on all obs. For example, using the 3rd obs in the data: -#' loo_3 <- loo_i(i = 3, llfun = llfun, data = fake_data, draws = fake_posterior, r_eff = NA) +#' loo_3 <- loo_i(i = 3, llfun = llfun, data = fake_data, draws = fake_posterior) #' print(loo_3$pointwise[, "elpd_loo"]) #' #' # Use loo.function method (default r_eff=1 is used as this posterior not obtained via MCMC) diff --git a/man/loo-glossary.Rd b/man/loo-glossary.Rd index 9f8173b6..084e3c75 100644 --- a/man/loo-glossary.Rd +++ b/man/loo-glossary.Rd @@ -91,23 +91,23 @@ is usually negligible. Thinning of MCMC draws can be used to improve the ratio ESS/S. \itemize{ \item If \eqn{k < min(1 - 1 / log10(S), 0.7)}, where \eqn{S} is the -sample size, PSIS estimate and the corresponding Monte +sample size, the PSIS estimate and the corresponding Monte Carlo standard error estimate are reliable. -\item If \eqn{1 - 1 / log10(S) <= k < 0.7}, PSIS estimate and the +\item If \eqn{1 - 1 / log10(S) <= k < 0.7}, the PSIS estimate and the corresponding Monte Carlo standard error estimate are not -reliable, but increasing (effective) sample size \eqn{S} above +reliable, but increasing the (effective) sample size \eqn{S} above 2200 may help (this will increase the sample size specific threshold \eqn{(1 - 1 / log10(2200) > 0.7} and then the bias specific threshold 0.7 dominates). -\item If \eqn{0.7 <= k < 1}, PSIS estimate and the corresponding Monte +\item If \eqn{0.7 <= k < 1}, the PSIS estimate and the corresponding Monte Carlo standard error have large bias and are not reliable. Increasing -sample size may reduce the variability in \eqn{k} estimate, which -may result in a lower \eqn{k} estimate, too. +the sample size may reduce the variability in the \eqn{k} estimate, which +may also result in a lower \eqn{k} estimate. \item If \eqn{k \geq 1}{k >= 1}, the target distribution is estimated to -have non-finite mean. PSIS estimate and the corresponding Monte -Carlo standard error are not well defined. Increasing sample size -may reduce the variability in \eqn{k} estimate, which -may result in a lower \eqn{k} estimate, too. +have non-finite mean. The PSIS estimate and the corresponding Monte +Carlo standard error are not well defined. Increasing the sample size +may reduce the variability in \eqn{k} estimate, which may also result in +a lower \eqn{k} estimate. } Pareto \eqn{k} is also useful as a measure of influence of an @@ -119,14 +119,12 @@ Section 6 of Gabry et al. (2019) for an example. \subsection{Interpreting \code{p_loo} when Pareto \code{k} is large}{ If \eqn{k < min(1 - 1 / log10(S), 0.7)} then we can also look at the \code{p_loo} estimate for some additional information about the problem: - \itemize{ \item If \verb{p_loo << p} (the total number of parameters in the model), then the model is likely to be misspecified. Posterior predictive checks (PPCs) are then likely to also detect the problem. Try using an overdispersed model, or add more structural information (nonlinearity, mixture model, etc.). - \item If \code{p_loo < p} and the number of parameters \code{p} is relatively large compared to the number of observations (e.g., \code{p>N/5}), it is likely that the model is so flexible or the population prior so weak that it’s @@ -134,7 +132,6 @@ difficult to predict the left out observation (even for the true model). This happens, for example, in the simulated 8 schools (in VGG2017), random effect models with a few observations per random effect, and Gaussian processes and spatial models with short correlation lengths. - \item If \code{p_loo > p}, then the model is likely to be badly misspecified. If the number of parameters \verb{p< Date: Wed, 31 Jan 2024 12:39:45 -0700 Subject: [PATCH 22/39] update Rd files --- man/loo.Rd | 15 ++++++++------- man/loo_subsample.Rd | 4 ++-- man/parallel_psis_list.Rd | 13 +++++++------ man/psis_approximate_posterior.Rd | 9 +++++---- man/update.psis_loo_ss.Rd | 4 ++-- 5 files changed, 24 insertions(+), 21 deletions(-) diff --git a/man/loo.Rd b/man/loo.Rd index c67e10af..128eaabb 100644 --- a/man/loo.Rd +++ b/man/loo.Rd @@ -61,16 +61,17 @@ the reported PSIS effective sample sizes and Monte Carlo error estimates can be over-optimistic. If the posterior draws are (near) independent then \code{r_eff=1} can be used. \code{r_eff} has to be a scalar (same value is used for all observations) or a vector with length equal to the number of -observations. The default value is 1. -\code{\link[=relative_eff]{relative_eff()}} helper functions for computing \code{r_eff}.} +observations. The default value is 1. See the \code{\link[=relative_eff]{relative_eff()}} helper +functions for help computing \code{r_eff}.} -\item{save_psis}{Should the \code{"psis"} object created internally by \code{loo()} be +\item{save_psis}{Should the \code{psis} object created internally by \code{loo()} be saved in the returned object? The \code{loo()} function calls \code{\link[=psis]{psis()}} -internally but by default discards the (potentially large) \code{"psis"} object +internally but by default discards the (potentially large) \code{psis} object after using it to compute the LOO-CV summaries. Setting \code{save_psis=TRUE} will add a \code{psis_object} component to the list returned by \code{loo}. -Currently this is only needed if you plan to use the \code{\link[=E_loo]{E_loo()}} function to -compute weighted expectations after running \code{loo}.} +This is useful if you plan to use the \code{\link[=E_loo]{E_loo()}} function to compute +weighted expectations after running \code{loo}. Several functions in the +\pkg{bayesplot} package also accept \code{psis} objects.} \item{cores}{The number of cores to use for parallelization. This defaults to the option \code{mc.cores} which can be set for an entire R session by @@ -259,7 +260,7 @@ llfun <- function(data_i, draws) { # Use the loo_i function to check that llfun works on a single observation # before running on all obs. For example, using the 3rd obs in the data: -loo_3 <- loo_i(i = 3, llfun = llfun, data = fake_data, draws = fake_posterior, r_eff = NA) +loo_3 <- loo_i(i = 3, llfun = llfun, data = fake_data, draws = fake_posterior) print(loo_3$pointwise[, "elpd_loo"]) # Use loo.function method (default r_eff=1 is used as this posterior not obtained via MCMC) diff --git a/man/loo_subsample.Rd b/man/loo_subsample.Rd index c494448b..dfde3eaf 100644 --- a/man/loo_subsample.Rd +++ b/man/loo_subsample.Rd @@ -62,8 +62,8 @@ the reported PSIS effective sample sizes and Monte Carlo error estimates can be over-optimistic. If the posterior draws are (near) independent then \code{r_eff=1} can be used. \code{r_eff} has to be a scalar (same value is used for all observations) or a vector with length equal to the number of -observations. The default value is 1. -\code{\link[=relative_eff]{relative_eff()}} helper functions for computing \code{r_eff}.} +observations. The default value is 1. See the \code{\link[=relative_eff]{relative_eff()}} helper +functions for help computing \code{r_eff}.} \item{save_psis}{Should the \code{"psis"} object created internally by \code{loo_subsample()} be saved in the returned object? See \code{\link[=loo]{loo()}} for details.} diff --git a/man/parallel_psis_list.Rd b/man/parallel_psis_list.Rd index 5863758a..f9c0224c 100644 --- a/man/parallel_psis_list.Rd +++ b/man/parallel_psis_list.Rd @@ -51,16 +51,17 @@ the reported PSIS effective sample sizes and Monte Carlo error estimates can be over-optimistic. If the posterior draws are (near) independent then \code{r_eff=1} can be used. \code{r_eff} has to be a scalar (same value is used for all observations) or a vector with length equal to the number of -observations. The default value is 1. -\code{\link[=relative_eff]{relative_eff()}} helper functions for computing \code{r_eff}.} +observations. The default value is 1. See the \code{\link[=relative_eff]{relative_eff()}} helper +functions for help computing \code{r_eff}.} -\item{save_psis}{Should the \code{"psis"} object created internally by \code{loo()} be +\item{save_psis}{Should the \code{psis} object created internally by \code{loo()} be saved in the returned object? The \code{loo()} function calls \code{\link[=psis]{psis()}} -internally but by default discards the (potentially large) \code{"psis"} object +internally but by default discards the (potentially large) \code{psis} object after using it to compute the LOO-CV summaries. Setting \code{save_psis=TRUE} will add a \code{psis_object} component to the list returned by \code{loo}. -Currently this is only needed if you plan to use the \code{\link[=E_loo]{E_loo()}} function to -compute weighted expectations after running \code{loo}.} +This is useful if you plan to use the \code{\link[=E_loo]{E_loo()}} function to compute +weighted expectations after running \code{loo}. Several functions in the +\pkg{bayesplot} package also accept \code{psis} objects.} \item{cores}{The number of cores to use for parallelization. This defaults to the option \code{mc.cores} which can be set for an entire R session by diff --git a/man/psis_approximate_posterior.Rd b/man/psis_approximate_posterior.Rd index 8cf1a03a..b09bb23f 100644 --- a/man/psis_approximate_posterior.Rd +++ b/man/psis_approximate_posterior.Rd @@ -40,13 +40,14 @@ the \code{.Rprofile} file to set \code{mc.cores} (using the \code{cores} argumen setting \code{mc.cores} interactively or in a script is fine). }} -\item{save_psis}{Should the \code{"psis"} object created internally by \code{loo()} be +\item{save_psis}{Should the \code{psis} object created internally by \code{loo()} be saved in the returned object? The \code{loo()} function calls \code{\link[=psis]{psis()}} -internally but by default discards the (potentially large) \code{"psis"} object +internally but by default discards the (potentially large) \code{psis} object after using it to compute the LOO-CV summaries. Setting \code{save_psis=TRUE} will add a \code{psis_object} component to the list returned by \code{loo}. -Currently this is only needed if you plan to use the \code{\link[=E_loo]{E_loo()}} function to -compute weighted expectations after running \code{loo}.} +This is useful if you plan to use the \code{\link[=E_loo]{E_loo()}} function to compute +weighted expectations after running \code{loo}. Several functions in the +\pkg{bayesplot} package also accept \code{psis} objects.} \item{log_q}{Deprecated argument name (the same as log_g).} } diff --git a/man/update.psis_loo_ss.Rd b/man/update.psis_loo_ss.Rd index c562a5b3..a8aecbd2 100644 --- a/man/update.psis_loo_ss.Rd +++ b/man/update.psis_loo_ss.Rd @@ -47,8 +47,8 @@ the reported PSIS effective sample sizes and Monte Carlo error estimates can be over-optimistic. If the posterior draws are (near) independent then \code{r_eff=1} can be used. \code{r_eff} has to be a scalar (same value is used for all observations) or a vector with length equal to the number of -observations. The default value is 1. -\code{\link[=relative_eff]{relative_eff()}} helper functions for computing \code{r_eff}.} +observations. The default value is 1. See the \code{\link[=relative_eff]{relative_eff()}} helper +functions for help computing \code{r_eff}.} \item{cores}{The number of cores to use for parallelization. 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