Clarification of assumptions required for dimension scaling results in paper

[matt-graham]

Submitting as part of JOSS review openjournals/joss-reviews#3397

ecological-inference/paper/paper.md

Line 68 in 2d47262

First, the Bayesian hierarchical methods implemented in `PyEI` rest on modern probabilistic programming tooling [@salvatier2016probabilistic] and gradient-based MCMC methods such as the No U-Turn Sampler (NUTS) [@hoffman2014no]. Using NUTS where possible should allow for faster convergence than existing implementations that rest primarily on Metropolis-Hastings and Gibbs sampling steps. Consider effective sample size, which is a measure of how the variance of the mean of drawn samples compare to the variance of i.i.d. samples from the posterior distribution (or, very roughly, how “effective” the samples are for computing the posterior mean, compared to i.i.d. samples) [@BDA3]. In Metropolis-Hastings, the number of evaluations of the log-posterior required for a given effective sample size scales linearly with the dimensionality of the parameter space, while in Hamiltonian Monte Carlo approaches such as NUTS, the number of required evaluations of the gradient of the log-posterior scales only as the fourth root of the dimension [@neal2011mcmc]. Reasonable scaling with the dimensionality of the parameter space is important in ecological inference, as that dimensionality is large when there are many precincts.

The claim regarding the scaling with dimension for Metropolis–Hastings versus Hamiltonian Monte Carlo (in the sentence in lines 65–69 in proof PDF) should ideally have a clarification that the result requires assumptions on the target distribution (in particular that it is a product of independent factors in each dimension) to hold. To avoid having to introduce technical details this could just simply be adding 'Under certain assumptions on the target posterior distribution,' at the beginning of the sentence.