Expert Knowledge Elicitation for Linear Models

This repository contains code for expert knowledge elicitation in sparse linear regression where the user can provide feedback about the covariates (coefficient value or relevance) with the goal of improving the predictions. The model uses a spike-and-slab prior on the regression coefficients and therefore it is especially useful for "small n, large p" problems, i.e., when the number of training samples is smaller than the number of dimensions.

Basic Usage

The script linreg_sns_ep.m contains the posterior inference and the function decision_policy.m has implementation of different query methods [1].

%Inputs: 
% x                covariates (n x m).
% y                target values (n x 1). 
% pr:              prior and other fixed model parameters (see the plate diagram).
% op:              options for the EP algorithm.
% w_feedbacks      values (1st column) and indices (2nd column) of feedback (n_w_feedbacks x 2)
% gamma_feedbacks  values (1st column, 0/1) and indices (2nd column) of feedback (n_gamma_feedbacks x 2)
% Outputs:
% fa         EP posterior approximation (struct)
% si         EP site terms (struct)
% converged  did EP converge or hit max_iter (1/0)

[fa, si, converged, subfunctions] = linreg_sns_ep(y, x, pr, op, w_feedbacks, gamma_feedbacks, si)

Example scenario

A "small n, large p" toy example: 100 dimensional data, only 10 first coefficients are non-zero, 10 training data is available. Feedback only on relevance of features.

%% "small n, large p" toy example
n = 10;      %number of training data
p = 100;     %number of coefficients or features
p_star = 10; %number of relevant coefficients
%unknown, true coefficient values (only first p_star are non-zero)
w_star = [randn(p_star, 1); zeros(p-p_star,1)];
%create n, p dimensional training data
x = randn(n,p);
y = normrnd(x*w_star, 1);
%create some test data
x_test = randn(1000,p);
y_test = normrnd(x_test*w_star, 1);
%initialize the inputs
pr  = struct('tau2', 1^2 , 'eta2',0.1^2,'p_u', 0.95, 'rho', p_star/p, ...
    'sigma2_prior',true,'sigma2_a',1,'sigma2_b',1 );
op = struct('damp',0.8, 'damp_decay',0.95, 'robust_updates',2, 'verbosity',0,...
    'max_iter',1000, 'threshold',1e-5, 'min_site_prec',1e-6);
%Assume we only have feedback about relevance of coefficients
gamma_feedbacks = [[ones(p_star,1);zeros(p-p_star,1)], [1:p]'];
%results with feedback (the proposed model [1])
[fa_fb, si, converged, subfunctions] = linreg_sns_ep(y, x, pr, op, [], gamma_feedbacks, []);
MSE_with_fb = mean((x_test*fa_fb.w.Mean- y_test).^2); 
%results without feedback (only spike and slab model)
[fa, si, converged, subfunctions] = linreg_sns_ep(y, x, pr, op, [], [], []);
MSE_without_fb = mean((x_test*fa.w.Mean- y_test).^2);
%ridge regression solution
w_ridge = inv(eye(p) + (x'*x)) * (x'*y);
MSE_ridge = mean((x_test*w_ridge- y_test).^2);

disp('Mean Squared Error on test data:')
disp(['The proposed method with user feedback:',num2str(MSE_with_fb)])
disp(['The proposed method without user feedback:',num2str(MSE_without_fb)])
disp(['Ridge regression:',num2str(MSE_ridge)])

More examples about sequential interaction with the user can be found in the main scripts.

Simulation and real data study

The starting point of the code is one of the main scripts. main.m runs the simulation for fixed number of training data and dimensions. In main_all.m the number of training data or dimensions can be arrays, e.g, 2:10:300. main_real_data.m is for real data. Your data should contain a matrix of X with size (n by p) and a matrix of Y with size (n by 1). In the code there is an example of using the method for Amazon and Yelp dataset.

Depending on the MODE parameter, the type of model and feedback changes:

       MODE =1: Feedback on weight values. Model: spike and slab prior         
       MODE =2: Feedback on relevance of features. Model: spike and slab prior

Mode 2 is the most realistic case and it is used for the real data (Amazon and Yelp dataset).

Citation

Full paper:

[1] Pedram Daee, Tomi Peltola, Marta Soare and Samuel Kaski (2017). Knowledge Elicitation via Sequential Probabilistic Inference for High-Dimensional Prediction. arXiv preprint: arXiv:1612.03328. [link].

Extended abstract:

[2] Pedram Daee, Tomi Peltola, Marta Soare and Samuel Kaski (2016). Probabilistic Expert Knowledge Elicitation of Feature Relevances in Sparse Linear Regression. In FILM 2016, NIPS Workshop on Future of Interactive Learning Machines. [Pdf]

Team

Pedram Daee	Tomi Peltola	Marta Soare

License

This project is licensed under the MIT License - see the LICENSE.md file for details