Bayesian projection approaches to variable selection and exploring model uncertainty

TL;DR

This paper introduces a Bayesian variable selection method based on L1-constrained projections, enabling model uncertainty exploration and structured variable selection, with proven asymptotic consistency.

Contribution

It extends Bayesian projection methods by incorporating L1 constraints, connecting to lasso techniques, and addressing structured models like ANOVA with hierarchical constraints.

Findings

Posterior distribution can assign exact zeros to some coefficients.

Method explores model uncertainty effectively.

Posterior concentrates on the true model asymptotically.

Abstract

A Bayesian approach to variable selection which is based on the expected Kullback-Leibler divergence between the full model and its projection onto a submodel has recently been suggested in the literature. Here we extend this idea by considering projections onto subspaces defined via some form of $L_1$ constraint on the parameter in the full model. This leads to Bayesian model selection approaches related to the lasso. In the posterior distribution of the projection there is positive probability that some components are exactly zero and the posterior distribution on the model space induced by the projection allows exploration of model uncertainty. We also consider use of the approach in structured variable selection problems such as ANOVA models where it is desired to incorporate main effects in the presence of interactions. Here we make use of projections related to the non-negative garotte which are able to respect the hierarchical constraints. We also prove a consistency result concerning the posterior distribution on the model induced by the projection, and show that for some projections related to the adaptive lasso and non-negative garotte the posterior distribution concentrates on the true model asymptotically.