Bayesian variable selection with shrinking and diffusing priors

TL;DR

This paper introduces a Bayesian variable selection method using hierarchical spike and slab priors with sample size-dependent variances, achieving strong consistency even with exponentially growing covariates.

Contribution

It presents a novel Bayesian approach with shrinking and diffusing priors that guarantees strong selection consistency in high-dimensional settings.

Findings

Achieves posterior probability convergence to the true model.

Comparable to $L_0$ penalty-based model selection asymptotically.

Demonstrates superior empirical performance over existing methods.

Abstract

We consider a Bayesian approach to variable selection in the presence of high dimensional covariates based on a hierarchical model that places prior distributions on the regression coefficients as well as on the model space. We adopt the well-known spike and slab Gaussian priors with a distinct feature, that is, the prior variances depend on the sample size through which appropriate shrinkage can be achieved. We show the strong selection consistency of the proposed method in the sense that the posterior probability of the true model converges to one even when the number of covariates grows nearly exponentially with the sample size. This is arguably the strongest selection consistency result that has been available in the Bayesian variable selection literature; yet the proposed method can be carried out through posterior sampling with a simple Gibbs sampler. Furthermore, we argue that the proposed method is asymptotically similar to model selection with the $L_0$ penalty. We also demonstrate through empirical work the fine performance of the proposed approach relative to some state of the art alternatives.