A majorization-minimization approach to variable selection using spike and slab priors

TL;DR

This paper introduces a novel optimization approach for Bayesian regression with spike and slab priors, improving variable selection accuracy and establishing theoretical consistency under certain conditions.

Contribution

It develops a majorization-minimization algorithm for MAP estimation with spike and slab priors, incorporating a tight approximation to the l0 norm and demonstrating theoretical and empirical advantages.

Findings

More accurate variable selection compared to benchmarks

Sign consistency established even when Irrepresentable Condition is violated

Extension to generalized linear models included

Abstract

We develop a method to carry out MAP estimation for a class of Bayesian regression models in which coefficients are assigned with Gaussian-based spike and slab priors. The objective function in the corresponding optimization problem has a Lagrangian form in that regression coefficients are regularized by a mixture of squared $l_2$ and $l_0$ norms. A tight approximation to the $l_0$ norm using majorization-minimization techniques is derived, and a coordinate descent algorithm in conjunction with a soft-thresholding scheme is used in searching for the optimizer of the approximate objective. Simulation studies show that the proposed method can lead to more accurate variable selection than other benchmark methods. Theoretical results show that under regular conditions, sign consistency can be established, even when the Irrepresentable Condition is violated. Results on posterior model consistency and estimation consistency, and an extension to parameter estimation in the generalized linear models are provided.