Consistent Group Selection with Bayesian High Dimensional Modeling

Abstract

In many applications with high dimensional covariates, the covariates are naturally structured into different groups which can be used to perform efficient statistical inference. We propose a Bayesian hierarchical model with a spike and slab prior specification to perform group selection in high dimensional linear regression models. While several penalization methods and more recently, some Bayesian approaches are proposed for group selection, theoretical properties of Bayesian approaches have not been studied extensively. In this paper, we provide novel theoretical results for group selection consistency under spike and slab priors which demonstrate that the proposed Bayesian approach has advantages compared to penalization approaches. Our theoretical results accommodate flexible conditions on the design matrix and can be applied to commonly used statistical models such as nonparametric additive models for which very limited theoretical results are available for the Bayesian methods. A shotgun stochastic search algorithm is adopted for the implementation of our proposed approach. We illustrate through simulation studies that the proposed method has better performance for group selection compared to a variety of existing methods.