On the Consistency of Bayesian Variable Selection for High Dimensional Linear Models

Abstract

First, good performance of Bayesian variable selection (BVS for short) in a variety of applications is introduced. Then, we will give a theoretical explanation why BVS works so well in linear models. We assume the true regression coefficients vector of the linear model is sparsity, in a sense that some regression coefficients are bounded from zero while the rest are exactly zero. In this case, under some conditions, BVS will show it can select the true model by means of giving a consistent estimate of the true regression coefficients vector.