Posterior consistency of [formula omitted]-prior for variable selection with a growing number of parameters

Abstract

Recently, Bayesian procedures based on mixtures of g-priors have been widely studied for the variable selection problem in linear models. Maruyama and George (2011) proposed an explicit Bayesian approach without integral representation and showed its posterior model selection consistency when the number of parameters, k, is fixed. Given that linear models with a growing number of parameters have also received increasing attention in practice, we further concentrate on its corresponding posterior model selection consistency when k grows with the sample size, n, at the rate of k=O(nb),0≤b≤1. Specifically, we consider the Bayesian approach with two most commonly used types of priors on the class of models and derive conditions under which the resulting Bayesian approaches achieve such consistency. In addition, we study the case for linear models with the non-normal errors. The proposed results are compared with the existing ones in the literature.