Consistency of Bayes factors under hyper g-priors with growing model size

In this paper, we examine Bayes factor consistency in the context of Bayesian variable selection for normal linear regression models. We take a hierarchical Bayesian approach using a hyper-g prior (Liang et al. (2008) JASA). There are two regimes for computing Bayes factors, which differ in the choice of the base model. For both these regimes, we study conditions under which Bayes factors are consistent when the number of all the potential regressors grows with sample size n. This situation is not fully understood in the current literature, but has gained increasing importance recently. In the present case, Bayes factors are not analytically tractable and are calculated via Laplace approximation. A rigorous justification for these high-dimensional Laplace approximations is also provided.