Consistency of objective Bayes factors as the model dimension grows

TL;DR

This paper investigates the consistency of Bayes factors in normal linear models where the number of regressors increases with sample size, analyzing the performance of Schwarz approximation and intrinsic priors.

Contribution

It extends understanding of Bayes factor consistency to high-dimensional models with growing parameters, comparing two key approximation methods.

Findings

Schwarz approximation is inconsistent when model dimension grows at rate O(n) under the alternative.

Bayes factor for intrinsic priors remains consistent for growth rates below O(n).

Intrinsic priors are robust for model selection in high-dimensional settings.

Abstract

In the class of normal regression models with a finite number of regressors, and for a wide class of prior distributions, a Bayesian model selection procedure based on the Bayes factor is consistent [Casella and Moreno J. Amer. Statist. Assoc. 104 (2009) 1261--1271]. However, in models where the number of parameters increases as the sample size increases, properties of the Bayes factor are not totally understood. Here we study consistency of the Bayes factors for nested normal linear models when the number of regressors increases with the sample size. We pay attention to two successful tools for model selection [Schwarz Ann. Statist. 6 (1978) 461--464] approximation to the Bayes factor, and the Bayes factor for intrinsic priors [Berger and Pericchi J. Amer. Statist. Assoc. 91 (1996) 109--122, Moreno, Bertolino and Racugno J. Amer. Statist. Assoc. 93 (1998) 1451--1460]. We find that the the Schwarz approximation and the Bayes factor for intrinsic priors are consistent when the rate of growth of the dimension of the bigger model is $O(n^b)$ for $b<1$. When $b=1$ the Schwarz approximation is always inconsistent under the alternative while the Bayes factor for intrinsic priors is consistent except for a small set of alternative models which is characterized.