Consistency of objective Bayes factors as the model dimension grows
Elías Moreno, F. Javier Girón, George Casella
The Annals of Statistics · 2010 · Volume 38 · Issue 4
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(nb) 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.
Keywords
62F05, 62J15, Bayes factors, BIC, intrinsic priors, linear models, multiplicity of parameters, rate of growth
Enable JavaScript for citation tools and additional paper interactions.