Bayes factor consistency in linear models when p grows with n

Abstract

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] This dissertation examines consistency of Bayes factors in the model comparison problem for linear models. Common approaches to Bayesian analysis of linear models use Zellner's g-prior, a partially conjugate normal prior on the model parameters indexed by a single parameter g. More generally, a hyper-prior can be placed on g, providing a mixture of g-priors. When comparing nested models, flat priors are often placed on the common parameters with the g-prior used for the other parameters, forcing the prior on g to be proper for a determinate Bayes factor. Even for the non-nested case, an "encompassing" approach comparing all models to a base model is often used, where the base model has a flat prior and the prior on g must be proper. In this dissertation, we consider the Jeffreys prior on g, an improper prior that is also the reference prior. We show consistency of the Bayes factor associated with the reference prior for g and a broad range of proper priors for the fixed model dimension case. We also discuss consistency and inconsistency for the Bayes factor associated with the reference prior for the growing dimension case. We obtain consistency and inconsistency depending on the limiting behavior of [rho/nu]. Laplace approximations are derived for bayes factors under different situations.