Consistency of Bayes factors for linear models

Abstract

The quality of a Bayes factor as variable selector crucially depends on the number of regressors, the sample size and the prior on the regression parameters, and hence it has to be established in a case-by-case basis. In this paper we analyze the consistency of a wide class of Bayes factors when the number of potential regressors grows as the sample size grows. We have found that when the number of regressors is finite some classes of priors yield inconsistency, and when the potential number of regressors grows at the same rate than the sample size different priors yield different degree of inconsistency. For moderate sample sizes, we evaluate the Bayes factors by comparing the posterior model probability. This gives valuable information to discriminate between the priors for the model parameters commonly used for variable selection.