Consistency of Bayesian Procedures for Variable Selection

It has long been known that for the comparison of pairwise nested models, a decision based on the Bayes factor produces a consistent model selector (in the frequentist sense). Here we go beyond the usual consistency for nested pairwise models, and show that for a wide class of prior distributions, including intrinsic priors, the corre-sponding Bayesian procedure for variable selection in normal regres-sion is consistent in the entire class of normal linear models. We also find that the asymptotics of the Bayes factors for intrinsic priors are equivalent to those of the Schwarz (BIC) criterion. On the other hand, the Jeffreys-Lindley paradox refers to the well-known fact that a point null hypothesis on the normal mean parameter is always ac-cepted when the variance of the conjugate prior goes to infinity. Thi