  Most of the consistency analyses of Bayesian procedures for variable
selection in regression refer to pairwise consistency, that is, consistency of
Bayes factors. However, variable selection in regression is carried out in a
given class of regression models where a natural variable selector is the
posterior probability of the models. In this paper we analyze the consistency
of the posterior model probabilities when the number of potential regressors
grows as the sample size grows. The novelty in the posterior model consistency
is that it depends not only on the priors for the model parameters through the
Bayes factor, but also on the model priors, so that it is a useful tool for
choosing priors for both models and model parameters. We have found that some
classes of priors typically used in variable selection yield posterior model
inconsistency, while mixtures of these priors improve this undesirable
behavior. For moderate sample sizes, we evaluate Bayesian pairwise variable
selection procedures by comparing their frequentist Type I and II error
probabilities. This provides valuable information to discriminate between the
priors for the model parameters commonly used for variable selection.
