Some Recent Developments in Bayesian Variable Selection

Abstract

After a general discussion of some basic issues in Bayesian model selection, we briefly review three fairly recent developments: (i) The median probability model (rather than the highest posterior probability model) is the model which is typically optimal for prediction in variable selection problems; discussion of this highlights the central role of the overall posterior probability with which a variable occurs in one of the candidate models, called the “posterior inclusion probability.” (ii) When the space of models is large, search strategies need to be carefully developed for exploration of the space; an appealing strategy is to perform a stochastic search that (roughly) adds or removes variables based on their current estimated posterior inclusion probability. (iii) Expected posterior priors provide a constructive solution to the problem of choosing prior distributions for model selection, priors that are appropriately ‘calibrated’ across the parameters of different models. Illustrations to linear mo...