Model choice: a minimum posterior predictive loss approach

Model choice is a fundamental and much discussed activity in the analysis of data sets. Hierarchical models introducing random effects can not be handled by classical methods. Bayesian approaches using predictive distributions can, though the formal solution, which includes Bayes factors as a special case, can be criticized. We propose a predictive criterion where the goal is good prediction of a replicate of the observed data but tempered by fidelity to the observed values. We obtain this criterion by minimizing posterior loss for a given model and then, for models under consideration, select the one which minimizes this criterion. For a broad range of losses, the criterion emerges approximately as a form partitioned into a goodness-of-fit term and a penalty term. In the context of generalized linear mixed effects models we obtain a penalized deviance criterion comprised of a piece which is a Bayesian deviance measure and a piece which is a penalty for model complexity. We illustrate ...