Model choice is a fundamental problem in data analysis. With interest in hierarchical models which typically arise as Bayesian specifications, we confine ourselves to Bayesian model choice criteria. If Y denotes the observed data and T(Y) is the criterion, our goal is to calibrate T(Y) in order to assess how large or small it is under a given model. In particular, if we have the distribution of T(Y), then we can compute any probabilities or determine any quantities of interest.
Apart from very special cases, analytic development of such distributions is intractable. Standard analytic approximations may be inapplicable if usual random effects are introduced at the various modeling levels. Indeed, calculation of T(Y) itself is often difficult enough.
We suggest a generic simulation-intensive approach for obtaining the distribution of T(Y) to arbitrary accuracy. We focus on various Bayes factors, e.g., the usual Bayes factor, the posterior Bayes factor and the pseudo-Bayes factor. We illustrate with a binomial regression example.