Bayesian model choice based on Monte Carlo estimates of posterior model probabilities

A range of approximate methods have been proposed for model choice based on Bayesian principles, given the problems involved in multiple integration in multi-parameter problems. Formal Bayesian model assessment is based on prior model probabilities P(M=j)P(M=j) and posterior model probabilities P(M=j|Y)P(M=j|Y) after observing the data. An approach is outlined here that produces posterior model probabilities and hence Bayes factor estimates but not marginal likelihoods. It uses a Monte Carlo approximation based on independent MCMC sampling of two or more different models. While parallel sampling of the models is not necessary, such a form of sampling facilitates model averaging and assessing the impact of individual observations on the overall estimated Bayes factor. Three worked examples used before in model choice studies illustrate application of the method.