Marginal likelihoods via power posteriors

Model choice plays an increasingly important role in Statistics. From a Bayesian perspective a crucial goal is to compute the marginal likelihood of the data for a given model. This however is typically a difficult task since it amounts to integrating over all model parameters. The aim of this paper is to illustrate how this may be achieved using ideas from thermodynamic integration or path sampling. We show how the marginal likelihood can be computed via MCMC methods on modified posterior distributions for each model. This then allows Bayes factors or posterior model probabilities to be calculated. We show that this approach requires very little tuning, and is straightforward to implement. The new methodology is illustrated in a variety of challenging statistical settings. This work represents joint collaboration with Tony Pettitt, QUT, Brisbane.