Bayes Factors : A Comparative

The problem of calculating posterior probabilities for a collection of competing models and associated Bayes factors continues to be a formidable challenge for applied Bayesian statisticians. Current approaches that take advantage of modern Markov chain Monte Carlo (MCMC) computing methods include those that attempt to sample over some form of the joint space created by the model indicators and the parameters for each model, others that sample over the model space alone, and still others that attempt to estimate the marginal likelihood of each model directly (since the collection of these is equivalent to the collection of model probabilities themselves). In this paper we review several of these methods, and subsequently compare them in the context of two examples, the rst a simple regression example, and the second a much more challenging hierarchical longitudinal model of the kind often encountered in biostatistical practice. We nd that the joint model-parameter space search methods perform adequately but can be diicult to program and tune, while the marginal likelihood methods are often less troublesome and require less in the way of additional coding. Our results suggest that the latter methods may be most appropriate for practitioners working in many standard model choice settings, while the former remain important for comparing large numbers of models, or models whose parameters cannot be easily updated in relatively few blocks. We caution however that all of the methods we compare require signiicant human and computer eeort, suggesting that less formal Bayesian model choice methods may ooer a more realistic alternative in many cases.