Estimation of Large Families of Bayes Factors from Markov Chain Output

We consider situations in Bayesian analysis where the prior is indexed by a hyperparameter taking on a continuum of values. We distinguish some arbitrary value of the hyperparameter, and consider the problem of estimating the Bayes factor for the model indexed by the hyperparameter vs. the model specified by the distinguished point, as the hyperparameter varies. We assume that we have Markov chain output from the posterior for a finite number of the priors, and develop a method for efficiently computing estimates of the entire family of Bayes factors. As an application of the ideas, we consider some commonly used hierarchical Bayesian models and show that the parametric assumptions in these models can be recast as assumptions regarding the prior. Therefore, our method can be used as a model selection criterion in a Bayesian framework. We illustrate our methodology through a detailed example involving Bayesian model selection.