Abstract: We consider families of semiparametric Bayesian models based on Dirichlet process mixtures, indexed by a multidimensional hyperparameter that includes the precision parameter. We wish to select the hyperparameter by considering Bayes factors. Our approach involves distinguishing some arbitrary value of the hyperparameter, and estimating the Bayes factor for the model indexed by the hyperparameter vs. the model indexed by the distinguished point, as the hyperparameter varies. The approach requires us to select a finite number of hyperparameter values, and for each get Markov chain Monte Carlo samples from the posterior distribution corresponding to the model indexed by that hyperparameter value. Implementation of the approach relies on a likelihood ratio formula for Dirichlet process models. Because we may view parametric models as limiting cases where the precision hyperparameter is infinite, the method also enables us to decide whether or not to use a semiparametric or an entirely parametric model. We illustrate the methodology through two detailed examples involving meta-analysis.
Key words and phrases: Bayes factors, Dirichlet processes, likelihood ratio formula, Markov chain Monte Carlo, model selection.