MARGINAL LIKELIHOOD FROM THE GIBBS OUTPUT

In the context of Bayes estimation via Gibbs sampling, with or without data augmentation, a simple approach is developed for computing the m arginal density of the sample data (marginal likelihood) given paramet er draws from the posterior distribution. Consequently, Bayes factors for model comparisons can be routinely computed as a by-product of the simulation. Hitherto, this calculation has proved extremely challengi ng. Our approach exploits the fact that the marginal density can be ex pressed as the prior times the likelihood function over the posterior density. This simple identity holds for any parameter value. An estima te of the posterior density is shown to be available if all complete c onditional densities used in the Gibbs sampler have closed-form expres sions. To improve accuracy, the posterior density is estimated at a hi gh density point, and the numerical standard error of resulting estima te is derived. The ideas are applied to probit regression and finite m ixture models.