Marginal Likelihood From the Metropolis–Hastings Output

This article provides a framework for estimating the marginal likelihood for the purpose of Bayesian model comparisons. The approach extends and completes the method presented in Chib (1995) by overcoming the problems associated with the presence of intractable full conditional densities. The proposed method is developed in the context of MCMC chains produced by the Metropolis–Hastings algorithm, whose building blocks are used both for sampling and marginal likelihood estimation, thus economizing on prerun tuning effort and programming. Experiments involving the logit model for binary data, hierarchical random effects model for clustered Gaussian data, Poisson regression model for clustered count data, and the multivariate probit model for correlated binary data, are used to illustrate the performance and implementation of the method. These examples demonstrate that the method is practical and widely applicable.