Bayesian Computation: From Posterior Densities to Bayes Factors, Marginal Likelihoods, and Posterior Model Probabilities

Abstract

Publisher Summary This chapter deals with the Bayesian computation. In Bayesian inference, a joint posterior distribution is available through the likelihood function and a prior distribution. One way to summarize a posterior distribution is to calculate and display marginal posterior densities because the marginal posterior densities provide complete information about parameters of interest. This chapter summarizes the current state of the art in the area of estimating marginal and full posterior densities and various applications of the posterior density estimation in computing Bayes factors, marginal likelihoods, and posterior model probabilities. This chapter provides a most updated overview on various Monte Carlo methods for computing marginal or full posterior densities, including the kernel density estimation, the conditional marginal density estimator (CMDE) the importance weighted marginal density estimation (IWMDE), the Gibbs stopper approach, and an approach based on the Metropolis–Hastings output. Finally, the development of an efficient and practically useful Monte Carlo method for this problem is a very challenging and important future project.