Basic Monte Carlo Methods for Estimating Posterior Quantities

Abstract

The fundamental goal of Bayesian computation is to compute posterior quantities of interest. When the posterior distribution π(¸|D) is high dimensional, one is typically driven to do multidimensional integration to evaluate marginal posterior summaries of the parameters. When a Bayesian model is complicated, analytical or exact numerical evaluation may fail to solve this computational problem. In this regard, the Monte Carlo (MC) method, in particular, the Markov chain Monte Carlo (MCMC) approach, may naturally serve as an alternative solution. Many of the MCMC sampling algorithms discussed in Chapter 2 can be applied here to generate an MCMC sample from the posterior distribution. The main objective of this chapter is to provide a comprehensive treatment of how to use an MCMC sample to obtain MC estimates of posterior quantities. In particular, we will present an overview of several basic MC methods for computing posterior quantities as well as assessing simulation accuracy of MC estimates. In addition, several related issues, including how to obtain more efficient MC estimates, such as the weighted MC estimates, and how to control simulation errors when computing MC estimates, will be addressed.