Introduction to Simulation and MCMC Methods

The purpose of this article is to provide an overview of Monte Carlo methods for generating variates from a target probability distribution that are based on Markov chains. These methods, called Markov chain Monte Carlo (MCMC) methods, are widely used to summarize complicated posterior distributions in Bayesian statistics and econometrics. This article begins with an intuitive explanation of the ideas and concepts that underlie popular algorithms such as the Metropolis-Hastings algorithm and multi-block algorithm. It provides the concept of a source or proposal density, which is used to supply a randomization step or an acceptance condition to determine if the candidate draw should be accepted. It is important to assess the performance of the sampling algorithm to determine the rate of mixing. Finally, this article offers an extensive discussion of marginal likelihood calculation using posterior simulator output.