Introduction to Markov Chain Monte Carlo Methods and Their Actuarial Applications

This paper introduces the readers of the Proceedings to an important class of computer based simulation techniques known as Markov chain Monte Carlo (MCMC) methods. General properties characterizing these methods will be discussed, but the main emphasis will be placed on one MCMC method known as the Gibbs sampler. The Gibbs sampler permits one to simulate realizations from complicated stochastic models in high dimensions by making use of the model’s associated full conditional distributions, which will generally have a much simpler and more manageable form. In its most extreme version, the Gibbs sampler reduces the analysis of a complicated multivariate stochastic model to the consideration of that model’s associated univariate full conditional distributions. In this paper, the Gibbs sampler will be illustrated with four examples. The first three of these examples serve as rather elementary yet instructive applications of the Gibbs sampler. The fourth example describes a reasonably sophisticated application of the Gibbs sampler in the important arena of credibility for classification ratemaking via hierarchical models, and involves the Bayesian prediction of frequency counts in workers compensation insurance.