Markov Chain Monte Carlo Methods

In Chapter 5, we introduced the use of simulation in Bayesian inference. Rejection sampling is a general method for simulating from an arbitrary posterior distribution, but it can be difficult to set up since it requires the construction of a suitable proposal density. Importance sampling and SIR algorithms are also general-purpose algorithms, but they also require proposal densities that may be difficult to find for high-dimensional problems. In this chapter, we illustrate the use of Markov chain Monte Carlo (MCMC) algorithms in summarizing posterior distributions. Markov chains are introduced in the discrete state space situation in Section 6.2. Through a simple random walk example, we illustrate some of the important properties of a special Markov chain, and we use R to simulate from the chain and move toward the stationary distribution.