Many clever methods have been devised for constructing and sampling from arbitrary posterior distributions. Markov chain simulation (also called Markov chain Monte Carlo, or MCMC) is a general method based on drawing values of θ from approximate distributions and then correcting those draws to better approximate the target posterior distribution, p(θ|y). The sampling is done sequentially, with the distribution of the sampled draws depending on the last value drawn; hence, the draws form a Markov chain. (As defined in probability theory, a Markov chain is a sequence of random variables θ1, θ2, . . ., for which, for any t, the distribution of θt given all previous θ’s depends only on the most recent value, θt−1.) The key to the method’s success, however, is not the Markov property but rather that the approximate distributions are improved at each step in the simulation, in the sense of converging to the target distribution. As we shall see in Section 11.2, the Markov property is helpful in proving this convergence.