THE BASIS OF an MCMC algorithm is the construction of a transition kernel (see Section 6.3), p(x, y), that has an invariant density equal to the target density. Given such a kernel, the process can be started at x0 to yield a draw x1 from p(x0, x1), x2 from p(x1, x2), …, and xG from p(xG−1, xG), where G is the desired number of simulations. After a transient period, the distribution of the xg is approximately equal to the target distribution. The question is how to find a kernel that has the target as its invariant distribution. It is remarkable that there is a general principle for finding such kernels, the Metropolis–Hastings (MH) algorithm. We first discuss a special case – the Gibbs algorithm or Gibbs sampler – and then explain a more general version of the MH algorithm.

It is important to distinguish between the number of simulated values G and the number of observations n in the sample of data that is being analyzed. The former may be made very large – the only restriction comes from computer time and capacity, but the number of observations is fixed at the time the data are collected. Larger values of G lead to more accurate approximations. MCMC algorithms provide an approximation to the exact posterior distribution of a parameter; that is, they approximate the posterior distribution of the parameters, taking the number of observations to be fixed at n.