Some "Simple" Theory of Markov Chain Monte Carlo1

It is sometimes the case that one has a “known” but unwieldy (joint) distribution for a large dimensional random vector Y , and would like to know something about the distribution (like, for example, the marginal distribution of Y1). While in theory it might be straightforward to compute the quantity of interest, in practice the necessary computation can be impossible. With the advent of cheap computing, given appropriate theory the possibility exists of simulating a sample of n realizations of Y and computing an empirical version of the quantity of interest to approximate the (intractable) theoretical one. This possibility is especially important to Bayesians, where X|θ ∼ Pθ and θ ∼ G produce a joint distribution for (X, θ) and then (in theory) a posterior distribution G(θ|X), the object of primary Bayesian interest. Now (in the usual kind of notation) g(θ|X = x) ∝ fθ(x)g(θ)