Markov chains for exploring posterior distributions " by LukeTierney

We congratulate Luke Tierney for this paper, which even before its appearance has done a valuable service in clarifying both theory and practice in this important area. For example, the discussion of combining strategies in Section 2.4 helped researchers break away from pure Gibbs sampling in 1991; it was, for example, part of the reasoning that lead to the \Metropolis-coupled" scheme of Geyer (1991) mentioned at the end of Section 2.3.3. Harris Recurrence. The discussion of Harris recurrence in Section 3.1 has been very helpful. Harris recurrence essentially says that there is no measure-theoretic pathology. Hence it seems that it should always be present in any sampler that can be run on a computer, even in the continuous approximation that treats the computer's real numbers as if they were the analyst's real numbers. The main point of Harris recurrence is that asymptotics do not then depend on the starting distribution because of the \split chain" construction (Nummelin, 1984, Proposition 4.8), something that was pointed out to us by Tierney. More precisely, the argument of the paragraph at the bottom of the page 135 in Nummelin (1984) shows that the rst regeneration time is a. s. nite (for any initial distribution) hence the sum up to the rst regeneration time is also a. s. nite and negligible when divided by p n. Typically irreducibility implies Harris recurrence. Corollaries 1 and 2 of Tier-ney's paper show this for Gibbs samplers and non-hybrid Metropolis-Hastings algorithms , respectively. The following theorem shows this for the other important case, variable-at-a-time Metropolis-Hastings algorithms. These are hybrid algorithms in the terminology of Tierney's Section 2.4 that update one coordinate at a time using a Metropolis-Hastings update as in the original example of Metropolis et al. (1953). Consider a variable-at-a-time Metropolis-Hastings algorithm on a subset of R d. We are trying to simulate from the distribution with density proportional to h(x) = h(x 1 ;. .. ; x d) with respect to Lebesgue measure on R d. The sampler procedes through the variables in order updating one at a time. It updates the ith variable by proposing a new value x i from a univariate density with respect to Lebesgue measure q i (x; x i) that depends on the current position x = (x 1 ;. .. ; x d) and accepts the proposal moving to the new position