How to Combine Fast Heuristic Markov Chain Monte Carlo with Slow Exact Sampling

Given a probability law π on a set S and a function g : S → R, suppose one wants to estimate the mean ḡ = ∫ g dπ. The Markov Chain Monte Carlo method consists of inventing and simulating a Markov chain with stationary distribution π. Typically one has no a priori bounds on the chain’s mixing time, so even if simulations suggest rapid mixing one cannot infer rigorous confidence intervals for ḡ. But suppose there is also a separate method which (slowly) gives samples exactly from π. Using n exact samples, one could immediately get a confidence interval of length O(n−1/2). But one can do better. Use each exact sample as the initial state of a Markov chain, and run each of these n chains for m steps. We show how to construct confidence intervals which are always valid, and which, if the (unknown) relaxation time of the chain is sufficiently small relative to m/n, have length O(n−1 log n) with high probability. 1 Background Let π be a given probability distribution on a set S. Given a function g : S → R, we want to estimate its mean ḡ := ∫ S g(s)π(ds). As we learn in elementary statistics, one can obtain an estimate for ḡ by taking samples from π and using the sample average g-value as an estimator. But algorithms which sample exactly from π may be prohibitively slow. This is the setting for the Markov chain Monte Carlo (MCMC) method, classical in statistical physics and over the last ten years studied extensively as statistical methodology [4, 7, 9, 12]. In MCMC one designs a Markov chain on state-space S to have stationary distribution π. Then the sample average g-value over a long run of the chain is a heuristic estimator of ḡ. Diagnostics for assessing 1This material is based upon work supported by the National Science Foundation under Grant No. 9970901