Mathematical Foundations of the Markov Chain Monte Carlo Method

The Markov chain Monte Carlo (MCMC) method exploits the idea that information about a set of combinatorial objects may be obtained by performing an appropriately deened random walk on those objects. In the area of statistical physics, MCMC algorithms have been in use for many years for the purpose of estimating various quantities of physical interest, often expectations of random variables on \conngurations" of a statistical model. The running time of MCMC algorithms depends on the rate at which the random walk converges to equilibrium; only when a condition of near-equilibrium has been achieved can the algorithm discover what \typical" objects are like. In the past decade or so, it has become possible to derive a priori bounds on the rate of convergence to equilibrium of random walks underlying MCMC algorithms of practical interest. In cases where a priori bounds cannot be derived, it may still be possible to conduct rigorously grounded experiments. Many of the main ideas and techniques are set out here, with the recent developments being discussed at greater length. Mathematical foundations of MCMC 1 Acknowledgements Two colleagues merit special acknowledgement for their contribution to this survey. Some of the results presented here were products of an extended period of collaboration with Alistair Sinclair, and in describing them I have freely plundered and adapted material from our joint articles. The experiment described in Section 7.2 was jointly undertaken with Vivek Gore, and is published here for the rst time. I also thank an anonymous referee for carefully reading and providing helpful comments on a draft of this chapter.