Micro-local analysis for the Metropolis algorithm

Abstract

We prove sharp rates of convergence to stationarity for a simple case of the Metropolis algorithm: the placement of a single disc of radius h randomly into the interval [−1 − h, 1 + h], with h > 0 small. We find good approximations for the top eigenvalues and eigenvectors. The analysis gives rigorous proof for the careful numerical work in ([DN04]). The micro-local techniques employed offer promise for the analysis of more realistic problems.

Tunnel effect for semiclassical random walks

“…More recently, Diaconis-Lebeau obtained first results on discrete time processes on continuous state space [5]. This approach was then further developed in [6] to get convergence results on the Metropolis algorithm on bounded domains of the Euclidean space.…”

Another Conversation with Persi Diaconis

“…I worked very hard for five years with wonderful analysts. We wrote papers [21,22] in the best math journals. But our theorems are basically useless as regards the real problem.…”

The Markov chain Monte Carlo revolution

“…The work is fairly technical but the big picture is fairly stable. It holds for natural walks on compact Riemannian manifolds [59] and in the detailed analysis of the one-dimensional hard disc problem [24,27].…”

“…The multiplier m(x) leads to a continuous spectrum. One of our discoveries [24,25,59] is that for many chains, this can be side-stepped and the basic outline above can be pushed through to give sharp useful bounds.…”

“…Modern PDE is an enormous subject with many more tools and ideas. Some of these, for example, the calculus of pseudo-differential operators and micro-local techniques, are just starting to make inroads into Markov chain convergence [24,25,59].…”