Yale University
Department of Statistics

Monday, November 6, 1995
Richard Tweedie
Department of Statistics, Colorado State University

"Geometric convergence rates for Markov chain Monte Carlo algorithms"

Seminar to be held in Room 107, 24 Hillhouse Ave at 4:15 pm

Markov chain Monte Carlo (MCMC) algorithms are becoming increasingly popular in allowing simulation in Bayesian statistical methods, and in a wide number of other areas. In this talk we describe applications of recent results on geometric ergodicity of Markov chains to MCMC algorithms, and in particular to Hastings and Metropolis algorithms. As a corollary of these conditions we can ensure that CLT's hold for natural estimators of moments of appropriate orders. For so-called Metropolis algorithms based on random walk candidate distributions in one dimension, we find necessary and sufficient conditions for convergence at a geometric rate to a prescribed distribution \pi: in essence convergence is geometric if and only if \pi has exponential tails. We also find necessary and sufficient conditions for such convergence when the algorithm uses independent candidate distribution. In higher dimensions, we show that geometric convergence properties hold for all distributions with density of the form \pi({x}) = h ({x}) \exp ( {p} ({x})) where h and p are polynomials, with p of degree \geq and satisfying an appropriate ``negative-definiteness'' property. Converse results, showing non-geometric convergence rates for chains where the rejection rate is not bounded from unity, are also given; these show that the negative-definiteness property is not redundant.