### Yale University

Department of Statistics

Seminar

#### 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.