Geometric Ergodicity of the Random Walk Metropolis with Position-Dependent Proposal Covariance

Samuel Livingstone

doi:10.3390/math9040341

Abstract

We consider a Metropolis–Hastings method with proposal N(x,hG(x)−1), where x is the current state, and study its ergodicity properties. We show that suitable choices of G(x) can change these ergodicity properties compared to the Random Walk Metropolis case N(x,hΣ), either for better or worse. We find that if the proposal variance is allowed to grow unboundedly in the tails of the distribution then geometric ergodicity can be established when the target distribution for the algorithm has tails that are heavier than exponential, in contrast to the Random Walk Metropolis case, but that the growth rate must be carefully controlled to prevent the rejection rate approaching unity. We also illustrate that a judicious choice of G(x) can result in a geometrically ergodic chain when probability concentrates on an ever narrower ridge in the tails, something that is again not true for the Random Walk Metropolis.

Highlights

Markov chain Monte Carlo (MCMC) methods are techniques for estimating expectations with respect to some probability measure π (·), which need not be normalised
A popular form of MCMC is the Metropolis–Hastings algorithm [1,2], where at each time step a ‘proposed’ move is drawn from some candidate distribution, and accepted with some probability, otherwise the chain stays at the current point
We begin with a result that emphasizes that a growing variance is a necessary requirement for geometric ergodicity in the heavy-tailed case

Summary

Introduction

Markov chain Monte Carlo (MCMC) methods are techniques for estimating expectations with respect to some probability measure π (·), which need not be normalised. Interest lies in finding choices of candidate distribution that will produce sensible estimators for expectations with respect to π (·) The quality of these estimators can be assessed in many different ways, but a common approach is to understand conditions on π (·) that will result in a chain which converges to its limiting distribution at a geometric rate. If such a rate can be established, a Central

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Conclusion