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

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,h sigma), 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.

Automated summary

This study investigates the impact of proposal distributions on the ergodicity of the Metropolis-Hastings method, showing that suitable choices can alter the ergodic properties of the algorithm. It is found that allowing the proposal variance to grow unboundedly in the tails of heavy-tailed distributions can establish geometric ergodicity, but the growth rate needs to be carefully controlled to avoid high rejection rates. Furthermore, a judicious choice of proposal distribution can lead to geometric ergodicity in scenarios where probability concentrates on narrower tails, which is not the case for the Random Walk Metropolis.