A Dirichlet Form approach to MCMC Optimal Scaling

TL;DR

This paper introduces Dirichlet form techniques to prove optimal scaling results for Metropolis-Hastings MCMC algorithms under weaker conditions, offering explicit infinite-dimensional constructions and convergence proofs.

Contribution

It presents a novel Dirichlet form approach that simplifies proofs and relaxes regularity assumptions for MCMC optimal scaling analysis.

Findings

Weaker regularity conditions for optimal scaling.

Explicit infinite-dimensional constructions.

Proofs of weak convergence to infinite-dimensional distributions.

Abstract

This paper develops the use of Dirichlet forms to deliver proofs of optimal scaling results for Markov chain Monte Carlo algorithms (specifically, Metropolis-Hastings random walk samplers) under regularity conditions which are substantially weaker than those required by the original approach (based on the use of infinitesimal generators). The Dirichlet form methods have the added advantage of providing an explicit construction of the underlying infinite-dimensional context. In particular, this enables us directly to establish weak convergence to the relevant infinite-dimensional distributions.