Diffusion limits of the random walk Metropolis algorithm in high dimensions

TL;DR

This paper extends the theoretical understanding of the random walk Metropolis algorithm's diffusion limits in high dimensions, especially for measures approximated from infinite-dimensional Hilbert spaces, aiding complexity analysis.

Contribution

It proves diffusion limits for the Metropolis algorithm in infinite-dimensional Hilbert spaces, broadening applicability beyond product-structured measures.

Findings

Diffusion limit established for high-dimensional measures from Hilbert space approximations.

Provides a framework for analyzing computational complexity of MCMC in infinite dimensions.

Enhances understanding of algorithm behavior in complex, high-dimensional settings.

Abstract

Diffusion limits of MCMC methods in high dimensions provide a useful theoretical tool for studying computational complexity. In particular, they lead directly to precise estimates of the number of steps required to explore the target measure, in stationarity, as a function of the dimension of the state space. However, to date such results have mainly been proved for target measures with a product structure, severely limiting their applicability. The purpose of this paper is to study diffusion limits for a class of naturally occurring high-dimensional measures found from the approximation of measures on a Hilbert space which are absolutely continuous with respect to a Gaussian reference measure. The diffusion limit of a random walk Metropolis algorithm to an infinite-dimensional Hilbert space valued SDE (or SPDE) is proved, facilitating understanding of the computational complexity of the algorithm.