Diffusion limits of the random walk Metropolis algorithm in high dimensions

Journal Article

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.

Full Text

Duke Authors

Cited Authors

  • Mattingly, JC; Pillai, NS; Stuart, AM

Published In

  • Annals of Applied Probability

Volume / Issue

  • 22 / 3

Start / End Page

  • 881 - 930

Digital Object Identifier (DOI)

  • 10.1214/10-AAP754