DIFFUSION LIMITS OF THE RANDOM WALK METROPOLIS ALGORITHM IN HIGH DIMENSIONS

• 成果类型: Article
• 署名作者: Mattingly, Jonathan C.; Pillai, Natesh S.; Stuart, Andrew M.
• 署名单位: Duke University; Duke University; Harvard University; University of Warwick
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/10-AAP754
• 发表日期: 2012
• 页码: 881-930
• 关键词: INVERSE PROBLEMS; approximation; chains; spdes

摘要

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.