Open Access
June 2012 Diffusion limits of the random walk Metropolis algorithm in high dimensions
Jonathan C. Mattingly, Natesh S. Pillai, Andrew M. Stuart
Ann. Appl. Probab. 22(3): 881-930 (June 2012). DOI: 10.1214/10-AAP754


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.


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Jonathan C. Mattingly. Natesh S. Pillai. Andrew M. Stuart. "Diffusion limits of the random walk Metropolis algorithm in high dimensions." Ann. Appl. Probab. 22 (3) 881 - 930, June 2012.


Published: June 2012
First available in Project Euclid: 18 May 2012

zbMATH: 1254.60081
MathSciNet: MR2977981
Digital Object Identifier: 10.1214/10-AAP754

Primary: 60H15 , 60J20 , 60J22 , 65C05 , 65C40

Keywords: Markov chain Monte Carlo , optimal convergence time , scaling limits , stochastic PDEs

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.22 • No. 3 • June 2012
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