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Diffusion limits of the random walk Metropolis algorithm in high dimensions

Mattingly, Jonathan C. and Pillai, Natesh S. and Stuart, Andrew M. (2012) Diffusion limits of the random walk Metropolis algorithm in high dimensions. Annals of Applied Probability, 22 (3). pp. 881-930. ISSN 1050-5164. doi:10.1214/10-AAP754. https://resolver.caltech.edu/CaltechAUTHORS:20160728-154635836

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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.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1214/10-AAP754DOIArticle
http://projecteuclid.org/euclid.aoap/1337347534#infoPublisherArticle
https://arxiv.org/abs/1003.4306arXivDiscussion Paper
Additional Information:© Institute of Mathematical Statistics, 2012. Received March 2010; revised November 2010. [JCM] Supported by NSF Grants DMS-04-49910 and DMS-08-54879. [AMS] Supported by EPSRC and ERC.
Funders:
Funding AgencyGrant Number
NSFDMS-04-49910
NSFDMS-08-54879
Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
European Research Council (ERC)UNSPECIFIED
Subject Keywords:Markov chain Monte Carlo, scaling limits, optimal convergence time, stochastic PDEs.
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Andrew StuartJ93
Issue or Number:3
Classification Code:MSC2010 subject classifications. 60J22, 60H15, 65C05, 65C40, 60J20.
DOI:10.1214/10-AAP754
Record Number:CaltechAUTHORS:20160728-154635836
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20160728-154635836
Official Citation:Mattingly, Jonathan C.; Pillai, Natesh S.; Stuart, Andrew M. Diffusion limits of the random walk Metropolis algorithm in high dimensions. Ann. Appl. Probab. 22 (2012), no. 3, 881--930. doi:10.1214/10-AAP754. http://projecteuclid.org/euclid.aoap/1337347534.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:69294
Collection:CaltechAUTHORS
Deposited By: Linda Taddeo
Deposited On:01 Aug 2016 22:23
Last Modified:11 Nov 2021 04:12

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