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Diffusion limits of the random walk Metropolis algorithm in high dimensions
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Mattingly, Jonathan C., Pillai, Natesh S. and Stuart, A. M.. (2012) Diffusion limits of the random walk Metropolis algorithm in high dimensions. The Annals of Applied Probability, Vol.22 (No.3). pp. 881930. ISSN 10505164

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Official URL: http://dx.doi.org/10.1214/10AAP754
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 highdimensional 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 infinitedimensional Hilbert space valued SDE (or SPDE) is proved, facilitating understanding of the computational complexity of the algorithm.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Mathematics 
Library of Congress Subject Headings (LCSH):  Diffusion processes, Monte Carlo method, Random walks (Mathematics), Hilbert space 
Journal or Publication Title:  The Annals of Applied Probability 
Publisher:  Institute of Mathematical Statistics 
ISSN:  10505164 
Official Date:  June 2012 
Volume:  Vol.22 
Number:  No.3 
Number of Pages:  50 
Page Range:  pp. 881930 
Identification Number:  10.1214/10AAP754 
Status:  Peer Reviewed 
Publication Status:  Published 
Access rights to Published version:  Restricted or Subscription Access 
Funder:  National Science Foundation (U.S.) (NSF), Engineering and Physical Sciences Research Council (EPSRC), European Research Council (ERC) 
Grant number:  DMS0449910 (NSF), DMS0854879 (NSF) 
References:  [1] BÉDARD, M. (2007).Weak convergence of Metropolis algorithms for noni.i.d. target distributions. 
URI:  http://wrap.warwick.ac.uk/id/eprint/48174 
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