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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. 881-930. doi:10.1214/10-AAP754 ISSN 1050-5164.
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Official URL: http://dx.doi.org/10.1214/10-AAP754
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: | Journal Article | ||||
|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||
| Divisions: | Faculty of Science, Engineering and Medicine > 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: | 1050-5164 | ||||
| Official Date: | June 2012 | ||||
| Dates: |
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| Volume: | Vol.22 | ||||
| Number: | No.3 | ||||
| Number of Pages: | 50 | ||||
| Page Range: | pp. 881-930 | ||||
| DOI: | 10.1214/10-AAP754 | ||||
| Status: | Peer Reviewed | ||||
| Publication Status: | Published | ||||
| Access rights to Published version: | Restricted or Subscription Access | ||||
| Date of first compliant deposit: | 22 December 2015 | ||||
| Date of first compliant Open Access: | 22 December 2015 | ||||
| Funder: | National Science Foundation (U.S.) (NSF), Engineering and Physical Sciences Research Council (EPSRC), European Research Council (ERC) | ||||
| Grant number: | DMS-04-49910 (NSF), DMS-08-54879 (NSF) |
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