The Annals of Applied Probability

Diffusion limits of the random walk Metropolis algorithm in high dimensions

Jonathan C. Mattingly, Natesh S. Pillai, and Andrew M. Stuart

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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.

Article information

Source
Ann. Appl. Probab., Volume 22, Number 3 (2012), 881-930.

Dates
First available in Project Euclid: 18 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1337347534

Digital Object Identifier
doi:10.1214/10-AAP754

Mathematical Reviews number (MathSciNet)
MR2977981

Zentralblatt MATH identifier
1254.60081

Subjects
Primary: 60J22: Computational methods in Markov chains [See also 65C40] 60H15: Stochastic partial differential equations [See also 35R60] 65C05: Monte Carlo methods 65C40: Computational Markov chains 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]

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

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. https://projecteuclid.org/euclid.aoap/1337347534


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