Diffusion Limits of the Random Walk Metropolis Algorithm in High Dimensions

Details Diffusion Limits of the Random Walk Metropolis Algorithm in High Dimensions Diffusion Limits of the Random Walk Metropolis Algorithm in High Dimensions

2010-03-22

arxiv.org/abs/1003.4306v3

Mathematics

Probability

31 pages, updated version. Improved assumptions in Sobolev spaces and clarity in a few places

Scientific paper

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 only 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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