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arXiv · 1003.4306

Diffusion limits of the random walk Metropolis algorithm in high dimensions

Jonathan C. Mattingly·Natesh S. Pillai·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.

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Jonathan C. Mattingly, Natesh S. Pillai, Andrew M. Stuart. 2012-10-04. Diffusion limits of the random walk Metropolis algorithm in high dimensions. https://doi.org/10.1214/10-aap754

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