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Paper No. 09-19

Download 09-19

JC Mattingly, NS Pillai, AM Stuart

SPDE limits of the random walk Metropolis algorithm in high dimensions

Abstract: Diffusion limits of MCMC methods in high dimensions provide a useful theoretical tool for studying efficiency. In particular they facilitate 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 to real applications. The purpose of this paper is to study diffusion limits for a class of naturally occuring 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 to an infinite dimensional Hilbert space valued SDE (or SPDE) is proved.