Journal
ANNALS OF APPLIED PROBABILITY
Volume 22, Issue 3, Pages 881-930Publisher
INST MATHEMATICAL STATISTICS
DOI: 10.1214/10-AAP754
Keywords
Markov chain Monte Carlo; scaling limits; optimal convergence time; stochastic PDEs
Categories
Funding
- NSF [DMS-04-49910, DMS-08-54879]
- EPSRC
- ERC
- Engineering and Physical Sciences Research Council [EP/F050798/1] Funding Source: researchfish
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1107070] Funding Source: National Science Foundation
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [0854879] Funding Source: National Science Foundation
- EPSRC [EP/F050798/1] Funding Source: UKRI
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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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