Spectral Gaps for a Metropolis-Hastings Algorithm in Infinite Dimensions

Details Spectral Gaps for a Metropolis-Hastings Algorithm in Infinite Dimensions Spectral Gaps for a Metropolis-Hastings Algorithm in Infinite Dimensions

2011-12-06

arxiv.org/abs/1112.1392v1

Mathematics

Probability

38 pages, 1 figure

Scientific paper

We study the problem of sampling high and infinite dimensional target measures arising in applications such as conditioned diffusions and inverse problems. We focus on those that arise from approximating measures on Hilbert spaces defined via a density with respect to a Gaussian reference measure. We consider the Metropolis-Hastings algorithm that adds an accept-reject mechanism to a Markov chain proposal in order to have the target measure as an ergodic invariant measure. We focus on cases where the proposal is either a Gaussian random walk (RWM) with covariance equal to that of the reference measure or an Ornstein-Uhlenbeck proposal (pCN) for which the reference measure is invariant. Abstract Previous results in terms of scaling and diffusion limits suggested that the pCN has a convergence rate that is independent of the dimension while the RWM method has undesirable dimension-dependent behaviour. We confirm this claim by showing dimension-independent Wasserstein spectral gap for pCN algorithm for a large class of target measures. In our setting this Wasserstein spectral gap implies an L^{2} -spectral gap. We use both spectral gaps to show that the ergodic average satisfies a strong law of large numbers, the central limit theorem and non-asymptotic bounds on the mean square error, all dimension independent. In contrast we show that the RWM algorithm applied to the reference measures degenerates as the dimension tends to infinity.

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