The Annals of Applied Probability

Spectral gaps for a Metropolis–Hastings algorithm in infinite dimensions

Martin Hairer, Andrew M. Stuart, and Sebastian J. Vollmer

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Abstract

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 make the chain reversible with respect to the target 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.

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 exhibiting a 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 nonasymptotic bounds on the mean square error, all dimension independent. In contrast we show that the spectral gap of the RWM algorithm applied to the reference measures degenerates as the dimension tends to infinity.

Article information

Source
Ann. Appl. Probab., Volume 24, Number 6 (2014), 2455-2490.

Dates
First available in Project Euclid: 26 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1409058037

Digital Object Identifier
doi:10.1214/13-AAP982

Mathematical Reviews number (MathSciNet)
MR3262508

Zentralblatt MATH identifier
1307.65002

Subjects
Primary: 65C40: Computational Markov chains 60B10: Convergence of probability measures 60J05: Discrete-time Markov processes on general state spaces 60J22: Computational methods in Markov chains [See also 65C40]

Keywords
Wasserstein spectral gaps $L^{2}$-spectral gaps Markov chain Monte Carlo in infinite dimensions weak Harris theorem random walk Metropolis

Citation

Hairer, Martin; Stuart, Andrew M.; Vollmer, Sebastian J. Spectral gaps for a Metropolis–Hastings algorithm in infinite dimensions. Ann. Appl. Probab. 24 (2014), no. 6, 2455--2490. doi:10.1214/13-AAP982. https://projecteuclid.org/euclid.aoap/1409058037


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