The Annals of Applied Probability

Error bounds for Metropolis–Hastings algorithms applied to perturbations of Gaussian measures in high dimensions

Andreas Eberle

Full-text: Open access

Abstract

The Metropolis-adjusted Langevin algorithm (MALA) is a Metropolis–Hastings method for approximate sampling from continuous distributions. We derive upper bounds for the contraction rate in Kantorovich–Rubinstein–Wasserstein distance of the MALA chain with semi-implicit Euler proposals applied to log-concave probability measures that have a density w.r.t. a Gaussian reference measure. For sufficiently “regular” densities, the estimates are dimension-independent, and they hold for sufficiently small step sizes $h$ that do not depend on the dimension either. In the limit $h\downarrow0$, the bounds approach the known optimal contraction rates for overdamped Langevin diffusions in a convex potential.

A similar approach also applies to Metropolis–Hastings chains with Ornstein–Uhlenbeck proposals. In this case, the resulting estimates are still independent of the dimension but less optimal, reflecting the fact that MALA is a higher order approximation of the diffusion limit than Metropolis–Hastings with Ornstein–Uhlenbeck proposals.

Article information

Source
Ann. Appl. Probab., Volume 24, Number 1 (2014), 337-377.

Dates
First available in Project Euclid: 9 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1389278728

Digital Object Identifier
doi:10.1214/13-AAP926

Mathematical Reviews number (MathSciNet)
MR3161650

Zentralblatt MATH identifier
1296.60195

Subjects
Primary: 60J22: Computational methods in Markov chains [See also 65C40]
Secondary: 60J05: Discrete-time Markov processes on general state spaces 65C40: Computational Markov chains 65C05: Monte Carlo methods

Keywords
Metropolis algorithm Markov chain Monte Carlo Langevin diffusion Euler scheme coupling contractivity of Markov kernels

Citation

Eberle, Andreas. Error bounds for Metropolis–Hastings algorithms applied to perturbations of Gaussian measures in high dimensions. Ann. Appl. Probab. 24 (2014), no. 1, 337--377. doi:10.1214/13-AAP926. https://projecteuclid.org/euclid.aoap/1389278728


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