The Annals of Applied Probability

Optimal scalings for local Metropolis–Hastings chains on nonproduct targets in high dimensions

Alexandros Beskos, Gareth Roberts, and Andrew Stuart

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Abstract

We investigate local MCMC algorithms, namely the random-walk Metropolis and the Langevin algorithms, and identify the optimal choice of the local step-size as a function of the dimension n of the state space, asymptotically as n→∞. We consider target distributions defined as a change of measure from a product law. Such structures arise, for instance, in inverse problems or Bayesian contexts when a product prior is combined with the likelihood. We state analytical results on the asymptotic behavior of the algorithms under general conditions on the change of measure. Our theory is motivated by applications on conditioned diffusion processes and inverse problems related to the 2D Navier–Stokes equation.

Article information

Source
Ann. Appl. Probab. Volume 19, Number 3 (2009), 863-898.

Dates
First available in Project Euclid: 15 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1245071013

Digital Object Identifier
doi:10.1214/08-AAP563

Mathematical Reviews number (MathSciNet)
MR2537193

Zentralblatt MATH identifier
1172.60328

Subjects
Primary: 60J22: Computational methods in Markov chains [See also 65C40]
Secondary: 65C40: Computational Markov chains

Keywords
Random-walk metropolis Langevin squared-jump-distance Gaussian law on Hilbert space Karhunen–Loève Navier–Stokes PDE diffusion

Citation

Beskos, Alexandros; Roberts, Gareth; Stuart, Andrew. Optimal scalings for local Metropolis–Hastings chains on nonproduct targets in high dimensions. Ann. Appl. Probab. 19 (2009), no. 3, 863--898. doi:10.1214/08-AAP563. https://projecteuclid.org/euclid.aoap/1245071013


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