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

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

Alexandros Beskos, Gareth Roberts, and Andrew Stuart

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

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.

Primary Subjects: 60J22

Secondary Subjects: 65C40

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

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Permanent link to this document: http://projecteuclid.org/euclid.aoap/1245071013
Digital Object Identifier: doi:10.1214/08-AAP563
Mathematical Reviews number (MathSciNet): MR2537193

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