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

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 $n\to \infty$. 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.

arXiv commentPublished in at http://dx.doi.org/10.1214/08-AAP563 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Optimal scaling for partially updating MCMC algorithms
• Optimal scaling of the random walk Metropolis on elliptically symmetric unimodal targets
• Weak convergence of Metropolis algorithms for non-i.i.d. target distributions
• Optimal Scaling of Mala for Nonlinear Regression

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