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

Details 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

2009-08-06

arxiv.org/abs/0908.0865v1

Annals of Applied Probability 2009, Vol. 19, No. 3, 863-898

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/08-AAP563 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/08-AAP563

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.

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