Optimal scaling for the transient phase of the random walk Metropolis algorithm: The mean-field limit

Abstract

We consider the random walk Metropolis algorithm on ${\mathbb{R}}^n$ with Gaussian proposals, and when the target probability measure is the $n$-fold product of a one-dimensional law. In the limit $n\to \infty$, it is well known (see [Ann. Appl. Probab. 7 (1997) 110-120]) that, when the variance of the proposal scales inversely proportional to the dimension $n$ whereas time is accelerated by the factor $n$, a diffusive limit is obtained for each component of the Markov chain if this chain starts at equilibrium. This paper extends this result when the initial distribution is not the target probability measure. Remarking that the interaction between the components of the chain due to the common acceptance/rejection of the proposed moves is of mean-field type, we obtain a propagation of chaos result under the same scaling as in the stationary case. This proves that, in terms of the dimension $n$, the same scaling holds for the transient phase of the Metropolis-Hastings algorithm as near stationarity. The diffusive and mean-field limit of each component is a diffusion process nonlinear in the sense of McKean. This opens the route to new investigations of the optimal choice for the variance of the proposal distribution in order to accelerate convergence to equilibrium (see [Optimal scaling for the transient phase of Metropolis-Hastings algorithms: The longtime behavior Bernoulli (2014) To appear]).

Published at http://dx.doi.org/10.1214/14-AAP1048 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (6)

• Optimal scalings for local Metropolis--Hastings chains on nonproduct targets in high dimensions
• Optimal scaling and diffusion limits for the Langevin algorithm in high dimensions
• Weak convergence of Metropolis algorithms for non-i.i.d. target distributions
• Optimal scaling for the transient phase of Metropolis Hastings algorithms: The longtime behavior
• Optimal scaling of random walk Metropolis algorithms with discontinuous target densities
• Optimal Scaling of Mala for Nonlinear Regression

Cited by in corpus (5)

• Propagation of chaos: a review of models, methods and applications. II. Applications
• Optimal scaling for the transient phase of Metropolis Hastings algorithms: The longtime behavior
• Non-local SPDE limits of spatially-correlated-noise driven spin systems derived to sample a canonical distribution
• Metropolis Monte Carlo sampling: convergence, localization transition and optimality
• Singular relaxation of a random walk in a box with a Metropolis Monte Carlo dynamics