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Optimal scaling of the random walk Metropolis algorithm under L p mean differentiability

Part of: Probabilistic methods, simulation and stochastic differential equations Limit theorems

Published online by Cambridge University Press:  30 November 2017

Alain Durmus ,
Sylvain Le Corff ,
Eric Moulines  and
Gareth O. Roberts
Alain Durmus*
Affiliation:
LTCI, CNRS and Télécom ParisTech
Sylvain Le Corff*
Affiliation:
Université Paris-Sud, CNRS and Université Paris-Saclay
Eric Moulines*
Affiliation:
Ecole Polytechnique
Gareth O. Roberts*
Affiliation:
University of Warwick
*
* Postal address: Télécom ParisTech, 46 rue Barrault, 75634 Paris, France. Email address: alain.durmus@telecom-paristech.fr
** Postal address: Département de Mathématiques, Université Paris-Sud, 91405 Orsay Cedex, France. Email address: sylvain.lecorff@math.u-psud.fr
*** Postal address: Centre de Mathématiques Appliquées, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France. Email address: eric.moulines@polytechnique.edu
**** Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Email address: gareth.o.roberts@warwick.ac.uk
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Abstract

In this paper we consider the optimal scaling of high-dimensional random walk Metropolis algorithms for densities differentiable in the L p mean but which may be irregular at some points (such as the Laplace density, for example) and/or supported on an interval. Our main result is the weak convergence of the Markov chain (appropriately rescaled in time and space) to a Langevin diffusion process as the dimension d goes to ∞. As the log-density might be nondifferentiable, the limiting diffusion could be singular. The scaling limit is established under assumptions which are much weaker than the one used in the original derivation of Roberts et al. (1997). This result has important practical implications for the use of random walk Metropolis algorithms in Bayesian frameworks based on sparsity inducing priors.

Keywords

Random walk MetropolisMarkov chain Monte Carlooptimal scalingL p mean differentiability

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 60F05: Central limit and other weak theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 4 , December 2017 , pp. 1233 - 1260
Copyright
Copyright © Applied Probability Trust 2017 

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Footnotes

The supplementary material for this article can be found at http://doi.org/10.1017/jpr.2017.61.

References

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