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

  • Alain Durmus (a1), Sylvain Le Corff (a2), Eric Moulines (a3) and Gareth O. Roberts (a4)
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.

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

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References
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[8] Durmus, A., Le Corff, S., Moulines, E. and Roberts, G. O. (2017). Optimal scaling of the random walk Metropolis algorithm under L p mean differentiability. Supplementary material. Available at http://doi.org/10.1017/jpr.2017.61.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
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