Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T23:52:16.506Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

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

References

[1] Bédard, M. (2007). Weak convergence of Metropolis algorithms for non-i.i.d. target distributions. Ann. Appl. Prob. 17, 1222–1244. CrossRefGoogle Scholar
[2] Bédard, M. (2008). Efficient sampling using Metropolis algorithms: applications of optimal scaling results. J. Comput. Graph. Statist. 17, 312–332. CrossRefGoogle Scholar
[3] Bédard, M. (2008). Optimal acceptance rates for Metropolis algorithms: moving beyond 0.234. Stoch. Process. Appl. 118, 2198–2222. Google Scholar
[4] Bédard, M. and Rosenthal, J. S. (2008). Optimal scaling of Metropolis algorithms: heading toward general target distributions. Canad. J. Statist. 36, 483–503. CrossRefGoogle Scholar
[5] Beskos, A., Roberts, G. and Stuart, A. (2009). Optimal scalings for local Metropolis–Hastings chains on nonproduct targets in high dimensions. Ann. Appl. Prob. 19, 863–898. Google Scholar
[6] Breyer, L. A. and Roberts, G. O. (2000). From Metropolis to diffusions: Gibbs states and optimal scaling. Stoch. Process. Appl. 90, 181–206. Google Scholar
[7] Cherny, A. S. and Engelbert, H.-J. (2005). Singular Stochastic Differential Equations (Lecture Notes Math. 1858). Springer, Berlin. CrossRefGoogle Scholar
[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. Google Scholar
[9] Jourdain, B., Lelièvre, T. and Miasojedow, B. (2015). Optimal scaling for the transient phase of the random walk Metropolis algorithm: the mean-field limit. Ann. Appl. Prob. 25, 2263–2300. CrossRefGoogle Scholar
[10] Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer, New York. CrossRefGoogle Scholar
[11] Metropolis, N. et al. (1953). Equations of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1091. Google Scholar
[12] Neal, P., Roberts, G. and Yuen, W. K. (2012). Optimal scaling of random walk Metropolis algorithms with discontinuous target densities. Ann. Appl. Prob. 22, 1880–1927. Google Scholar
[13] Petrov, V. V. (1995). Limit Theorems of Probability Theory (Oxford Stud. Prob. 4). Oxford University Press. Google Scholar
[14] Roberts, G. O. and Rosenthal, J. S. (2014). Minimising MCMC variance via diffusion limits, with an application to simulated tempering. Ann. Appl. Prob. 24, 131–149. Google Scholar
[15] Roberts, G. O., Gelman, A. and Gilks, W. R. (1997). Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann. Appl. Prob. 7, 110–120. Google Scholar
[16] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes and Martingales, Vol. 2, Itô Calculus. Cambridge University Press. Google Scholar
[17] Sherlock, C. and Roberts, G. (2009). Optimal scaling of the random walk Metropolis on elliptically symmetric unimodal targets. Bernoulli 15, 774–798. CrossRefGoogle Scholar
Supplementary material: PDF

Durmus et al. supplementary material

Durmus et al. supplementary material 1

Download Durmus et al. supplementary material(PDF)
PDF 270.9 KB