Optimal scaling of the Random Walk Metropolis algorithm under Lp mean differentiability

Alain Durmus (1) , Sylvain Le Corff (2) , Éric Moulines (3, 4) , Gareth O. O. Roberts (5)

1 LTCI - Laboratoire Traitement et Communication de l'Information
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay
3 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
4 XPOP - Modélisation en pharmacologie de population
5 Department of Statistics [Warwick]

This paper considers the optimal scaling problem for high-dimensional random walk Metropolis algorithms for densities which are differentiable in Lp mean but which may be irregular at some points (like the Laplace density for example) and/or are 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 infinity. Because the log-density might be non-differentiable, 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 [6]. This result has important practical implications for the use of random walk Metropolis algorithms in Bayesian frameworks based on sparsity inducing priors.

Mots clés: Random walk Metropolis; Optimal scaling; Markov chain Monte Carlo; Lp mean differentiability

Domaines: Probabilités [math.PR]; Statistiques [math.ST]

Alain Durmus, Sylvain Le Corff, Éric Moulines, Gareth O. O. Roberts. Optimal scaling of the Random Walk Metropolis algorithm under Lp mean differentiability. Journal of Applied Probability, 2017, 54 (4), pp.1233 -1260. ⟨10.1017/jpr.2017.61⟩. ⟨hal-01298922⟩