Pillai, Natesh S., Stuart, A. M. and Thiéry, Alexandre H. (2011) Optimal scaling and diffusion limits for the Langevin algorithm in high dimensions. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology. Working papers, Vol.2011 (No.8).

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Abstract

The Metropolis-adjusted Langevin (MALA) algorithm is a sampling algorithm which
makes local moves by incorporating information about the gradient of the target density. In this
paper we study the efficiency of MALA on a natural class of target measures supported on an infinite
dimensional Hilbert space. These natural measures have density with respect to a Gaussian random
field measure and arise in many applications such as Bayesian nonparametric statistics and the theory
of conditioned diffusions. We prove that, started at stationarity, a suitably interpolated and scaled
version of the Markov chain corresponding to MALA converges to an infinite dimensional diffusion
process. Our results imply that, in stationarity, the MALA algorithm applied to an N-dimensional
approximation of the target will take O(N1/3) steps to explore the invariant measure. As a by-product
of the diffusion limit it also follows that the MALA algorithm is optimized at an average acceptance
probability of 0.574. Until now such results were proved only for targets which are products of one
dimensional distributions, or for variants of this situation. Our result is the first derivation of scaling
limits for the MALA algorithm to target measures which are not of the product form. As a consequence the rescaled MALA algorithm converges weakly to an infinite dimensional Hilbert space valued
diffusion, and not to a scalar diffusion. The diffusion limit is proved by showing that a drift-martingale
decomposition of the Markov chain, suitably scaled, closely resembles an Euler-Maruyama discretization of the putative limit. An invariance principle is proved for the Martingale and a continuous
mapping argument is used to complete the proof.

Item Type:	Working or Discussion Paper (Working Paper)
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH):	Langevin equations, Hilbert space, Diffusion processes
Series Name:	Working papers
Publisher:	University of Warwick. Centre for Research in Statistical Methodology
Place of Publication:	Coventry
Official Date:	2011
Dates:	Date Event 2011 Published
Volume:	Vol.2011
Number:	No.8
Number of Pages:	29
Institution:	University of Warwick
Status:	Not Peer Reviewed
Access rights to Published version:	Open Access
Funder:	Engineering and Physical Sciences Research Council (EPSRC), University of Warwick. Centre for Research in Statistical Methodology, European Research Council (ERC)

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