Pillai, Natesh S., 1981-, 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).

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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
Date:	2011
Volume:	Vol.2011
Number:	No.8
Number of Pages:	29
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)
References:	[Bed07] M. Bedard. Weak convergence of Metropolis algorithms for non-i.i.d. target distributions. Ann. Appl. Probab., 17(4):1222{1244, 2007. [Bed09] M. Bedard. On the optimal scaling problem of metropolis algorithms for hierarchical target distributions. 2009. preprint. [Ber86] E. Berger. Asymptotic behaviour of a class of stochastic approximation procedures. Probab. Theory Relat. Fields, 71(4):517{552, 1986. [BPR+11] A. Beskos, N. Pillai, G.O. Roberts, J.-M. Sanz-Serna, and A.M. Stuart. Optimal tuning of hybrid monte-carlo. Submitted, 2011. [BPS04] Laird Arnault Breyer, Mauro Piccioni, and Sergio Scarlatti. Optimal scaling of MaLa for non- linear regression. Ann. Appl. Probab., 14(3):1479{1505, 2004. [BPSSA] A. Beskos, F. Pinski, J.M. Sanz-Serna, and A.M.Stuart. Hybrid Monte-Carlo on hilbert spaces. Stoch. Proc. Applics. [BR00] L. A. Breyer and G. O. Roberts. From Metropolis to diffusions: Gibbs states and optimal scaling. Stochastic Process. Appl., 90(2):181{206, 2000. [BRS09] A. Beskos, G.O. Roberts, and A.M. Stuart. Optimal scalings for local Metropolis-Hastings chains on non-product targets in high dimensions. Ann. Appl. Probab., 19(3):863{898, 2009. [BRSV08] A. Beskos, G.O. Roberts, A.M. Stuart, and J. Voss. An MCMC method for diffusion bridges. Stochastics and Dynamics, 8(3):319{350, 2008. [BS09] A. Beskos and A.M. Stuart. MCMC methods for sampling function space. In Invited Lectures, Sixth International Congress on Industrial and Applied Mathematics, ICIAM07, Editors Rolf Jeltsch and Gerhard Wanner, pages 337{364. European Mathematical Society, 2009. [CRR05] O.F. Christensen, G.O. Roberts, and J.S. Rosenthal. Scaling limits for the transient phase of local Metropolis{Hastings algorithms. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 67(2):253{268, 2005. [DPZ92] G. Da Prato and J. Zabczyk. Stochastic equations in infinite dimensions, volume 44 of Encyclo- pedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1992. [Dur96] Richard Durrett. Stochastic Calculus: A Practical Introduction. CRC Press; Subsequent edition (21 Jun 1996), 1996. [EK86] S. N. Ethier and T. G. Kurtz. Markov processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons Inc., New York, 1986. Characterization and convergence. [HAVW05] M. Hairer, A.M.Stuart, J. Voss, and P. Wiberg. Analysis of SPDEs arising in path sampling. Part i: the gaussian case. Comm. Math. Sci., 3:587{603, 2005. [HSV07] M. Hairer, A. M. Stuart, and J. Voss. Analysis of SPDEs arising in path sampling. PartII: the nonlinear case. Ann. Appl. Probab., 17(5-6):1657{1706, 2007. [HSV10] M. Hairer, A. M. Stuart, and J. Voss. Signal processing problems on function space: Bayesian formulation, stochastic pdes and effective mcmc methods. The Oxford Handbook of Nonlinear Filtering, Editors D. Crisan and B. Rozovsky, 2010. To Appear. [MPS09] J.C. Mattingly, N.S. Pillai, and A.M. Stuart. SPDE Limits of the Random Walk Metropolis Algorithm in High Dimensions. 2009. [MRTT53] N. Metropolis, A.W. Rosenbluth, M.N. Teller, and E. Teller. Equations of state calculations by fast computing machines. J. Chem. Phys., 21:1087{1092, 1953. [RC04] C. P. Robert and G. Casella. Monte Carlo statistical methods. Springer Texts in Statistics. Springer-Verlag, New York, second edition, 2004. [RGG97] G. O. Roberts, A. Gelman, and W. R. Gilks. Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann. Appl. Probab., 7(1):110{120, 1997. [RR98] G. O. Roberts and J. S. Rosenthal. Optimal scaling of discrete approximations to Langevin diffusions. J. R. Stat. Soc. Ser. B Stat. Methodol., 60(1):255{268, 1998. [RR01] G. O. Roberts and J. S. Rosenthal. Optimal scaling for various Metropolis-Hastings algorithms. Statist. Sci., 16(4):351{367, 2001. [SFR10] Chris Sherlock, Paul Fearnhead, and Gareth O. Roberts. The random walk metropolis : linking theory and practice through a case study. Statistical Science, 25(2):172{190, 2010. [Stu10] A.M. Stuart. Inverse problems: a Bayesian perspective. Acta Numerica, To appear, 2010.
URI:	http://wrap.warwick.ac.uk/id/eprint/34878

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