Beskos, Alexandros, Roberts, Gareth O., Stuart, A. M. and Voss, Jochen (2008) MCMC methods for diffusion bridges. Stochastics and Dynamics, Vol.8 (No.3). pp. 319-350. doi:10.1142/S0219493708002378 ISSN 0219-4937.

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Abstract

We present and study a Langevin MCMC approach for sampling nonlinear diffusion bridges. The method is based on recent theory concerning stochastic partial differential equations (SPDEs) reversible with respect to the target bridge, derived by applying the Langevin idea on the bridge pathspace. In the process, a Random-Walk Metropolis algorithm and an Independence Sampler are also obtained. The novel algorithmic idea of the paper is that proposed moves for the MCMC algorithm are determined by discretising the SPDEs in the time direction using an implicit scheme, parametrised by theta is an element of [0,1]. We show that the resulting infinite-dimensional MCMC sampler is well-defined only if theta = 1/2, when the MCMC proposals have the correct quadratic variation. Previous Langevin-based MCMC methods used explicit schemes, corresponding to. = 0. The significance of the choice theta = 1/2 is inherited by the finite-dimensional approximation of the algorithm used in practice. We present numerical results illustrating the phenomenon and the theory that explains it. Diffusion bridges (with additive noise) are representative of the family of laws defined as a change of measure from Gaussian distributions on arbitrary separable Hilbert spaces; the analysis in this paper can be readily extended to target laws from this family and an example from signal processing illustrates this fact.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics Q Science > QC Physics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Statistics
Library of Congress Subject Headings (LCSH):	Sampling (Statistics), Langevin equations, Markov processes, Monte Carlo method, Gaussian measures, Diffusion, Stochastic partial differential equations
Journal or Publication Title:	Stochastics and Dynamics
Publisher:	World Scientific Publishing Co. Pte. Ltd.
ISSN:	0219-4937
Official Date:	September 2008
Dates:	Date Event September 2008 UNSPECIFIED
Volume:	Vol.8
Number:	No.3
Number of Pages:	32
Page Range:	pp. 319-350
DOI:	10.1142/S0219493708002378
Status:	Peer Reviewed
Publication Status:	Published
Version or Related Resource:	Presented at: Conference on Random Dynamical Systems and Stochastic Dynamics held in Honor of Ludwig Arnold, Univ Paderborn, Paderborn, Germany, Jul 06, 2007
Type of Event:	Conference

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