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MCMC Methods for Diffusion Bridges

Beskos, Alexandros and Roberts, Gareth and Stuart, Andrew and Voss, Jochen (2008) MCMC Methods for Diffusion Bridges. Stochastics and Dynamics, 8 (3). pp. 319-350. ISSN 1793-6799. doi:10.1142/S0219493708002378.

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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 θ ∈ [0,1]. We show that the resulting infinite-dimensional MCMC sampler is well-defined only if θ = 1/2, when the MCMC proposals have the correct quadratic variation. Previous Langevin-based MCMC methods used explicit schemes, corresponding to θ = 0. The significance of the choice θ = 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:Article
Related URLs:
URLURL TypeDescription
Voss, Jochen0000-0001-7740-8811
Additional Information:© 2008 World Scientific. The computing facilities for producing Fig.1-4 were provided by the Centre for Scientific Computing of the University of Warwick. The authors are grateful to EPSRC and ONR for financial support.
Funding AgencyGrant Number
Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
Office of Naval Research (ONR)UNSPECIFIED
Subject Keywords:Diffusion Bridge; MCMC; Langevin Sampling; Gaussian Measure; SDE on Hilbert Space; Implicit Euler Scheme; Quadratic Variation
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Andrew StuartJ74
Issue or Number:3
Classification Code:AMS Subject Classification: 65C05, 60H35
Record Number:CaltechAUTHORS:20160805-165106874
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:69496
Deposited By: Linda Taddeo
Deposited On:09 Aug 2016 00:04
Last Modified:11 Nov 2021 04:15

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