Alexandros Beskos, Gareth O. Roberts, Andrew M. Stuart and Jochen Voss:
MCMC Methods for Diffusion Bridges.
Stochastics and Dynamics, vol. 8, no. 3, pp. 319–350,
2008.
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,
parameterised 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.
Citations
I believe that this work is cited in the following texts.
If you know of any more citations, please let me know.
M. Hairer, A.M. Stuart and J. Voss:
Signal Processing Problems on Function Space: Bayesian Formulation, Stochastic PDEs and Effective MCMC Methods.
Pages 833–873 in The Oxford Handbook of Nonlinear Filtering,
Dan Crisan and Boris Rozovsky (editors), Oxford University Press, 2011.
link, preprint:pdf, more…
J. Voss:
The Effect of Finite Element Discretisation on the Stationary Distribution of SPDEs.
Submitted, 2011.
arXiv:1110.4653, more…
A. Beskos and A.M. Stuart:
Computational complexity of Metropolis-Hastings methods in high dimensions.
Pages 61–72 in Proceedings of MCQMC08,
Pierre L'Ecuyer and Art B. Owen (editors), 2010.
link
O. Stramer, M. Bognar and P. Schneider:
Bayesian Inference for Discretely Sampled Markov Processes with Closed-Form Likelihood Expansions.
Journal of Financial Econometrics, vol. 8, no. 4,
pp. 450–480, 2010.
DOI:10.1093/jjfinec/nbp027
D. White and A.M. Stuart:
Green's Functions by Monte Carlo.
Pages 627–637 in Proceedings of MCQMC08,
Pierre L'Ecuyer and Art B. Owen (editors), 2010.
link
A.M. Stuart:
Inverse problems: A Bayesian perspective.
Acta Numerica, vol. 19, pp. 451–559, 2010.
DOI:10.1017/S0962492910000061
M. Lin, R. Chen and P. Mykland:
On generating Monte Carlo samples of continuous diffusion bridges.
J. Amer. Statist. Assoc., vol. 105, no. 490, pp. 820–838,
2010.
DOI:10.1198/jasa.2010.tm09057
A. Beskos, G.O. Roberts and A.M. Stuart:
Optimal scalings for local Metropolis-Hastings chains on nonproduct targets in high dimensions.
Ann. Appl. Probab., vol. 19, no. 3, pp. 863–898, 2009.
MR2537193
M. Hairer, A.M. Stuart and J. Voss:
Sampling Conditioned Diffusions.
Pages 159–186 in Trends in Stochastic Analysis,
Cambridge University Press,
vol. 353 of London Mathematical Society Lecture Note Series, 2009.
link, preprint:pdf, more…
S.L. Cotter, M. Dashti, J.C. Robinson and A.M. Stuart:
Bayesian inverse problems for functions and applications to fluid mechanics.
Inverse Problems, vol. 25, 2009.
DOI:10.1088/0266-5611/25/11/115008
A. Beskos and A.M. Stuart:
MCMC methods for sampling function space.
Pages 337–364 in Proceedings of the 6th International Congress on Industrial and Applied Mathematicians (Zürich, 2007),
Rolf Jeltsch and Gerhard Wanner (editors), 2009.
J.C. Mattingly, N.S. Pillai and A.M. Stuart:
SPDE Limits of the Random Walk Metropolis Algorithm in High Dimensions.
Preprint, 2009.
link
M. Hairer, A.M. Stuart and J. Voss:
Analysis of SPDEs Arising in Path Sampling, Part II: The Nonlinear Case.
Annals of Applied Probability, vol. 17, no. 5,
pp. 1657–1706, 2007.
DOI:10.1214/07-AAP441, arXiv:math/0601092, more…
P. Plechac and M. Rousset:
Implicit Mass-Matrix Penalization of Hamiltonian dynamics with application to exact sampling of stiff systems.
Preprint.
arXiv:0905.4737