Computational Statistics

, Volume 27, Issue 1, pp 149–176 | Cite as

Variational Markov chain Monte Carlo for Bayesian smoothing of non-linear diffusions

  • Yuan Shen
  • Dan Cornford
  • Manfred Opper
  • Cedric Archambeau
Original Paper


In this paper we develop set of novel Markov chain Monte Carlo algorithms for Bayesian smoothing of partially observed non-linear diffusion processes. The sampling algorithms developed herein use a deterministic approximation to the posterior distribution over paths as the proposal distribution for a mixture of an independence and a random walk sampler. The approximating distribution is sampled by simulating an optimized time-dependent linear diffusion process derived from the recently developed variational Gaussian process approximation method. The novel diffusion bridge proposal derived from the variational approximation allows the use of a flexible blocking strategy that further improves mixing, and thus the efficiency, of the sampling algorithms. The algorithms are tested on two diffusion processes: one with double-well potential drift and another with SINE drift. The new algorithm’s accuracy and efficiency is compared with state-of-the-art hybrid Monte Carlo based path sampling. It is shown that in practical, finite sample applications the algorithm is accurate except in the presence of large observation errors and low observation densities, which lead to a multi-modal structure in the posterior distribution over paths. More importantly, the variational approximation assisted sampling algorithm outperforms hybrid Monte Carlo in terms of computational efficiency, except when the diffusion process is densely observed with small errors in which case both algorithms are equally efficient.


Stochastic dynamic systems Data assimilation Bridge sampling 


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  1. Alexander F, Eyink G, Restrepo J (2005) Accelerated Monte Carlo for optimal estimation of time series. J Stat Phys 119: 1331–1345CrossRefMATHGoogle Scholar
  2. Andrieu C, de Freitas D, Doucet A, Jordan M (2003) An introduction to MCMC for machine learning. Mach Learn 50: 5–43CrossRefMATHGoogle Scholar
  3. Andrieu C, Doucet A, Holenstein R (2010) Particle Markov Chain Monte Carlo methods. J R Statist Soc B 72: 1–33MathSciNetCrossRefGoogle Scholar
  4. Archambeau C, Cornford D, Opper M, Shawe-Tayler J (2007) Gaussian Process approximations of stochastic differential equations. J Mach Learn Res Workshop and Conference Proceedings 1: 1–16Google Scholar
  5. Archambeau C, Opper M, Shen Y, Cornford D, Shawe-Tayler J (2008) Variational inference for diffusion processes. In: Platt C, Koller D, Singer Y, Roweis S (eds) Neural information processing systems (NIPS), vol 20. The MIT Press, Cambridge, pp 17–24Google Scholar
  6. Beskos A, Papaspiliopoulous O, Roberts GO, Fearnhead P (2006) Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes. J R Statist Soc B 68: 333–382CrossRefMATHGoogle Scholar
  7. Beskos A, Papaspiliopoulous O, Roberts GQ (2008) A factorisation of diffusion measure and finite sample path construction. Methodol Comput Appl Probab 10: 85–104MathSciNetCrossRefMATHGoogle Scholar
  8. de Freitas N, H⌽jen-S⌽rensen P, Jordan M, Russell S (2001) Variational MCMC. In: Proceedings of the 17th annual conference on uncertainty in artificial intelligence. Morgan Kaufmann Publishers Inc., San Francisco, CA, pp 120–127Google Scholar
  9. Derber J (1989) A variational continuous assimilation technique. Mon Wea Rev 117: 2437–2446CrossRefGoogle Scholar
  10. Duane S, Kennedy AD, Pendleton BJ, Roweth D (1987) Hybrid Monte Carlo. Phys Lett B 55: 2774–2777CrossRefGoogle Scholar
  11. Durham GB, Gallant AR (2002) Numerical techniques for maximum likelihood estimation of continuous-time diffusion process. J Bus Econom Stat 20: 297–338MathSciNetCrossRefGoogle Scholar
  12. Elerian O, Chib S, Shephard N (2001) Likelihood inference for discretely observed nonlinear diffusions. Econometrica 69: 959–993MathSciNetCrossRefMATHGoogle Scholar
  13. Eraker B (2001) Markov Chain Monte Carlo analysis of diffusion models with application to finance. J Bus Econ Statist 19: 177–191MathSciNetCrossRefGoogle Scholar
  14. Evensen G (1994) Sequential data assimilation with a non-linear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J Geophys Res 99: 10,143–10,162CrossRefGoogle Scholar
  15. Evensen G (2000) An ensemble Kalman smoother for nonlinear dynamics. Mon Wea Rev 128: 1852–1867CrossRefGoogle Scholar
  16. Eyink GL, Restrepo JM, Alexander FJ (2004) A mean-field approximation in data assimilation for nonlinear dynamics. Physica D 194: 347–368MathSciNetCrossRefGoogle Scholar
  17. Golightly A, Wilkinson GJ (2006) Bayesian sequential inference for nonlinear multivariate diffusions. Stat Comput 16: 323–338MathSciNetCrossRefGoogle Scholar
  18. Golightly A, Wilkinson GJ (2008) Bayesian inference for nonlinear multivariate diffusion models observed with error. Comput Stat Data Anal 52: 1674–1693MathSciNetCrossRefMATHGoogle Scholar
  19. Haario H, Laine M, Mira A, Saksman E (2006) Dram: efficient adaptive MCMC. Stat Comput 16: 339–354MathSciNetCrossRefGoogle Scholar
  20. Hastings WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57: 97–109CrossRefMATHGoogle Scholar
  21. Honerkamp J (1994) Stochastic dynamical systems. VCH, WeinheimGoogle Scholar
  22. Jazwinski AH (1970) Stochastic processes and filtering theory. Academic Press, New YorkMATHGoogle Scholar
  23. Julier SJ, Uhlmann J, Durrant-Whyte H (2000) A new method for the nonlinear tranformation of means and covariances in filters and estimators. IEEE Trans Autom Control 45: 477–482MathSciNetCrossRefMATHGoogle Scholar
  24. Kalman RE, Bucy R (1961) New results in linear filtering and prediction theory. J Basic Eng D 83: 95–108MathSciNetCrossRefGoogle Scholar
  25. Kalnay E (2003) Atmospheric modelling, data assimilation and predictability. Cambridge University Press, CambridgeGoogle Scholar
  26. Kitagawa G (1987) Non-Gaussian state space modelling of non-stationary time series. J Am Stat Assoc 82: 503–514MathSciNetGoogle Scholar
  27. Kitagawa G (1996) Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. J Comput Graph Stat 5: 1–25MathSciNetCrossRefGoogle Scholar
  28. Klöden PE, Platen E (1992) Numerical solution of stochastic differential equations. Spinger, BerlinMATHGoogle Scholar
  29. Kushner HJ (1967) Dynamical equations for optimal filter. J Differ Equ 3: 179–190MathSciNetCrossRefMATHGoogle Scholar
  30. Liu JS (2001) Monte Carlo strategies in scientific computing. Spinger, BerlinMATHGoogle Scholar
  31. Miller RN, Carter EF, Blue ST (1999) Data assimilation into nonlinear stochastic models. Tellus A 51: 167–194CrossRefGoogle Scholar
  32. Mira A (2001) On Metropolis-Hastings algorithms with delayed rejection. Metron LIX: 231–241MathSciNetGoogle Scholar
  33. Ozaki T (1992) A bridge between nonlinear time series models and nonlinear stochastic dynamical systems: a local linearization approach. Stat Sinica 2: 113–135MathSciNetMATHGoogle Scholar
  34. Papaspiliopolous O, Roberts GO, Skold M (2003) Non-centered parameterisations for hierarchical models and data augmentation. In: Bayesian Statistics 7, pp 307–326Google Scholar
  35. Pardoux E (1982) équations du filtrage non linéaire de la prédiction et du lissage. Stochastics 6: 193–231MathSciNetCrossRefMATHGoogle Scholar
  36. Rabier F, Jarvinen H, Klinker E, Mahfouf JF, Simmons A (2000) The ecmwf operational implementation of four-dimensional variational assimilation. Part i: experimental results with simplified physics. Quart J Roy Met Soc 126: 1143–1170CrossRefGoogle Scholar
  37. Roberts GQ, Stramer O (2001) On inferencee for partially observed non-linear diffusion models using Metropolis-Hasting algorithm. Biometrika 88: 603–621MathSciNetCrossRefMATHGoogle Scholar
  38. Stuart AM, Voss J, Winberg P (2004) Conditional path sampling of SDEs and the Langevin MCMC method. Commun Math Sci 2: 685–697MathSciNetMATHGoogle Scholar
  39. Wan E, van der Merwe R (2001) The unscented Kalman filter. In: Haykin S (ed) Kalman filtering and neural networks. Wiley, New York, pp 207–219Google Scholar
  40. Wilkinson D (2006) Stochastic modelling for systems biology. Chapman & Hall/CRC, Boca RatonMATHGoogle Scholar
  41. Wilkinson DJ, Golightly A (2010) Markov Chain Monte Carlo algorithms for SDE parameter estimation. In: Learning and inference in computational systems biology, pp 253–276Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Yuan Shen
    • 1
  • Dan Cornford
    • 1
  • Manfred Opper
    • 2
  • Cedric Archambeau
    • 3
  1. 1.Non-linearity and Complexity Research GroupAston UniversityBirminghamUK
  2. 2.Artificial Intelligence GroupTechnical University BerlinBerlinGermany
  3. 3.Department of Computer ScienceUniversity College LondonLondonUK

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