A Comparison of Variational and Markov Chain Monte Carlo Methods for Inference in Partially Observed Stochastic Dynamic Systems
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In recent work we have developed a novel variational inference method for partially observed systems governed by stochastic differential equations. In this paper we provide a comparison of the Variational Gaussian Process Smoother with an exact solution computed using a Hybrid Monte Carlo approach to path sampling, applied to a stochastic double well potential model. It is demonstrated that the variational smoother provides us a very accurate estimate of mean path while conditional variance is slightly underestimated. We conclude with some remarks as to the advantages and disadvantages of the variational smoother.
- Honerkamp, J. (1994). Stochastic dynamical systems. New York: VCH.
- Wilkinson, D. J. (2006). Stochastic modelling for system biology. Boca Raton: Chapman & Hall/CRC.
- Kalnay, E. (2003). Atmospheric modeling, data assimilation and predictability. Cambridge: Cambridge University Press.
- Anderson, B. D. O., & Moore, J. B. (2005). Optimal filtering. Mineola: Dover.
- Kushner, H. J. (1967). Dynamical equations for optimal filter. Journal of Differential Equations, 3, 179–190. CrossRef
- Stratonovich, R. L. (1960) Conditional markov processes. Theory of Probability and Its Applications, 5, 156–178. CrossRef
- Pardoux, E. (1982). Équations du filtrage non linéaire de la prédiction et du lissage. Stochastics, 6, 193–231.
- Kalman, R. E., & Bucy, R. S. (1961). New results in linear filtering and prediction theory. Journal of Basic Engineering, 83D, 95–108.
- Shumway, R. H., & Stoffer, D. S. (2000). Time series analysis and its applications. New York: Springer.
- Archambeau, C., Cornford, D., Opper, M., & Shawe-Tayler, J. (2007). Gaussian process approximations of stochastic differential equations. Journal of Machine Learning Research Workshop and Conference Proceedings, 1, 1–16.
- Klöden, P. E., & Platen, E. (1992). Numerical solution of stochastic differential equations. Berlin: Spinger.
- Andrieu, C., De Freitas, N., Doucet, A., & Jordan, M. I. (2003). An introduction to MCMC for machine learning. Machine Learning, 50, 5–43. CrossRef
- Hürzler, M. (1998). Statistical methods for general state-space models. PhD Thesis Nr. 12674, ETH Zürich.
- Alexander, F. J., Eyink, G. L., & Restrepo, J. M. (2005). Accelerated Monte Carlo for optimal estimation of time series. Journal of Statistical Physics, 119, 1331–1345. CrossRef
- Gelb, A. (1974). Applied optimal estimation. Cambridge: MIT.
- Evensen, G. (1992). Using the extended Kalman filter with a multilayer quasi-geostrophic ocean model. Journal of Geophysical Research, 97, 17905–17924. CrossRef
- Evensen, G. (1994). Sequential data assimilation with a non-linear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. Journal of Geophysical Research, 99, 10143–10162. CrossRef
- Kitagawa, G. (1987). Non-Gaussian state space modelling of non-stationary time series. Journal of the American Statistical Association, 82, 503–514.
- Kullback, S., & Leibler, R. A. (1951). On information and sufficiency. Annal of Mathematical Statistics, 22, 79–86. CrossRef
- Jaakkola, T. S. (2001). Tutorial on variational approximation methods. In D. Saad, & M. Opper (Eds.), Advanced mean field methods. Cambridge: MIT.
- Crisan, D., Del Moral, P., & Lyons, T. J. (1999). Interacting particle systems approximations of the Kushner-Stratonovich equation. Advances in Applied Probability, 31, 819–838. CrossRef
- Rasmussen, C. E., & Williams, C. K. I. (2006). Gaussian process for machine learning. Cambridge: MIT.
- Miller, R. N., Carter, E. F, & Blue, S. T. (1999). Data assimilation into nonlinear stochastic models. Tellus, 51A, 167–194.
- Chan, G., & Wood, A. T. A. (1999). Simulation of stationary Gaussian vector fields. Statistics and Computing, 22, 265–268. CrossRef
- Eyink, G. L., Restrepo, J. M., & Alexander, F. J. (2004). A mean-field approximation in data assimilation for nonlinear dynamics. Physica D, 194, 347–368. CrossRef
- Julier, S. J., Uhlmann, J., & Durrant-Whyte, H. F. (2000). A new method for the nonlinear tranformation of means and covariances in filters and estimators. IEEE Transactions on Automatic Control, 45, 477–482. CrossRef
- Kitagawa, G. (1994). The two-filter formula for smoothing and an implementation of the Gaussian-sum smoother. Annals of the Institute of Statistical Mathematics, 46(4), 605–623. CrossRef
- Alspach, D. L., & Sorenson, H. W. (1972). Nonlinear Bayesian estimation using Gaussian sum approximations. IEEE Transactions On Automatic Control, 17(4) 439–448. CrossRef
- A Comparison of Variational and Markov Chain Monte Carlo Methods for Inference in Partially Observed Stochastic Dynamic Systems
Journal of Signal Processing Systems
Volume 61, Issue 1 , pp 51-59
- Cover Date
- Print ISSN
- Online ISSN
- Springer US
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- Data assimilation
- Signal processing
- Nonlinear smoothing
- Variational approximation
- Bayesian computation
- Industry Sectors
- Author Affiliations
- 1. Neural Computing Research Group, Aston University, Birmingham, UK
- 2. Department of Computer Science, University College London, London, UK
- 3. Artificial Intelligence Group, Technical University Berlin, Berlin, Germany