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

Abstract

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.

Keywords

Stochastic dynamic systems Data assimilation Bridge sampling 

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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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