Estimating parameters in stochastic systems: a variational Bayesian approach

Michail Vrettas, Dan Cornford, Manfred Opper

Research output: Contribution to journalArticlepeer-review


This work is concerned with approximate inference in dynamical systems, from a variational Bayesian perspective. When modelling real world dynamical systems, stochastic differential equations appear as a natural choice, mainly because of their ability to model the noise of the system by adding a variation of some stochastic process to the deterministic dynamics. Hence, inference in such processes has drawn much attention. Here a new extended framework is derived that is based on a local polynomial approximation of a recently proposed variational Bayesian algorithm. The paper begins by showing that the new extension of this variational algorithm can be used for state estimation (smoothing) and converges to the original algorithm. However, the main focus is on estimating the (hyper-) parameters of these systems (i.e. drift parameters and diffusion coefficients). The new approach is validated on a range of different systems which vary in dimensionality and non-linearity. These are the Ornstein–Uhlenbeck process, the exact likelihood of which can be computed analytically, the univariate and highly non-linear, stochastic double well and the multivariate chaotic stochastic Lorenz ’63 (3D model). As a special case the algorithm is also applied to the 40 dimensional stochastic Lorenz ’96 system. In our investigation we compare this new approach with a variety of other well known methods, such as the hybrid Monte Carlo, dual unscented Kalman filter, full weak-constraint 4D-Var algorithm and analyse empirically their asymptotic behaviour as a function of observation density or length of time window increases. In particular we show that we are able to estimate parameters in both the drift (deterministic) and the diffusion (stochastic) part of the model evolution equations using our new methods.
Original languageEnglish
Pages (from-to)1877–1900
Number of pages24
JournalPhysica D
Issue number23
Early online date18 Sept 2011
Publication statusPublished - 15 Nov 2011

Bibliographical note

NOTICE: this is the author’s version of a work that was accepted for publication in Physica D: Nonlinear Phenomena. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Vrettas, M, Cornford, D & Opper, M, 'Estimating parameters in stochastic systems: a variational Bayesian approach', PHYSICA D-NONLINEAR PHENOMENA, vol 240, no. 23, pp. 1877–1900., (2011) DOI


  • Bayesian inference
  • variational techniques
  • dynamical systems
  • stochastic differential equations
  • parameter estimation


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