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Parameter estimation in stochastic differential equations with Markov chain Monte Carlo and non-linear Kalman filtering

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Abstract

This paper is concerned with parameter estimation in linear and non-linear Itô type stochastic differential equations using Markov chain Monte Carlo (MCMC) methods. The MCMC methods studied in this paper are the Metropolis–Hastings and Hamiltonian Monte Carlo (HMC) algorithms. In these kind of models, the computation of the energy function gradient needed by HMC and gradient based optimization methods is non-trivial, and here we show how the gradient can be computed with a linear or non-linear Kalman filter-like recursion. We shall also show how in the linear case the differential equations in the gradient recursion equations can be solved using the matrix fraction decomposition. Numerical results for simulated examples are presented and discussed in detail.

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Notes

  1. Originally called hybrid Monte Carlo (see, Neal 2011).

  2. The Matlab codes can be obtained from the corresponding author on request.

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Acknowledgments

We would like to acknowledge Aki Vehtari, Antti Solonen, Jouni Hartikainen, and Arno Solin for their contributions. We thank the Department of Mathematics and Physics in Lappeenranta University of Technology, the Department of Biomedical Engineering and Computational Science in Aalto University and the Academy of Finland for their financial support.

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Authors and Affiliations

  1. Department of Mathematics and Physics, Lappeenranta University of Technology, P.O.Box 20, 53851 , Lappeenranta, Finland

    Isambi S. Mbalawata & Heikki Haario

  2. Department of Biomedical Engineering and Computational Science, Aalto University, P.O.Box 12200, 00076 , Aalto, Finland

    Simo Särkkä

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  1. Isambi S. Mbalawata
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  2. Simo Särkkä
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  3. Heikki Haario
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Correspondence to Isambi S. Mbalawata.

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Mbalawata, I.S., Särkkä, S. & Haario, H. Parameter estimation in stochastic differential equations with Markov chain Monte Carlo and non-linear Kalman filtering. Comput Stat 28, 1195–1223 (2013). https://doi.org/10.1007/s00180-012-0352-y

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  • DOI: https://doi.org/10.1007/s00180-012-0352-y

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