Posterior inference on parameters of stochastic differential equations via non-linear Gaussian filtering and adaptive MCMC
This article is concerned with Bayesian estimation of parameters in non-linear multivariate stochastic differential equation (SDE) models occurring, for example, in physics, engineering, and financial applications. In particular, we study the use of adaptive Markov chain Monte Carlo (AMCMC) based numerical integration methods with non-linear Kalman-type approximate Gaussian filters for parameter estimation in non-linear SDEs. We study the accuracy and computational efficiency of gradient-free sigma-point approximations (Gaussian quadratures) in the context of parameter estimation, and compare them with Taylor series and particle MCMC approximations. The results indicate that the sigma-point based Gaussian approximations lead to better approximations of the parameter posterior distribution than the Taylor series, and the accuracy of the approximations is comparable to that of the computationally significantly heavier particle MCMC approximations.
KeywordsStochastic differential equation Parameter estimation Gaussian approximation Non-linear Kalman filter Adaptive Markov chain Monte Carlo
The authors would like to thank Marko Laine for his valuable help in AMCMC methods, and the anonymous referees for their suggestions for improving the paper.
- Archambeau, C., Opper, M.: Approximate inference for continuous-time Markov processes. In: Inference and Learning in Dynamic Models. Cambridge University Press, Cambridge (2011) Google Scholar
- Doucet, A., Pitt, M., Kohn, R.: Efficient implementation of Markov chain Monte Carlo when using an unbiased likelihood estimator (2012). arXiv:1210.1871
- Fort, G., Moulines, E., Priouret, P.: Convergence of adaptive MCMC algorithms: ergodicity and law of large number. Tech. Rep., LTCI (2011) Google Scholar
- Jeisman, J.: Estimation of the Parameters of Stochastic Differential Equations. Doctoral dissertation, Queensland University of Technology (2005) Google Scholar
- Jones, C.S.: A simple Bayesian method for the analysis of diffusion processes. Working paper, University of Pennsylvania (1998) Google Scholar
- Kloeden, P.E., Platen, E.: Numerical Solution to Stochastic Differential Equations. Springer, Berlin (1999) Google Scholar
- Särkkä, S.: Recursive Bayesian inference on stochastic differential equations. Doctoral dissertation, Helsinki University of Technology (2006) Google Scholar
- Särkkä, S., Solin, A.: On continuous-discrete cubature Kalman filtering. In: Proceedings of SYSID 2012, pp. 1210–1215 (2012) Google Scholar
- Stramer, O., Bognar, M., Schneider, P.: Bayesian inference for discretely sampled Markov processes with closed-form likelihood expansions. J. Financ. Econ. 8(4), 450–480 (2010) Google Scholar
- Van Kampen, N.G.: Stochastic processes in physics and chemistry, 3rd edn. Elsevier, Amsterdam (2007) Google Scholar