• Bernoulli
  • Volume 14, Number 1 (2008), 155-179.

Sequential Monte Carlo smoothing with application to parameter estimation in nonlinear state space models

Jimmy Olsson, Olivier Cappé, Randal Douc, and Éric Moulines

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This paper concerns the use of sequential Monte Carlo methods (SMC) for smoothing in general state space models. A well-known problem when applying the standard SMC technique in the smoothing mode is that the resampling mechanism introduces degeneracy of the approximation in the path space. However, when performing maximum likelihood estimation via the EM algorithm, all functionals involved are of additive form for a large subclass of models. To cope with the problem in this case, a modification of the standard method (based on a technique proposed by Kitagawa and Sato) is suggested. Our algorithm relies on forgetting properties of the filtering dynamics and the quality of the estimates produced is investigated, both theoretically and via simulations.

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Bernoulli, Volume 14, Number 1 (2008), 155-179.

First available in Project Euclid: 8 February 2008

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EM algorithm exponential family particle filters sequential Monte Carlo methods state space models stochastic volatility model


Olsson, Jimmy; Cappé, Olivier; Douc, Randal; Moulines, Éric. Sequential Monte Carlo smoothing with application to parameter estimation in nonlinear state space models. Bernoulli 14 (2008), no. 1, 155--179. doi:10.3150/07-BEJ6150.

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  • Andrieu, C. and Doucet, A. (2003). Online expectation–maximization type algorithms for parameter estimation in general state space models. In, Proc. IEEE Int. Conf. Acoust., Speech, Signal Process. 6 VI69–VI72.
  • Cappé, O., Moulines, E. and Rydén, T. (2005)., Inference in Hidden Markov Models. New York: Springer.
  • Del Moral, P. (2004)., Feynman–Kac Formulae. Genealogical and Interacting Particle Systems with Applications. New York: Springer.
  • Del Moral, P. and Doucet, A. (2003). On a class of genealogical and interacting Metropolis models., Séminaire de Probabilités XXXVII (J. Azéma, M. Emery, M. Ledoux and M. Yor, eds.). Lecture Notes in Math. 1832 415–446. Berlin: Springer.
  • Douc, R., Fort, G., Moulines, E. and Priouret, P. (2007). Forgetting of the initial distribution for hidden Markov models. Available at,
  • Doucet, A., de Freitas, N. and Gordon, N. (2001)., Sequential Monte Carlo Methods in Practice. New York: Springer.
  • Doucet, A., Godsill, J. and West, M. (2004). Monte Carlo smoothing for nonlinear time series., J. Amer. Statist. Assoc. 99 156–168.
  • Fort, G. and Moulines, É. (2003). Convergence of the Monte Carlo expectation maximization for curved exponential families., Ann. Statist. 31 1220–1259.
  • Hull, J. and White, A. (1987). The pricing of options on assets with stochastic volatilities., J. Finance 42 281–300.
  • Jaquier, E., Polson, N.G. and Rossi, P.E. (1994). Bayesian analysis of stochastic volatility models (with discussion)., J. Bus. Econom. Statist. 12 371–417.
  • Kitagawa, G. and Sato, S. (2001). Monte Carlo smoothing and self-organising state-space model. In, Sequential Monte Carlo Methods in Practice (A. Doucet, N. de Freitas and N. Gordon, eds.) 178–195. New York: Springer.
  • Kleptsyna, M.L. and Veretennikov, A.Y. (2007). On discrete time ergodic filters with wrong initial conditions. Technical report, Univ. Main and Univ., Leeds.
  • Pitt, M.K. and Shephard, N. (1999). Filtering via simulation: Auxiliary particle filters., J. Amer. Statist. Assoc. 94 590–599.
  • Ristic, B., Arulampalam, M. and Gordon, A. (2004)., Beyond the Kalman Filter: Particle Filters for Target Tracking. Atrech House Radar Library.
  • Wei, G.C.G. and Tanner, M.A. (1991). A Monte Carlo implementation of the EM algorithm and the poor man’s data augmentation algorithms., J. Amer. Statist. Assoc. 85 699–704.