Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-24T06:47:22.241Z Has data issue: false hasContentIssue false
  • English
  • Français

On the two-filter approximations of marginal smoothing distributions in general state-space models

Part of: Parametric inference Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  20 March 2018

Thi Ngoc Minh Nguyen ,
Sylvain Le Corff  and
Eric Moulines
Thi Ngoc Minh Nguyen*
Affiliation:
CNRS and Télécom ParisTech
Sylvain Le Corff*
Affiliation:
Université Paris-Sud, CNRS and Université Paris-Saclay
Eric Moulines*
Affiliation:
École Polytechnique
*
* Postal address: LTCI, Télécom ParisTech, 46, rue Barrault, 75013 Paris, France.
** Postal address: Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France. Email address: sylvain.le-corff@u-psud.fr
*** Postal address: Centre de Mathématiques Appliquées, École Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

A prevalent problem in general state-space models is the approximation of the smoothing distribution of a state conditional on the observations from the past, the present, and the future. The aim of this paper is to provide a rigorous analysis of such approximations of smoothed distributions provided by the two-filter algorithms. We extend the results available for the approximation of smoothing distributions to these two-filter approaches which combine a forward filter approximating the filtering distributions with a backward information filter approximating a quantity proportional to the posterior distribution of the state, given future observations.

Keywords

Sequential Monte Carlotwo-filter smoothingstate-space modelparticle filter

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 62F12: Asymptotic properties of estimators
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 1 , March 2018 , pp. 154 - 177
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bresler, Y. (1986). Two-filter formulae for discrete-time non-linear Bayesian smoothing. Internat J. Control 43, 629–641. Google Scholar
[2] Briers, M., Doucet, A. and Maskell, S. (2010). Smoothing algorithms for state-space models. Ann. Inst. Statist. Math. 62, 61–89. Google Scholar
[3] Cappé, O. (2011). Online EM algorithm for hidden Markov models. J. Comput. Graph. Statist. 20, 728–749. Google Scholar
[4] Cappé, O., Moulines, É. and Rydén, T. (2005). Inference in Hidden Markov Models. Springer, New York. CrossRefGoogle Scholar
[5] Del Moral, P. (2004). Feynman–Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Springer, New York. Google Scholar
[6] Del Moral, P., Doucet, A. and Singh, S. (2010). A backward particle interpretation of Feynman–Kac formulae. Math. Model. Numer. Anal. 44, 947–975. Google Scholar
[7] Douc, R. and Moulines, E. (2008). Limit theorems for weighted samples with applications to sequential Monte Carlo methods. Ann. Statist. 36, 2344–2376. Google Scholar
[8] Douc, R., Moulines, E. and Stoffer, D. S. (2014). Nonlinear Time Series: Theory, Methods, and Applications with R Examples. Chapman & Hall/CRC, Boca Raton, FL. CrossRefGoogle Scholar
[9] Douc, R., Garivier, A., Moulines, E. and Olsson, J. (2011). Sequential Monte Carlo smoothing for general state space hidden Markov models. Ann. Appl. Prob. 21, 2109–2145. CrossRefGoogle Scholar
[10] Doucet, A., de Freitas, N. and Gordon, N. (eds) (2001). Sequential Monte Carlo Methods in Practice. Springer, New York. Google Scholar
[11] Doucet, A., Godsill, S. and Andrieu, C. (2000). On sequential Monte Carlo sampling methods for Bayesian filtering. Statist. Comput. 10, 197–208. CrossRefGoogle Scholar
[12] Dubarry, C. and Le Corff, S. (2013). Non-asymptotic deviation inequalities for smoothed additive functionals in nonlinear state-space models. Bernoulli 19, 2222–2249. Google Scholar
[13] Fearnhead, P., Wyncoll, D. and Tawn, J. (2010). A sequential smoothing algorithm with linear computational cost. Biometrika 97, 447–464. Google Scholar
[14] Godsill, S. J., Doucet, A. and West, M. (2004). Monte Carlo smoothing for nonlinear times series. J. Amer. Statist. Assoc. 99, 156–168. Google Scholar
[15] Gordon, N. J., Salmond, D. J. and Smith, A. F. M. (1993). Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proc. F Radar Signal Process. 140, 107–113. Google Scholar
[16] Hürzeler, M. and Künsch, H. R. (1998). Monte Carlo approximations for general state-space models. J. Comput. Graph. Statist. 7, 175–193. Google Scholar
[17] Jacob, P. E., Murray, L. M. and Rubenthaler, S. (2013). Path storage in the particle filter. Statist. Comput. 25, 487–496. Google Scholar
[18] Kantas, N. et al. (2015). On particle methods for parameter estimation in state-space models. Statist. Sci. 30, 328–351. CrossRefGoogle Scholar
[19] Kitagawa, G. (1994). The two-filter formula for smoothing and an implementation of the Gaussian-sum smoother. Ann. Inst. Statist. Math. 46, 605–623. CrossRefGoogle Scholar
[20] Kitagawa, G. (1996). Monte-Carlo filter and smoother for non-Gaussian nonlinear state space models. J. Comput. Graph. Statist. 5, 1–25. Google Scholar
[21] Künsch, H. R. (2005). Recursive Monte Carlo filters: algorithms and theoretical analysis. Ann. Statist. 33, 1983–2021. Google Scholar
[22] Le Corff, S. and Fort, G. (2013). Convergence of a particle-based approximation of the block online expectation maximization algorithm. ACM Trans. Model. Comput. Simul. 23, 2. Google Scholar
[23] Le Corff, S. and Fort, G. (2013). Online expectation maximization based algorithms for inference in hidden Markov models. Electron. J. Statist. 7, 763–792. CrossRefGoogle Scholar
[24] Lee, A. and Whiteley, N. (2016). Variance estimation in the particle filter. Preprint. Available at https://arxiv.org/abs/1509.00394. Google Scholar
[25] Mongillo, G. and Deneve, S. (2008). Online learning with hidden Markov models. Neural Comput. 20, 1706–1716. Google Scholar
[26] Olsson, J. and Douc, R. (2017). Numerically stable online estimation of variance in particle filters. Preprint. Available at https://arxiv.org/abs/1701.01001. Google Scholar
[27] Olsson, J. and Westerborn, J. (2017). Efficient particle-based online smoothing in general hidden Markov models: the PaRIS algorithm. Bernoulli 23, 1951–1996. Google Scholar
[28] Persing, A. and Jasra, A. (2013). Likelihood computation for hidden Markov models via generalized two-filter smoothing. Statist. Prob. Lett. 83, 1433–1442. CrossRefGoogle Scholar
[29] Pitt, M. K. and Shephard, N. (1999). Filtering via simulation: auxiliary particle filters. J. Amer. Statist. Assoc. 94, 590–599. Google Scholar
[30] Poyiadjis, G., Doucet, A. and Singh, S. S. (2011). Particle approximations of the score and observed information matrix in state space models with application to parameter estimation. Biometrika 98, 65–80. Google Scholar