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Uniform convergence over time of a nested particle filtering scheme for recursive parameter estimation in state-space Markov models

Published online by Cambridge University Press:  17 November 2017

Dan Crisan*
Affiliation:
Imperial College London
Joaquín Míguez*
Affiliation:
Universidad Carlos III de Madrid
*
* Postal address: Department of Mathematics, Imperial College London, 180 Queens Gate, London SW7 2BZ, UK. Email address: d.crisan@imperial.ac.uk
** Postal address: Department of Signal Theory & Communications, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Madrid, Spain. Email address: joaquin.miguez@uc3m.es

Abstract

We analyse the performance of a recursive Monte Carlo method for the Bayesian estimation of the static parameters of a discrete-time state-space Markov model. The algorithm employs two layers of particle filters to approximate the posterior probability distribution of the model parameters. In particular, the first layer yields an empirical distribution of samples on the parameter space, while the filters in the second layer are auxiliary devices to approximate the (analytically intractable) likelihood of the parameters. This approach relates the novel algorithm to the recent sequential Monte Carlo square method, which provides a nonrecursive solution to the same problem. In this paper we investigate the approximation of integrals of real bounded functions with respect to the posterior distribution of the system parameters. Under assumptions related to the compactness of the parameter support and the stability and continuity of the sequence of posterior distributions for the state-space model, we prove that the Lp norms of the approximation errors vanish asymptotically (as the number of Monte Carlo samples generated by the algorithm increases) and uniformly over time. We also prove that, under the same assumptions, the proposed scheme can asymptotically identify the parameter values for a class of models. We conclude the paper with a numerical example that illustrates the uniform convergence results by exploring the accuracy and stability of the proposed algorithm operating with long sequences of observations.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Andrieu, C., Doucet, A. and Holenstein, R. (2010). Particle Markov chain Monte Carlo methods. J. R. Statist. Soc. B 72, 269342. CrossRefGoogle Scholar
[2] Andrieu, C., Doucet, A., Singh, S. S. and Tadić, V. B. (2004). Particle methods for change detection, system identification, and control. Proc. IEEE 92, 423438. Google Scholar
[3] Bain, A. and Crisan, D. (2009). Fundamentals of Stochastic Filtering. Springer, New York. Google Scholar
[4] Cappé, O., Godsill, S. J. and Moulines, E. (2007). An overview of existing methods and recent advances in sequential Monte Carlo. Proc. IEEE 95, 899924. Google Scholar
[5] Carvalho, C. M., Johannes, M. S., Lopes, H. F. and Polson, N. G. (2010). Particle learning and smoothing. Statist. Sci. 25, 88106. Google Scholar
[6] Chen, R. and Liu, J. S. (2000). Mixture Kalman filters. J. R. Statist. Soc. B 62, 493508. Google Scholar
[7] Chopin, N., Jacob, P. E. and Papaspiliopoulos, O. (2013). SMC2: an efficient algorithm for sequential analysis of state space models. J. R. Statist. Soc. B 75, 397426. Google Scholar
[8] Chorin, A. J. and Krause, P. (2004). Dimensional reduction for a Bayesian filter. Proc. Nat. Acad. Sci. USA 101, 1501315017. Google Scholar
[9] Crisan, D. (2001). Particle filters—a theoretical perspective. In Sequential Monte Carlo Methods in Practice, Springer, New York, pp. 1741. CrossRefGoogle Scholar
[10] Crisan, D. and Míguez, J. (2017). Nested particle filters for online parameter estimation in discrete-time state-space Markov models. Preprint. Available at https://arxiv.org/abs/1308.1883v3. Google Scholar
[11] Crisan, D. and Míguez, J. (2017). Online appendix: Uniform convergence over time of a nested particle filtering scheme for recursive parameter estimation in state-space Markov models. Supplementary material. Available at http://doi.org/10.1017/apr.2017.38. Google Scholar
[12] Del Moral, P. (2004). Feynman–Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Springer, New York. Google Scholar
[13] Del Moral, P. and Guionnet, A. (2001). On the stability of interacting processes with applications to filtering and genetic algorithms. Ann. Inst. H. Poincaré Prob. Statist. 37, 155194. CrossRefGoogle Scholar
[14] Del Moral, P., Jasra, A. and Zhou, Y. (2017). Biased online parameter inference for state-space models. Methodol. Comput. Appl. Prob. 19, 727749. CrossRefGoogle Scholar
[15] Diaconis, P. and Freedman, D. (1986). On the consistency of Bayes estimates. Ann. Statist. 14, 167. Google Scholar
[16] Doucet, A., de Freitas, N. and Gordon, N. (2001). An introduction to sequential Monte Carlo methods. In Sequential Monte Carlo Methods in Practice, Springer, New York, pp. 314. Google Scholar
[17] Doucet, A., de Freitas, N. and Gordon, N. (eds) (2001). Sequential Monte Carlo Methods in Practice. Springer, New York. CrossRefGoogle Scholar
[18] Doucet, A., Godsill, S. and Andrieu, C. (2000). On sequential Monte Carlo sampling methods for Bayesian filtering. Statist. Comput. 10, 197208. Google Scholar
[19] Gordon, N. J., Salmond, D. J. and Smith, A. F. M. (1993). Novel approach to nonlinear non-Gaussian Bayesian state estimation. IEE Proc. F 140, 107113. Google Scholar
[20] Kantas, N. et al. (2015). On particle methods for parameter estimation in state-space models. Statist. Sci. 30, 328351. CrossRefGoogle Scholar
[21] Kitagawa, G. (1996). Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. J. Comput. Graph. Statist. 1, 125. Google Scholar
[22] Kitagawa, G. (1998). A self-organizing state-space model. J. Amer. Statist. Assoc. 93, 12031215. Google Scholar
[23] Künsch, H. R. (2005). Recursive Monte Carlo filters: algorithms and theoretical analysis. Ann. Statist. 33, 19832021. Google Scholar
[24] Künsch, H. R. (2013). Particle filters. Bernoulli 19, 13911403. Google Scholar
[25] LeGland, F. and Mevel, L. (1997). Recursive estimation in hidden Markov models. In Proceedings of the 36th IEEE Conference on Decision and Control, IEEE, New York, pp. 34683473. Google Scholar
[26] Liu, J. and West, M. (2001). Combined parameter and state estimation in simulation-based filtering. In Sequential Monte Carlo Methods in Practice, Springer, Now York, pp. 197223. Google Scholar
[27] Liu, J. S. and Chen, R. (1998). Sequential Monte Carlo methods for dynamic systems. J. Amer. Statist. Assoc. 93, 10321044. CrossRefGoogle Scholar
[28] Lorenz, E. N. (1963). Deterministic nonperiodic flow. J. Atmospheric Sci. 20, 130141. Google Scholar
[29] Papavasiliou, A. (2005). A uniformly convergent adaptive particle filter. J. Appl. Prob. 42, 10531068. Google Scholar
[30] Papavasiliou, A. (2006). Parameter estimation and asymptotic stability in stochastic filtering. Stoch. Process. Appl. 116, 10481065. Google Scholar
[31] Pitt, M. K. and Shephard, N. (2001). Auxiliary variable based particle filters. In Sequential Monte Carlo Methods in Practice, Springer, New York, pp. 273293. Google Scholar
[32] Pomeau, Y. and Manneville, P. (1980). Intermittent transition to turbulence in dissipative dynamical systems. Commun. Math. Phys. 74, 189197. Google Scholar
[33] 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, 6580. Google Scholar
[34] Ristic, B., Arulampalam, S. and Gordon, N. (2004). Beyond the Kalman Filter: Particle Filters for Tracking Applications. Artech House, Boston, MA. Google Scholar
[35] Storvik, G. (2002). Particle filters for state-space models with the presence of unknown static parameters. IEEE Trans. Signal Process. 50, 281289. Google Scholar
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