Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-24T02:13:19.276Z Has data issue: false hasContentIssue false
  • English
  • Français

On the asymptotic variance in the central limit theorem for particle filters

Published online by Cambridge University Press:  03 July 2012

Benjamin Favetto
Benjamin Favetto*
Affiliation:
Laboratoire MAP5, Université Paris Descartes, U.F.R. de Mathématique et Informatique, CNRS UMR 8145, 45, rue des Saints-Pères, 75270 Paris Cedex 06, France. Benjamin.Favetto@mi.parisdescartes.fr
Get access

Abstract

Particle filter algorithms approximate a sequence of distributions by a sequence of empirical measures generated by a population of simulated particles. In the context of Hidden Markov Models (HMM), they provide approximations of the distribution of optimal filters associated to these models. For a given set of observations, the behaviour of particle filters, as the number of particles tends to infinity, is asymptotically Gaussian, and the asymptotic variance in the central limit theorem depends on the set of observations. In this paper we establish, under general assumptions on the hidden Markov model, the tightness of the sequence of asymptotic variances when considered as functions of the random observations as the number of observations tends to infinity. We discuss our assumptions on examples and provide numerical simulations.

Keywords

Hidden Markov ModelParticle filterCentral Limit TheoremAsymptotic varianceTightnessSequential Monte-Carlo.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 16 , 2012 , pp. 151 - 164
Copyright
© EDP Sciences, SMAI, 2012

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

Références

Atar, R. and Zeitouni, O., Exponential stability for nonlinear filtering. Ann. Inst. Henri Poincaré 33 (1997) 697–725. Google Scholar
O. Cappé, E. Moulines and T. Ryden, Inference in Hidden Markov Models, in Springer Series in Statistics. Springer-Verlag New York, Inc., Secaucus, NJ, USA (2005).
Chaleyat-Maurel, M. and Genon-Catalot, V., Computable infinite-dimensional filters with applications to discretized diffusion processes. Stoc. Proc. Appl. 116 (2006) 1447–1467. Google Scholar
Chopin, N., Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. Ann. Statist. 32 (2004) 2385–2411. Google Scholar
E.B. Davies, Heat kernels and spectral theory, in Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge 92 (1989).
P. Del Moral, Feynman-Kac formulae, Genealogical and interacting particle systems with applications. Probab. Appl. Springer-Verlag, New York (2004).
Del Moral, P. and Guionnet, A., On the stability of interacting processes with applications to filtering and genetic algorithms. Ann. Inst. Henri Poincaré 37 (2001) 155–194. Google Scholar
P. Del Moral and J. Jacod, Interacting particle filtering with discrete observations, in Sequential Monte Carlo methods in practice, Springer, New York. Stat. Eng. Inf. Sci. (2001) 43–75.
P. Del Moral and J. Jacod, Interacting particle filtering with discrete-time observations : asymptotic behaviour in the Gaussian case, in Stochastics in finite and infinite dimensions, Birkhäuser Boston, Boston, MA. Trends Math. (2001) 101–122.
Douc, R., Guillin, A. and Najim, J., Moderate deviations for particle filtering. Ann. Appl. Probab. 15 (2005) 587–614. Google Scholar
Douc, R., Fort, G., Moulines, E. and Priouret, P., Forgetting of the initial distribution for hidden Markov models. Stoc. Proc. Appl. 119 (2009) 1235–1256. Google Scholar
A. Doucet, N. de Freitas and N. Gordon, Sequential Monte Carlo methods in practice, Stat. Eng. Inform. Sci. Springer-Verlag, New York (2001).
H.R. Künsch, State space and hidden Markov models, in Complex Stochastic Systems. Eindhoven (1999); Chapman & Hall/CRC, Boca Raton, FL. Monogr. Statist. Appl. Probab. 87 (2001) 109–173.
Künsch, H.R., Recursive Monte Carlo filters : algorithms and theoretical analysis. Ann. Statist. 33 (2005) 1983–2021. Google Scholar
Oudjane, N. and Rubenthaler, S., Stability and uniform particle approximation of nonlinear filters in case of non ergodic signals. Stoch. Anal. Appl. 23 (2005) 421–448. Google Scholar
C.P. Robert and G. Casella, Monte Carlo statistical methods, 2nd edition, in Springer Texts in Statistics. Springer-Verlag, New York (2004).
Van Handel, R., Uniform time average consistency of Monte Carlo particle filters. Stoc. Proc. Appl. 119 (2009) 3835–3861. Google Scholar