Forgetting of the initial condition for the filter in general state-space hidden Markov chain: a coupling approach

Randal Douc (Institut Telecom/ Telecom SudParis)
Eric Moulines (Institut Telecom/ Telecom ParisTech)
Yaacov Ritov (Hebrew University Jerusalem)

Abstract


We give simple conditions that ensure exponential forgetting of the initial conditions of the filter for general state-space hidden Markov chain. The proofs are based on the coupling argument applied to the posterior Markov kernels. These results are useful both for filtering hidden Markov models using approximation methods (e.g., particle filters) and for proving asymptotic properties of estimators. The results are general enough to cover models like the Gaussian state space model, without using the special structure that permits the application of the Kalman filter.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 27-49

Publication Date: January 14, 2009

DOI: 10.1214/EJP.v14-593

References

  1. Atar, R. and Zeitouni, O., Exponential stability for nonlinear filtering, Ann. Inst. H. Poincar'e Probab. Statist., 1997 Math. Review 97k:93065
  2. Baum, L.E., Petrie, T.P., Soules, G., and Weiss, N. A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. Ann. Math. Statist. Math. Review 44 #4816
  3. Budhiraja, A. and Ocone, D., Exponential stability of discrete-time filters for bounded observation noise. Systems Control Lett., 1997 Math. Review 98c:93110
  4. Budhiraja, A. and Ocone, D., Exponential stability in discrete-time filtering for non-ergodic signals, Stochastic Process. Appl.,1999 Math. Review 2000d:94010
  5. Cappe, O., Moulines, E., and Ryden, T., Inference in Hidden Markov Models, Springer, 2005. Math. Review 2006e:60002
  6. Chigansky, P. and Liptser, R., Stability of nonlinear filters in nonmixing case. Ann. Appl. Probab.,2004 Math. Review 2005h:62265
  7. Chigansky, P. and Liptser, R., On a role of predictor in the filtering stability. Electron. Comm. Probab.,2006 Math. Review 2007k:60118
  8. Del Moral, P., Feynman-Kac Formulae. Genealogical and Interacting Particle Systems with Applications. Probability and its Applications, Springer, 2004 Math. Review 2005f:60003
  9. Del Moral, P. and Guionnet, A., Large deviations for interacting particle systems: applications to non-linear filtering. Stoch. Proc. App., 1998 Math. Review 99k:60071
  10. Del Moral, P., Ledoux, M., and Miclo, L., On contraction properties of Markov kernels., Probab. Theory Related Fields, 2003 Math. Review 2004d:60202
  11. Douc, R., Moulines, E., and Ryden, T., Asymptotic properties of the maximum likelihood estimator in autoregressive models with {Markov regime. Ann. Statist., 2004 Math. Review 2005h:62226
  12. Griffeath, D. . A maximal coupling for Markov chains. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete (1974/75) Math. Review 51 #6996
  13. LeGland, F. and Oudjane, N., A robustification approach to stability and to uniform particle approximation of nonlinear filters: the example of pseudo-mixing signals. Stochastic Process. Appl., 2003 Math. Review 2004i:93184
  14. Lindvall, T. Lectures on the Coupling Method. Wiley, New-York, 1992. Math. Review 94c:60002
  15. Ocone, D. and Pardoux, E., Asymptotic stability of the optimal filter with respect to its initial condition. SIAM J. Control, 1996 Math. Review 97e:60073
  16. Oudjane, N. and Rubenthaler, S., Stability and uniform particle approximation of nonlinear filters in case of non ergodic signals. Stoch. Anal. Appl., 2005 Math. Review 2005m:93153
  17. Thorisson, H. . Coupling, Stationarity and Regeneration. Probability and its Applications. Springer-Verlag, New-York, 2000 Math. Review 2001b:60003
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.