Abstract
Partial non-Gaussian state-space models include many models of interest while keeping a convenient analytical structure. In this paper, two problems related to partial non-Gaussian models are addressed. First, we present an efficient sequential Monte Carlo method to perform Bayesian inference. Second, we derive simple recursions to compute posterior Cramér-Rao bounds (PCRB). An application to jump Markov linear systems (JMLS) is given.
Similar content being viewed by others
References
Anderson, B. D. O. and Moore, J. B. (1979). Optimal Filtering, Prentice-Hall, Englewood Cliffs, New Jersey.
Bar-Shalom, Y. and Li, X. R. (1995). Multitarget-multisensor Tracking: Principles and Techniques, YBS Publishing.
Bergman, N. (1999). Recursive Bayesian estimation: Navigation and tracking applications, Ph.D. Thesis, Department of Automatic Control, University of Linköping, Sweden.
Bergman, N., Ljung, L. and Gustafsson, F. (1999). Terrain navigation using Bayesian statistics, IEEE Control Systems Magazine, 19, 33-40.
Borobsky, B. Z. and Zakai, M. (1975). A lower bound on the estimation error for Markov processes, IEEE Trans. Automat. Control, 20(6), 785-788.
Carter, C. K. and Kohn, R. (1994). On Gibbs sampling for state space models, Biometrika, 81(3), 541-553.
Carter, C. K. and Kohn, R. (1996). Markov chain Monte Carlo methods in conditionally Gaussian state space models, Biometrika, 83, 589-601.
Crisan, D. (2001). Particle filters—A theoretical perspective, Sequential Monte Carlo Methods in Practice (eds. A. Doucet, De Freitas and N. J. Gordon), ISBN: 0-387-95146-6, Springer, New York.
Crisan, D. and Doucet, A. (2000). Convergence of sequential Monte Carlo methods, CUED/F-INFENG/TR381, Cambridge University.
Crisan, D., Del Moral, P. and Lyons, T. (1999). Discrete filtering using branching and interacting particle systems, Markov Processes and Related Fields, 5(3), 293-318.
Del Moral, P. and Guionnet, A. (1998). On the Stability of Measure Valued Processes. Applications to Non Linear Filtering and Interacting Particle Systems, Laboratoire de Statistiques et Probabilités, Toulouse.
Doerschuk, P. C. (1995). Cramér-Rao bounds for discrete-time nonlinear filtering problems, IEEE Trans. Automat. Control, 40(8), 1465-1469.
Doucet, A. (1997). Monte Carlo methods for Bayesian estimation of hidden Markov models, Ph.D. Thesis, Department of Electrical Engineering, University of Paris-Sud, Orsay, France (in French).
Doucet, A. and Andrieu, C. (1999). Iterative algorithms for state estimation of jump Markov linear systems, CUED/F-INFENG/TR353, Cambridge University.
Doucet, A., de Freitas, J. F. G. and Gordon N. J. (eds.) (2001). Sequential Monte Carlo Methods in Practice, ISBN: 0-387-95146-6, Springer, New York.
Doucet, A., Godsill S. J. and Andrieu, C. (2000). On sequential Monte Carlo sampling methods for Bayesian filtering, Statist. Comput., 10(3), 197-208.
Früwirth-Schnatter, S. (1994). Data augmentation and dynamic linear models, J. Time Ser. Anal., 15, 183-202.
Galdos, J. I. (1980). A Cramér-Rao bound for discrete-time nonlinear filtering problems, IEEE Trans. Automat. Control, 25(1), 117-119.
Gilks, W. R. and Berzuini, C. (1998). Following a moving target—Monte Carlo inference for dynamic Bayesian models (submitted to J. Roy. Statist. Soc. Ser. B).
Gill, R. D. and Levit, B. Y. (1995). Applications of the Van Trees inequality: a Bayesian Cramér Rao bound, Bernoulli, 1, 59-79.
Gordon, N. J., Salmond, D. J. and Smith, A. F. M. (1993). Novel approach to nonlinear/non-Gaussian Bayesian state estimation, IEE Proceedings-F, 140(2), 107-113.
Higuchi, T. (1997). Monte Carlo filter using the genetic algorithm operators, J. Statist. Comput. Simulation, 59(1), 1-23.
Kerr, T. (1989). Status of Cramér-Rao lower bounds for nonlinear filtering, IEEE Transactions on Aerospace and Electronic Systems, 25, 590-601.
Kitagawa, G. (1996). Monte Carlo filter and smoother for non-Gaussian nonlinear state space models, J. Comput. Graph. Statist., 5(1), 1-25.
Kitagawa, G. and Gersch, W. (1996). Smoothness Priors Analysis of Time Series, Springer, New York.
Liu, J. S. and Chen, R. (1998). Sequential Monte Carlo methods for dynamic systems, J. Amer. Statist. Assoc., 93, 1032-1044.
Pitt, M. K. and Shephard, N. (1999). Filtering via simulation: auxiliary particle filters, J. Amer. Statist. Assoc., 94, 590-599.
Robert, C. P. and Casella, G. (1999). Monte Carlo Statistical Methods, Springer, New York.
Shephard, N. (1994). Partial non-Gaussian times series models, Biometrika, 81, 115-131.
Tichavský, P., Muravchik, C. and Nehorai, A. (1998). Posterior Cramér-Rao bounds for discrete-time nonlinear filtering. IEEE Transactions on Signal Processing, 46(5), 1386-1396.
van Trees, H. L. (1968). Detection, Estimation and Modulation Theory, Wiley, New York.
West, M. (1993a). Approximating posterior distributions by mixtures, J. Roy. Statist. Soc. Ser. B, 55, 409-422.
West, M. (1993b). Mixture models, Monte Carlo, Bayesian updating and dynamic models, Computing Science and Statistics, 24, 325-333.
West, M. and Harrison, P. J. (1997). Bayesian Forecasting and Dynamic Models, 2nd ed. Springer, New York.
Author information
Authors and Affiliations
About this article
Cite this article
Bergman, N., Doucet, A. & Gordon, N. Optimal Estimation and Cramér-Rao Bounds for Partial Non-Gaussian State Space Models. Annals of the Institute of Statistical Mathematics 53, 97–112 (2001). https://doi.org/10.1023/A:1017920621802
Issue Date:
DOI: https://doi.org/10.1023/A:1017920621802