The Annals of Statistics

Iterated filtering

Edward L. Ionides, Anindya Bhadra, Yves Atchadé, and Aaron King

Full-text: Open access


Inference for partially observed Markov process models has been a longstanding methodological challenge with many scientific and engineering applications. Iterated filtering algorithms maximize the likelihood function for partially observed Markov process models by solving a recursive sequence of filtering problems. We present new theoretical results pertaining to the convergence of iterated filtering algorithms implemented via sequential Monte Carlo filters. This theory complements the growing body of empirical evidence that iterated filtering algorithms provide an effective inference strategy for scientific models of nonlinear dynamic systems. The first step in our theory involves studying a new recursive approach for maximizing the likelihood function of a latent variable model, when this likelihood is evaluated via importance sampling. This leads to the consideration of an iterated importance sampling algorithm which serves as a simple special case of iterated filtering, and may have applicability in its own right.

Article information

Ann. Statist. Volume 39, Number 3 (2011), 1776-1802.

First available in Project Euclid: 25 July 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M09: Non-Markovian processes: estimation

Dynamic systems sequential Monte Carlo filtering importance sampling state space model partially observed Markov process


Ionides, Edward L.; Bhadra, Anindya; Atchadé, Yves; King, Aaron. Iterated filtering. Ann. Statist. 39 (2011), no. 3, 1776--1802. doi:10.1214/11-AOS886.

Export citation


  • Anderson, J. L. and Collins, N. (2007). Scalable implementations of ensemble filter algorithms for data assimilation. Journal of Atmospheric and Oceanic Technology 24 1452–1463.
  • Andrieu, C., Doucet, A. and Holenstein, R. (2010). Particle Markov chain Monte Carlo methods. J. R. Stat. Soc. Ser. B Stat. Methodol. 72 269–342.
  • Andrieu, C., Moulines, É. and Priouret, P. (2005). Stability of stochastic approximation under verifiable conditions. SIAM J. Control Optim. 44 283–312.
  • Arulampalam, M. S., Maskell, S., Gordon, N. and Clapp, T. (2002). A tutorial on particle filters for online nonlinear, non-Gaussian Bayesian tracking. IEEE Trans. Signal Process. 50 174–188.
  • Bailey, N. T. J. (1955). Some problems in the statistical analysis of epidemic data. J. Roy. Statist. Soc. Ser. B 17 35–58; Discussion 58–68.
  • Bartlett, M. S. (1960). Stochastic Population Models in Ecology and Epidemiology. Methuen, London.
  • Beyer, H.-G. (2001). The Theory of Evolution Strategies. Springer, Berlin.
  • Bjørnstad, O. N. and Grenfell, B. T. (2001). Noisy clockwork: Time series analysis of population fluctuations in animals. Science 293 638–643.
  • Bretó, C., He, D., Ionides, E. L. and King, A. A. (2009). Time series analysis via mechanistic models. Ann. Appl. Stat. 3 319–348.
  • Cappé, O., Godsill, S. and Moulines, E. (2007). An overview of existing methods and recent advances in sequential Monte Carlo. Proceedings of the IEEE 95 899–924.
  • Cappé, O., Moulines, E. and Rydén, T. (2005). Inference in Hidden Markov Models. Springer, New York.
  • Cauchemez, S. and Ferguson, N. M. (2008). Likelihood-based estimation of continuous-time epidemic models from time-series data: Application to measles transmission in London. Journal of the Royal Society Interface 5 885–897.
  • Celeux, G., Marin, J.-M. and Robert, C. P. (2006). Iterated importance sampling in missing data problems. Comput. Statist. Data Anal. 50 3386–3404.
  • Chopin, N. (2004). Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. Ann. Statist. 32 2385–2411.
  • Crisan, D. and Doucet, A. (2002). A survey of convergence results on particle filtering methods for practitioners. IEEE Trans. Signal Process. 50 736–746.
  • Del Moral, P. and Jacod, J. (2001). Interacting particle filtering with discrete observations. In Sequential Monte Carlo Methods in Practice (A. Doucet, N. de Freitas and N. J. Gordon, eds.) 43–75. Springer, New York.
  • Diggle, P. J. and Gratton, R. J. (1984). Monte Carlo methods of inference for implicit statistical models. J. Roy. Statist. Soc. Ser. B 46 193–227.
  • Durbin, J. and Koopman, S. J. (2001). Time Series Analysis by State Space Methods. Oxford Statist. Sci. Ser. 24. Oxford Univ. Press, Oxford.
  • Ergün, A., Barbieri, R., Eden, U. T., Wilson, M. A. and Brown, E. N. (2007). Construction of point process adaptive filter algorithms for neural systems using sequential Monte Carlo methods. IEEE Trans. Biomed. Eng. 54 419–428.
  • Fernández-Villaverde, J. and Rubio-Ramírez, J. F. (2007). Estimating macroeconomic models: A likelihood approach. Rev. Econom. Stud. 74 1059–1087.
  • Ferrari, M. J., Grais, R. F., Bharti, N., Conlan, A. J. K., Bjornstad, O. N., Wolfson, L. J., Guerin, P. J., Djibo, A. and Grenfell, B. T. (2008). The dynamics of measles in sub-Saharan Africa. Nature 451 679–684.
  • Godsill, S., Vermaak, J., Ng, W. and Li, J. (2007). Models and algorithms for tracking of maneuvering objects using variable rate particle filters. Proceedings of the IEEE 95 925–952.
  • Grenfell, B. T., Bjornstad, O. N. and Finkenstädt, B. F. (2002). Dynamics of measles epidemics: Scaling noise, determinism, and predictability with the TSIR model. Ecological Monographs 72 185–202.
  • Grimmett, G. R. and Stirzaker, D. R. (1992). Probability and Random Processes, 2nd ed. The Clarendon Press/Oxford Univ. Press, New York.
  • He, D., Ionides, E. L. and King, A. A. (2010). Plug-and-play inference for disease dynamics: Measles in large and small towns as a case study. Journal of the Royal Society Interface 7 271–283.
  • Ionides, E. L., Bretó, C. and King, A. A. (2006). Inference for nonlinear dynamical systems. Proc. Natl. Acad. Sci. USA 103 18438–18443.
  • Jensen, J. L. and Petersen, N. V. (1999). Asymptotic normality of the maximum likelihood estimator in state space models. Ann. Statist. 27 514–535.
  • Johannes, M., Polson, N. and Stroud, J. (2009). Optimal filtering of jump diffusions: Extracting latent states from asset prices. Review of Financial Studies 22 2759–2799.
  • Johansen, A. M., Doucet, A. and Davy, M. (2008). Particle methods for maximum likelihood estimation in latent variable models. Stat. Comput. 18 47–57.
  • Keeling, M. and Ross, J. (2008). On methods for studying stochastic disease dynamics. Journal of the Royal Society Interface 5 171–181.
  • Kendall, B. E., Briggs, C. J., Murdoch, W. W., Turchin, P., Ellner, S. P., McCauley, E., Nisbet, R. M. and Wood, S. N. (1999). Why do populations cycle? A synthesis of statistical and mechanistic modeling approaches. Ecology 80 1789–1805.
  • Kiefer, J. and Wolfowitz, J. (1952). Stochastic estimation of the maximum of a regression function. Ann. Math. Statist. 23 462–466.
  • King, A. A., Ionides, E. L., Pascual, M. and Bouma, M. J. (2008). Inapparent infections and cholera dynamics. Nature 454 877–880.
  • Kitagawa, G. (1998). A self-organising state-space model. J. Amer. Statist. Assoc. 93 1203–1215.
  • Kushner, H. J. and Clark, D. S. (1978). Stochastic Approximation Methods for Constrained and Unconstrained Systems. Appl. Math. Sci. 26. Springer, New York.
  • Kushner, H. J. and Yin, G. G. (2003). Stochastic Approximation and Recursive Algorithms and Applications, 2nd ed. Applications of Mathematics: Stochastic Modelling and Applied Probability 35. Springer, New York.
  • Laneri, K., Bhadra, A., Ionides, E. L., Bouma, M., Yadav, R., Dhiman, R. and Pascual, M. (2010). Forcing versus feedback: Epidemic malaria and monsoon rains in NW India. PLoS Comput. Biol. 6 e1000898.
  • Liu, J. and West, M. (2001). Combined parameter and state estimation in simulation-based filtering. In Sequential Monte Carlo Methods in Practice (A. Doucet, N. de Freitas and N. J. Gordon, eds.) 197–223. Springer, New York.
  • Maryak, J. L. and Chin, D. C. (2008). Global random optimization by simultaneous perturbation stochastic approximation. IEEE Trans. Automat. Control 53 780–783.
  • Morton, A. and Finkenstädt, B. F. (2005). Discrete time modelling of disease incidence time series by using Markov chain Monte Carlo methods. J. Roy. Statist. Soc. Ser. C 54 575–594.
  • Newman, K. B., Fernández, C., Thomas, L. and Buckland, S. T. (2009). Monte Carlo inference for state-space models of wild animal populations. Biometrics 65 572–583.
  • Qian, Z. and Shapiro, A. (2006). Simulation-based approach to estimation of latent variable models. Comput. Statist. Data Anal. 51 1243–1259.
  • Reuman, D. C., Desharnais, R. A., Costantino, R. F., Ahmad, O. S. and Cohen, J. E. (2006). Power spectra reveal the influence of stochasticity on nonlinear population dynamics. Proc. Natl. Acad. Sci. USA 103 18860–18865.
  • Rogers, L. C. G. and Williams, D. (1994). Diffusions, Markov Processes, and Martingales. Vol. 1: Foundations, 2nd ed. Wiley, Chichester.
  • Spall, J. C. (2003). Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control. Wiley-Interscience, Hoboken, NJ.
  • Toni, T., Welch, D., Strelkowa, N., Ipsen, A. and Stumpf, M. P. (2008). Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems. Journal of the Royal Society Interface 6 187–202.
  • West, M. and Harrison, J. (1997). Bayesian Forecasting and Dynamic Models, 2nd ed. Springer, New York.