The Annals of Statistics
- Ann. Statist.
- Volume 39, Number 3 (2011), 1776-1802.
Iterated filtering
Edward L. Ionides, Anindya Bhadra, Yves Atchadé, and Aaron King
Full-text: Access has been disabled (more information)
Abstract
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
Source
Ann. Statist. Volume 39, Number 3 (2011), 1776-1802.
Dates
First available in Project Euclid: 25 July 2011
Permanent link to this document
http://projecteuclid.org/euclid.aos/1311600283
Digital Object Identifier
doi:10.1214/11-AOS886
Mathematical Reviews number (MathSciNet)
MR2850220
Zentralblatt MATH identifier
1220.62103
Subjects
Primary: 62M09: Non-Markovian processes: estimation
Keywords
Dynamic systems sequential Monte Carlo filtering importance sampling state space model partially observed Markov process
Citation
Ionides, Edward L.; Bhadra, Anindya; Atchadé, Yves; King, Aaron. Iterated filtering. Ann. Statist. 39 (2011), no. 3, 1776--1802. doi:10.1214/11-AOS886. http://projecteuclid.org/euclid.aos/1311600283.
References
- 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.Mathematical Reviews (MathSciNet): MR2758115
Digital Object Identifier: doi:10.1111/j.1467-9868.2009.00736.x - Andrieu, C., Moulines, É. and Priouret, P. (2005). Stability of stochastic approximation under verifiable conditions. SIAM J. Control Optim. 44 283–312.Mathematical Reviews (MathSciNet): MR2177157
Zentralblatt MATH: 1083.62073
Digital Object Identifier: doi:10.1137/S0363012902417267 - 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.Mathematical Reviews (MathSciNet): MR73090
- 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.Mathematical Reviews (MathSciNet): MR2159833
- 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.Mathematical Reviews (MathSciNet): MR2236856
- Chopin, N. (2004). Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. Ann. Statist. 32 2385–2411.Mathematical Reviews (MathSciNet): MR2153989
Zentralblatt MATH: 1079.65006
Digital Object Identifier: doi:10.1214/009053604000000698
Project Euclid: euclid.aos/1107794873 - 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.Mathematical Reviews (MathSciNet): MR1847786
- 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.Mathematical Reviews (MathSciNet): MR781880
- 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.Mathematical Reviews (MathSciNet): MR1199812
- 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.Mathematical Reviews (MathSciNet): MR1714719
Zentralblatt MATH: 0952.62023
Digital Object Identifier: doi:10.1214/aos/1018031205
Project Euclid: euclid.aos/1018031205 - 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.Mathematical Reviews (MathSciNet): MR2416438
Digital Object Identifier: doi:10.1007/s11222-007-9037-8 - 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.Zentralblatt MATH: 1178.05100
- 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.Mathematical Reviews (MathSciNet): MR499560
- 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.Mathematical Reviews (MathSciNet): MR2137255
Zentralblatt MATH: 05188699
Digital Object Identifier: doi:10.1111/j.1467-9876.2005.05366.x - 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.Mathematical Reviews (MathSciNet): MR2297520
- 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.Mathematical Reviews (MathSciNet): MR1331599
- 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.Mathematical Reviews (MathSciNet): MR1482232

- You have access to this content.
- You have partial access to this content.
- You do not have access to this content.
More like this
- Theory of segmented particle filters
Chan, Hock Peng, Heng, Chiang-Wee, and Jasra, Ajay, Advances in Applied Probability, 2016 - Online Expectation Maximization based algorithms for inference in Hidden Markov Models
Le Corff, Sylvain and Fort, Gersende, Electronic Journal of Statistics, 2013 - The General Projected Normal Distribution of Arbitrary Dimension: Modeling and Bayesian Inference
Hernandez-Stumpfhauser, Daniel, Breidt, F. Jay, and van der Woerd, Mark J., Bayesian Analysis, 2017
- Theory of segmented particle filters
Chan, Hock Peng, Heng, Chiang-Wee, and Jasra, Ajay, Advances in Applied Probability, 2016 - Online Expectation Maximization based algorithms for inference in Hidden Markov Models
Le Corff, Sylvain and Fort, Gersende, Electronic Journal of Statistics, 2013 - The General Projected Normal Distribution of Arbitrary Dimension: Modeling and Bayesian Inference
Hernandez-Stumpfhauser, Daniel, Breidt, F. Jay, and van der Woerd, Mark J., Bayesian Analysis, 2017 - Profile-kernel likelihood inference with diverging number of parameters
Lam, Clifford and Fan, Jianqing, The Annals of Statistics, 2008 - A Problem in Particle Physics and Its Bayesian Analysis
Landon, Joshua, Lee, Frank X., and Singpurwalla, Nozer D., Statistical Science, 2011 - Recursive Monte Carlo filters: Algorithms and theoretical analysis
Künsch, Hans R., The Annals of Statistics, 2005 - Exact and Approximate Bayesian Inference for Low Integer-Valued Time Series Models with Intractable Likelihoods
Drovandi, Christopher C., Pettitt, Anthony N., and McCutchan, Roy A., Bayesian Analysis, 2016 - Importance sampling on coalescent histories. II: Subdivided population models
De Iorio, Maria and Griffiths, Robert C., Advances in Applied Probability, 2004 - Recursive Learning for Sparse Markov Models
Xiong, Jie, Jääskinen, Väinö, and Corander, Jukka, Bayesian Analysis, 2016 - Statistical testing of covariate effects in conditional copula models
Acar, Elif F., Craiu, Radu V., and Yao, Fang, Electronic Journal of Statistics, 2013
