Statistical Science

On Particle Methods for Parameter Estimation in State-Space Models

Nikolas Kantas, Arnaud Doucet, Sumeetpal S. Singh, Jan Maciejowski, and Nicolas Chopin

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Abstract

Nonlinear non-Gaussian state-space models are ubiquitous in statistics, econometrics, information engineering and signal processing. Particle methods, also known as Sequential Monte Carlo (SMC) methods, provide reliable numerical approximations to the associated state inference problems. However, in most applications, the state-space model of interest also depends on unknown static parameters that need to be estimated from the data. In this context, standard particle methods fail and it is necessary to rely on more sophisticated algorithms. The aim of this paper is to present a comprehensive review of particle methods that have been proposed to perform static parameter estimation in state-space models. We discuss the advantages and limitations of these methods and illustrate their performance on simple models.

Article information

Source
Statist. Sci., Volume 30, Number 3 (2015), 328-351.

Dates
First available in Project Euclid: 10 August 2015

Permanent link to this document
https://projecteuclid.org/euclid.ss/1439220716

Digital Object Identifier
doi:10.1214/14-STS511

Mathematical Reviews number (MathSciNet)
MR3383884

Zentralblatt MATH identifier
1332.62096

Keywords
Bayesian inference maximum likelihood inference particle filtering Sequential Monte Carlo state-space models

Citation

Kantas, Nikolas; Doucet, Arnaud; Singh, Sumeetpal S.; Maciejowski, Jan; Chopin, Nicolas. On Particle Methods for Parameter Estimation in State-Space Models. Statist. Sci. 30 (2015), no. 3, 328--351. doi:10.1214/14-STS511. https://projecteuclid.org/euclid.ss/1439220716


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References

  • [1] Alspach, D. and Sorenson, H. (1972). Nonlinear Bayesian estimation using Gaussian sum approximations. IEEE Trans. Automat. Control 17 439–448.
  • [2] Andrieu, C., De Freitas, J. F. G. and Doucet, A. (1999). Sequential MCMC for Bayesian model selection. In Proc. IEEE Workshop Higher Order Statistics 130–134. IEEE, New York.
  • [3] 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.
  • [4] Andrieu, C., Doucet, A. and Tadić, V. B. (2005). On-line parameter estimation in general state-space models. In Proc. 44th IEEE Conf. on Decision and Control 332–337. IEEE, New York.
  • [5] Benveniste, A., Métivier, M. and Priouret, P. (1990). Adaptive Algorithms and Stochastic Approximations. Applications of Mathematics (New York) 22. Springer, Berlin.
  • [6] Briers, M., Doucet, A. and Maskell, S. (2010). Smoothing algorithms for state-space models. Ann. Inst. Statist. Math. 62 61–89.
  • [7] Briers, M., Doucet, A. and Singh, S. S. (2005). Sequential auxiliary particle belief propagation. In Proc. Conf. Fusion. Philadelphia, PA.
  • [8] Cappé, O. (2009). Online sequential Monte Carlo EM algorithm. In Proc. 15th IEEE Workshop on Statistical Signal Processing 37–40. IEEE, New York.
  • [9] Cappé, O. (2011). Online EM algorithm for hidden Markov models. J. Comput. Graph. Statist. 20 728–749.
  • [10] Cappé, O. and Moulines, E. (2009). On-line expectation–maximization algorithm for latent data models. J. R. Stat. Soc. Ser. B. Stat. Methodol. 71 593–613.
  • [11] Cappé, O., Moulines, E. and Rydén, T. (2005). Inference in Hidden Markov Models. Springer, New York.
  • [12] Carpenter, J., Clifford, P. and Fearnhead, P. (1999). An improved particle filter for non-linear problems. IEE Proceedings—Radar, Sonar and Navigation 146 2–7.
  • [13] Carvalho, C. M., Johannes, M. S., Lopes, H. F. and Polson, N. G. (2010). Particle learning and smoothing. Statist. Sci. 25 88–106.
  • [14] Cérou, F., Del Moral, P. and Guyader, A. (2011). A nonasymptotic theorem for unnormalized Feynman–Kac particle models. Ann. Inst. Henri Poincaré, B Probab. Stat. 47 629–649.
  • [15] Chen, R. and Liu, J. S. (2000). Mixture Kalman filters. J. R. Stat. Soc. Ser. B. Stat. Methodol. 62 493–508.
  • [16] Chopin, N. (2002). A sequential particle filter method for static models. Biometrika 89 539–551.
  • [17] Chopin, N. (2004). Central limit theorem for sequential Monte Carlo methods and its application to Bayesian inference. Ann. Statist. 32 2385–2411.
  • [18] Chopin, N., Iacobucci, A., Marin, J. M., Mengersen, K., Robert, C. P., Ryder, R. and Schäufer, C. (2011). On particle learning. In Bayesian Statistics 9 (J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D., Heckerman A. F. M. Smith and M., West, eds.) 317–360. Oxford Univ. Press, Oxford.
  • [19] Chopin, N., Jacob, P. E. and Papaspiliopoulos, O. (2013). $\mathrm{SMC}^{2}$: An efficient algorithm for sequential analysis of state space models. J. R. Stat. Soc. Ser. B. Stat. Methodol. 75 397–426.
  • [20] Coquelin, P. A., Deguest, R. and Munos, R. (2009). Sensitivity analysis in HMMs with application to likelihood maximization. In Proc. 22th Conf. NIPS. Vancouver.
  • [21] Dahlin, J., Lindsten, F. and Schön, T. B. (2015). Particle Metropolis–Hastings using gradient and Hessian information. Stat. Comput. 25 81–92.
  • [22] DeJong, D. N., Liesenfeld, R., Moura, G. V., Richard, J.-F. and Dharmarajan, H. (2013). Efficient likelihood evaluation of state-space representations. Rev. Econ. Stud. 80 538–567.
  • [23] Del Moral, P. (2004). Feynman–Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Springer, New York.
  • [24] Del Moral, P., Doucet, A. and Singh, S. S. (2009). Forward smoothing using sequential Monte Carlo. Technical Report 638, CUED-F-INFENG, Cambridge Univ. Preprint. Available at arXiv:1012.5390.
  • [25] Del Moral, P., Doucet, A. and Singh, S. S. (2010). A backward particle interpretation of Feynman–Kac formulae. ESAIM Math. Model. Numer. Anal. 44 947–975.
  • [26] Del Moral, P., Doucet, A. and Singh, S. S. (2015). Uniform stability of a particle approximation of the optimal filter derivative. SIAM J. Control Optim. 53 1278–1304.
  • [27] Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. Ser. B 39 1–38.
  • [28] Douc, R., Garivier, A., Moulines, E. and Olsson, J. (2011). Sequential Monte Carlo smoothing for general state space hidden Markov models. Ann. Appl. Probab. 21 2109–2145.
  • [29] Douc, R., Moulines, E. and Ritov, Y. (2009). Forgetting of the initial condition for the filter in general state-space hidden Markov chain: A coupling approach. Electron. J. Probab. 14 27–49.
  • [30] Doucet, A., De Freitas, J. F. G. and Gordon, N. J., eds. (2001). Sequential Monte Carlo Methods in Practice. Springer, New York.
  • [31] Doucet, A., Godsill, S. J. and Andrieu, C. (2000). On sequential Monte Carlo sampling methods for Bayesian filtering. Stat. Comput. 10 197–208.
  • [32] Doucet, A. and Johansen, A. M. (2011). A tutorial on particle filtering and smoothing: Fifteen years later. In The Oxford Handbook of Nonlinear Filtering 656–704. Oxford Univ. Press, Oxford.
  • [33] Doucet, A., Pitt, M. K., Deligiannidis, G. and Kohn, R. (2015). Efficient implementation of Markov chain Monte Carlo when using an unbiased likelihood estimator. Biometrika 102 295–313.
  • [34] Elliott, R. J., Aggoun, L. and Moore, J. B. (1995). Hidden Markov Models: Estimation and Control. Applications of Mathematics (New York) 29. Springer, New York.
  • [35] Elliott, R. J., Ford, J. J. and Moore, J. B. (2000). On-line consistent estimation of hidden Markov models. Technical report, Dept. Systems Engineering, Australian National Univ., Canberra.
  • [36] Fearnhead, P. (2002). Markov chain Monte Carlo, sufficient statistics, and particle filters. J. Comput. Graph. Statist. 11 848–862.
  • [37] Fearnhead, P. and Meligkotsidou, L. (2014). Augmentation schemes for particle MCMC. Preprint. Available at arXiv:1408.6980.
  • [38] Fearnhead, P., Wyncoll, D. and Tawn, J. (2010). A sequential smoothing algorithm with linear computational cost. Biometrika 97 447–464.
  • [39] Fernández-Villaverde, J. and Rubio-Ramírez, J. F. (2007). Estimating macroeconomic models: A likelihood approach. Rev. Econ. Stud. 74 1059–1087.
  • [40] Flury, T. and Shephard, N. (2009). Learning and filtering via simulation: Smoothly jittered particle filters. Economics Series Working Papers 469.
  • [41] Flury, T. and Shephard, N. (2011). Bayesian inference based only on simulated likelihood: Particle filter analysis of dynamic economic models. Econometric Theory 27 933–956.
  • [42] Ford, J. J. (1998). Adaptive hidden Markov model estimation and applications. Ph.D. thesis, Dept. Systems Engineering, Australian National Univ., Canberra. Available at http://infoeng.rsise.anu.edu.au/files/jason_ford_thesis.pdf.
  • [43] Fulop, A. and Li, J. (2013). Efficient learning via simulation: A marginalized resample–move approach. J. Econometrics 176 146–161.
  • [44] Gilks, W. R. and Berzuini, C. (2001). Following a moving target—Monte Carlo inference for dynamic Bayesian models. J. R. Stat. Soc. Ser. B. Stat. Methodol. 63 127–146.
  • [45] Godsill, S. J., Doucet, A. and West, M. (2004). Monte Carlo smoothing for nonlinear times series. J. Amer. Statist. Assoc. 99 156–168.
  • [46] Gordon, N. J., Salmond, D. J. and Smith, A. F. M. (1993). Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proc. F, Comm., Radar, Signal. Proc. 140 107–113.
  • [47] Higuchi, T. (2001). Self-organizing time series model. In Sequential Monte Carlo Methods in Practice. Stat. Eng. Inf. Sci. 429–444. Springer, New York.
  • [48] Hürzeler, M. and Künsch, H. R. (1998). Monte Carlo approximations for general state-space models. J. Comput. Graph. Statist. 7 175–193.
  • [49] Hürzeler, M. and Künsch, H. R. (2001). Approximating and maximising the likelihood for a general state-space model. In Sequential Monte Carlo Methods in Practice. Stat. Eng. Inf. Sci. 159–175. Springer, New York.
  • [50] Ionides, E. L., Bhadra, A., Atchadé, Y. and King, A. (2011). Iterated filtering. Ann. Statist. 39 1776–1802.
  • [51] Ionides, E. L., Bretó, C. and King, A. A. (2006). Inference for nonlinear dynamical systems. Proc. Natl. Acad. Sci. USA 103 18438–18443.
  • [52] Kim, S., Shephard, N. and Chib, S. (1998). Stochastic volatility: Likelihood inference and comparison with ARCH models. Rev. Econ. Stud. 65 361–393.
  • [53] Kitagawa, G. (1996). Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. J. Comput. Graph. Statist. 5 1–25.
  • [54] Kitagawa, G. (1998). A self-organizing state-space model. J. Amer. Statist. Assoc. 93 1203–1215.
  • [55] Kitagawa, G. (2014). Computational aspects of sequential Monte Carlo filter and smoother. Ann. Inst. Statist. Math. 66 443–471.
  • [56] Kitagawa, G. and Sato, S. (2001). Monte Carlo smoothing and self-organising state-space model. In Sequential Monte Carlo Methods in Practice. Stat. Eng. Inf. Sci. 177–195. Springer, New York.
  • [57] Klaas, M., Briers, M., De Freitas, N., Doucet, A., Maskell, S. and Lang, D. (2006). Fast particle smoothing: If I had a million particles. In Proc. International Conf. Machine Learning 481–488. Pittsburgh, PA.
  • [58] Künsch, H. R. (2013). Particle filters. Bernoulli 19 1391–1403.
  • [59] Lee, A. (2008). Towards smoother multivariate particle filters. M.Sc. Computer Science, Univ. British Columbia, Vancouver, BC.
  • [60] Lee, A. and Whiteley, N. (2014). Forest resampling for distributed sequential Monte Carlo. Preprint. Available at arXiv:1406.6010.
  • [61] Lee, D. S. and Chia, K. K. (2002). A particle algorithm for sequential Bayesian parameter estimation and model selection. IEEE Trans. Signal Process. 50 326–336.
  • [62] Le Corff, S. and Fort, G. (2013). Online expectation maximization based algorithms for inference in hidden Markov models. Electron. J. Stat. 7 763–792.
  • [63] Le Corff, S. and Fort, G. (2013). Convergence of a particle-based approximation of the block online expectation maximization algorithm. ACM Trans. Model. Comput. Simul. 23 Art. 2, 22.
  • [64] Le Gland, F. and Mevel, M. (1997). Recursive estimation in hidden Markov models. In Proc. 36th IEEE Conf. Decision and Control 3468–3473. San Diego, CA.
  • [65] Lin, M., Chen, R. and Liu, J. S. (2013). Lookahead strategies for sequential Monte Carlo. Statist. Sci. 28 69–94.
  • [66] Lindsten, F., Jordan, M. I. and Schön, T. B. (2014). Particle Gibbs with ancestor sampling. J. Mach. Learn. Res. 15 2145–2184.
  • [67] Liu, J. and West, M. (2001). Combined parameter and state estimation in simulation-based filtering. In Sequential Monte Carlo Methods in Practice. Springer, New York.
  • [68] Liu, J. S. (2001). Monte Carlo Strategies in Scientific Computing. Springer, New York.
  • [69] Liu, J. S. and Chen, R. (1998). Sequential Monte Carlo methods for dynamic systems. J. Amer. Statist. Assoc. 93 1032–1044.
  • [70] Lopes, H. F., Carvalho, C. M., Johannes, M. S. and Polson, N. G. (2011). Particle learning for sequential Bayesian computation. In Bayesian Statistics 9 (J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D., Heckerman A. F. M. Smith and M., West, eds.). Oxford Univ. Press, Oxford.
  • [71] Lopes, H. F. and Tsay, R. S. (2011). Particle filters and Bayesian inference in financial econometrics. J. Forecast. 30 168–209.
  • [72] Malik, S. and Pitt, M. K. (2011). Particle filters for continuous likelihood evaluation and maximisation. J. Econometrics 165 190–209.
  • [73] Nemeth, C., Fearnhead, P. and Mihaylova, L. (2013). Particle approximations of the score and observed information matrix for parameter estimation in state space models with linear computational cost. Preprint. Available at arXiv:1306.0735.
  • [74] Olsson, J., Cappé, O., Douc, R. and Moulines, E. (2008). Sequential Monte Carlo smoothing with application to parameter estimation in nonlinear state space models. Bernoulli 14 155–179.
  • [75] Olsson, J. and Westerborn, J. (2014). Efficient particle-based online smoothing in general hidden Markov models: The PaRIS algorithm. Preprint. Available at arXiv:1412.7550.
  • [76] Oudjane, N. and Rubenthaler, S. (2005). Stability and uniform particle approximation of nonlinear filters in case of non ergodic signals. Stoch. Anal. Appl. 23 421–448.
  • [77] Paninski, L., Ahmadian, Y., Ferreira, D. G., Koyama, S., Rad, K. R., Vidne, M., Vogelstein, J. and Wu, W. (2010). A new look at state-space models for neural data. J. Comput. Neurosci. 29 107–126.
  • [78] Pitt, M. K. and Shephard, N. (1999). Filtering via simulation: Auxiliary particle filters. J. Amer. Statist. Assoc. 94 590–599.
  • [79] Pitt, M. K., Silva, R. d. S., Giordani, P. and Kohn, R. (2012). On some properties of Markov chain Monte Carlo simulation methods based on the particle filter. J. Econometrics 171 134–151.
  • [80] Polson, N. G., Stroud, J. R. and Müller, P. (2008). Practical filtering with sequential parameter learning. J. R. Stat. Soc. Ser. B. Stat. Methodol. 70 413–428.
  • [81] Poyiadjis, G., Doucet, A. and Singh, S. S. (2011). Particle approximations of the score and observed information matrix in state space models with application to parameter estimation. Biometrika 98 65–80.
  • [82] Schön, T. B., Wills, A. and Ninness, B. (2011). System identification of nonlinear state-space models. Automatica J. IFAC 47 39–49.
  • [83] Sherlock, C., Thiery, A. H., Roberts, G. O. and Rosenthal, J. S. (2015). On the efficiency of pseudo-marginal random walk Metropolis algorithms. Ann. Statist. 43 238–275.
  • [84] Storvik, G. (2002). Particle filters in state space models with the presence of unknown static parameters. IEEE Trans. Signal Process. 50 281–289.
  • [85] Taghavi, E., Lindsten, F., Svensson, L. and Schön, T. B. (2013). Adaptive stopping for fast particle smoothing. In Proc. IEEE ICASSP 6293–6297. Vancouver, BC.
  • [86] Vercauteren, T., Toledo, A. and Wang, X. (2005). Online Bayesian estimation of hidden Markov models with unknown transition matrix and applications to IEEE 802.11 networks. In Proc. IEEE ICASSP, Vol. IV 13–16. Philadelphia, PA.
  • [87] West, M. and Harrison, J. (1997). Bayesian Forecasting and Dynamic Models, 2nd ed. Springer, New York.
  • [88] Westerborn, J. and Olsson, J. (2014). Efficient particle-based online smoothing in general hidden Markov models. In Proc. IEEE ICASSP 8003–8007. Florence.
  • [89] Whiteley, N. (2010). Discussion of Particle Markov chain Monte Carlo methods. J. Royal Stat. Soc. 72 306–307.
  • [90] Whiteley, N. (2013). Stability properties of some particle filters. Ann. Appl. Probab. 23 2500–2537.
  • [91] Whiteley, N., Andrieu, C. and Doucet, A. (2010). Efficient Bayesian inference for switching state–space models using discrete particle Markov chain Monte Carlo methods. Preprint. Available at arXiv:1011.2437.
  • [92] Whiteley, N. and Lee, A. (2014). Twisted particle filters. Ann. Statist. 42 115–141.
  • [93] Wilkinson, D. J. (2012). Stochastic Modelling for Systems Biology, 2nd ed. CRC Press, Boca Raton, FL.
  • [94] Yildirim, S., Singh, S. S. and Doucet, A. (2013). An online expectation–maximization algorithm for changepoint models. J. Comput. Graph. Statist. 22 906–926.
  • [95] Yuan, Y.-x. (2008). Step-sizes for the gradient method. In Third International Congress of Chinese Mathematicians. Part 1, 2. AMS/IP Stud. Adv. Math., 42, Pt. 1 2 785–796. Amer. Math. Soc., Providence, RI.