Statistical Science

Particle Learning and Smoothing

Carlos M. Carvalho, Michael S. Johannes, Hedibert F. Lopes, and Nicholas G. Polson

Full-text: Open access


Particle learning (PL) provides state filtering, sequential parameter learning and smoothing in a general class of state space models. Our approach extends existing particle methods by incorporating the estimation of static parameters via a fully-adapted filter that utilizes conditional sufficient statistics for parameters and/or states as particles. State smoothing in the presence of parameter uncertainty is also solved as a by-product of PL. In a number of examples, we show that PL outperforms existing particle filtering alternatives and proves to be a competitor to MCMC.

Article information

Statist. Sci. Volume 25, Number 1 (2010), 88-106.

First available: 3 August 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)


Carvalho, Carlos M.; Johannes, Michael S.; Lopes, Hedibert F.; Polson, Nicholas G. Particle Learning and Smoothing. Statistical Science 25 (2010), no. 1, 88--106. doi:10.1214/10-STS325.

Export citation


  • Briers, M., Doucet, A. and Maskell, S. (2010). Smoothing algorithms for state-space models. Ann. Inst. Statist. Math. 62 61–89.
  • Cappé, O., Godsill, S. and Moulines, E. (2007). An overview of existing methods and recent advances in sequential Monte Carlo. IEEE Proceedings 95 899–924.
  • Carlin, B., Polson, N. G. and Stoffer, D. (1992). A Monte Carlo approach to nonnormal and nonlinear state-space modeling. J. Amer. Statist. Assoc. 87 493–500.
  • Carter, C. and Kohn, R. (1994). On Gibbs sampling for state space models. Biometrika 82 339–350.
  • Carvalho, C. M. and Lopes, H. F. (2007). Simulation-based sequential analysis of Markov switching stochastic volatility models. Comput. Statist. Data Anal. 51 4526–4542.
  • Carvalho, C. M., Lopes, H. F., Polson, N. G. and Taddy, M. (2009). Particle learning for general mixtures. Working paper, Univ. Chicago Booth School of Business.
  • Chen, R. and Liu, J. (2000). Mixture Kalman filters. J. Roy. Statist. Soc. Ser. B 62 493–508.
  • Doucet, A., de Freitas, J. and Gordon, N. (2001). Sequential Monte Carlo Methods in Practice. Springer, New York.
  • Fearnhead, P. (2002). Markov chain Monte Carlo, sufficient statistics, and particle filters. J. Comput. Graph. Statist. 11 848–862.
  • Fearnhead, P., Wyncoll, D. and Tawn, J. (2008). A sequential smoothing algorithm with linear computational cost. Working paper, Dept. Mathematics and Statistics, Lancaster Univ.
  • Frühwirth-Schnatter, S. (1994). Applied state space modelling of non-Gaussian time series using integration-based Kalman filtering. Statist. Comput. 4 259–269.
  • Gamerman, D. and Lopes, H. F. (2006). Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference. Chapman & Hall/CRC Press, Boca Raton, FL.
  • Godsill, S. J., Doucet, A. and West, M. (2004). Monte Carlo smoothing for nonlinear time series. J. Amer. Statist. Assoc. 99 156–168.
  • Gordon, N., Salmond, D. and Smith, A. F. M. (1993). Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proceedings-F 140 107–113.
  • Johannes, M. and Polson, N. G. (2008). Exact particle filtering and learning. Working paper, Univ. Chicago Booth School of Business.
  • Johannes, M., Polson, N. G. and Yae, S. M. (2008). Nonlinear filtering and learning. Working paper, Univ. Chicago Booth School of Business.
  • Kalman, R. E. (1960). A new approach to linear filtering and prediction problems. Transactions of the ASME—Journal of Basic Engineering 82 35–45.
  • Liu, J. and Chen, R. (1998). Sequential Monte Carlo methods for dynamic systems. J. Amer. Statist. Assoc. 93 1032–1044.
  • Liu, J. and West, M. (2001). Combined parameters and state estimation in simulation-based filtering. In Sequential Monte Carlo Methods in Practice (A. Doucet, N. de Freitas and N. Gordon, eds.). Springer, New York.
  • Lopes, H. F. and Polson, N. G. (2010). Extracting SP500 and NASDAQ volatility: The credit crisis of 2007–2008. In Handbook of Applied Bayesian Analysis (A. O’Hagan and M. West, eds.) 319–342. Oxford Univ. Press, Oxford.
  • Lopes, H. F. and Tsay, R. E. (2010). Bayesian analysis of financial time series via particle filters. J. Forecast. To appear.
  • Lopes, H. F., Carvalho, C. M., Johannes, M. and Polson, N. G. (2010). 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.
  • Papaspiliopoulos, O. and Roberts, G. (2008). Stability of the Gibbs sampler for Bayesian hierarchical models. Ann. Statist. 36 95–117.
  • Pitt, M. and Shephard, N. (1999). Filtering via simulation: Auxiliary particle filters. J. Amer. Statist. Assoc. 94 590–599.
  • Polson, N. G., Stroud, J. and Müller, P. (2008). Practical filtering with sequential parameter learning. J. Roy. Statist. Soc. Ser. B 70 413–428.
  • Prado, R. and Lopes, H. F. (2010). Sequential parameter learning and filtering in structured autoregressive models. Working paper, Univ. Chicago Booth School of Business.
  • Storvik, G. (2002). Particle filters in state space models with the presence of unknown static parameters. IEEE Trans. Signal Process. 50 281–289.
  • West, M. (1986). Bayesian model monitoring. J. Roy. Statist. Soc. Ser. B 48 70–78.
  • West, M. and Harrison, J. (1997). Bayesian Forecasting and Dynamic Models, 2nd ed. Springer, New York.