Skip to main content
SpringerLink
Log in

Static-parameter estimation in piecewise deterministic processes using particle Gibbs samplers

  • Special Issue: Bayesian Inference and Stochastic Computation
  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

We develop particle Gibbs samplers for static-parameter estimation in discretely observed piecewise deterministic process (PDPs). PDPs are stochastic processes that jump randomly at a countable number of stopping times but otherwise evolve deterministically in continuous time. A sequential Monte Carlo (SMC) sampler for filtering in PDPs has recently been proposed. We first provide new insight into the consequences of an approximation inherent within that algorithm. We then derive a new representation of the algorithm. It simplifies ensuring that the importance weights exist and also allows the use of variance-reduction techniques known as backward and ancestor sampling. Finally, we propose a novel Gibbs step that improves mixing in particle Gibbs samplers whose SMC algorithms make use of large collections of auxiliary variables, such as many instances of SMC samplers. We provide a comparison between the two particle Gibbs samplers for PDPs developed in this paper. Simulation results indicate that they can outperform reversible-jump MCMC approaches.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Andrieu, C., Roberts, G. O. (2009). The pseudo-marginal approach for efficient Monte Carlo computations. Annals of Statistics, 37(2), 697–725.

    Google Scholar 

  • Andrieu, C., Doucet, A., Holenstein, R. (2010). Particle Markov chain Monte Carlo methods. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72(3), 269–342, with discussion.

    Google Scholar 

  • Andrieu, C., Lee, A., Vihola, M. (2013). Uniform ergodicity of the iterated conditional SMC and geometric ergodicity of particle Gibbs samplers. arXiv, preprint arXiv:1312.6432.

  • Beaumont, M. A. (2003). Estimation of population growth or decline in genetically monitored populations. Genetics, 164(3), 1139–1160.

    Google Scholar 

  • Bunch, P., Godsill, S. J. (2013). Particle smoothing algorithms for variable rate models. IEEE Transactions on Signal Processing, 61(7), 1663–1675.

    Google Scholar 

  • Carlin, B. P., Chib, S. (1995). Bayesian model choice via Markov chain Monte Carlo methods. Journal of the Royal Statistical Society Series B (Statistical Methodology), 57(3), 473–484.

    Google Scholar 

  • Carpenter, J., Clifford, P., Fearnhead, P. (1999). An improved particle filter for nonlinear problems. IEE Proceedings, Radar, Sonar and Navigation, 146(1), 2–7.

    Google Scholar 

  • Centanni, S., Minozzo, M. (2006a). Estimation and filtering by reversible jump MCMC for a doubly stochastic Poisson model for ultra-high-frequency financial data. Statistical Modelling, 6(2), 97–118.

    Google Scholar 

  • Centanni, S., Minozzo, M. (2006b). A Monte Carlo approach to filtering for a class of marked doubly stochastic Poisson processes. Journal of the American Statistical Association, 101(476), 1582–1597.

  • Chopin, N. (2002). A sequential particle filter method for static models. Biometrika, 89(3), 539–552.

    Article  MATH  MathSciNet  Google Scholar 

  • Chopin, N., Singh, S. S. (2013). On the particle Gibbs sampler. arXiv, preprint arXiv:1304.1887.

  • Chopin, N., Jacob, P. E., Papaspiliopoulos, O. (2013). SMC\(^{2}\): an efficient algorithm for sequential analysis of state space models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75(3), 397–426.

    Google Scholar 

  • Dassios, A., Jang, J. (2003). Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity. Finance and Stochastics, 7(1), 73–95.

    Google Scholar 

  • Del Moral, P. (1995). Nonlinear filtering using random particles. Theory of Probability and Its Applications, 40(4), 690–701.

  • Del Moral, P. (1996). Nonlinear filtering: Interacting particle resolution. Markov Processes and Related Fields, 2(4), 555–580.

    MATH  MathSciNet  Google Scholar 

  • Del Moral, P. (2004). Feynman–Kac formulae: Genealogical and interacting particle systems with applications. Berlin: Springer.

    Book  Google Scholar 

  • Del Moral, P., Doucet, A., Jasra, A. (2006). Sequential Monte Carlo samplers. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68(3), 411–436.

    Google Scholar 

  • Del Moral, P., Doucet, A., Jasra, A. (2007). Sequential Monte Carlo for Bayesian computation. In J. Bernardo, M. Bayarri, J. Berger, A. Dawid, D. Heckerman, A. Smith, M. West (Eds.), Bayesian statistics 8 (pp. 115–148). USA: Oxford University Press.

  • Del Moral, P., Doucet, A., Jasra, A. (2012). On adaptive resampling strategies for sequential Monte Carlo methods. Bernoulli, 18(1), 252–278.

    Google Scholar 

  • Douc, R., Cappé, O., Moulines, E. (2005). Comparison of resampling schemes for particle filtering. In Proceedings of the 4th international symposium on image and signal processing and analysis (ISPA 2005) (pp. 64–69). Zagreb, Croatia: IEEE.

  • Doucet, A., Johansen, A. M. (2011). A tutorial on particle filtering and smoothing: Fifteen years later. In Crisan, D., Rozovskii, B. (eds). The Oxford handbook of nonlinear filtering, Oxford handbooks (Chap. 24, pp. 656–704). Oxford: Oxford University Press.

  • Doucet, A., de Freitas, N., Gordon, N. (Eds.). (2001). Sequential Monte Carlo methods in practice. Statistics for engineering and information science. Berlin: Springer.

  • Godsill, S. J. (2001). On the relationship between Markov chain Monte Carlo methods for model uncertainty. Journal of Computational and Graphical Statistics, 10(2), 230–248.

    Article  MathSciNet  Google Scholar 

  • Godsill, S. J., Vermaak, J. (2004). Models and algorithms for tracking using trans-dimensional sequential Monte Carlo. In Proceedings of the IEEE international conference on acoustics, speech, and signal processing, 2004 (ICASSP’04) (Vol. 3, pp. 976–979).

  • Gordon, N. J., Salmond, D. J., Smith, A. F. M. (1993). Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proceedings F, Radar and Signal Processing, 140(2), 107–113.

  • Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82(4), 711–732.

    Article  MATH  MathSciNet  Google Scholar 

  • Johansen, A. M., Doucet, A. (2008). A note on auxiliary particle filters. Statistics and Probability Letters, 78(12), 1498–1504.

    Google Scholar 

  • Kantas, N., Doucet, A., Singh, S. S., Maciejowski, J. M. (2009). An overview of sequential Monte Carlo methods for parameter estimation in general state-space models. In 15th IFAC symposium on system identification (pp. 774–785).

  • Kitagawa, G. (1996). Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. Journal of Computational and Graphical Statistics, 5(1), 1–25.

    MathSciNet  Google Scholar 

  • Kitagawa, G. (1998). A self-organizing state-space model. Journal of the American Statistical Association, 93(443), 1203–1215.

    Google Scholar 

  • Kong, A., Liu, J. S., Wong, W. H. (1994). Sequential imputations and Bayesian missing data problems. Journal of the American Statistical Association, 89(425), 278–288.

    Google Scholar 

  • Lee, A., Murray, L. M., Johansen, A. M. (2014) (in prep.) Resampling in conditional SMC algorithms.

  • Lindsten, F., Schön, T. B. (2012). On the use of backward simulation in the particle Gibbs sampler. In Proceedings of the 37th IEEE international conference on acoustics, speech, and signal processing (ICASSP), Kyoto, Japan.

  • Lindsten, F., Schön, T. B. (2013). Backward simulation methods for Monte Carlo statistical inference. Foundations and Trends in Machine Learning, 6(1), 1–143.

    Google Scholar 

  • Lindsten, F., Jordan, M. I., Schön, T. B. (2012). Ancestor sampling for particle Gibbs. In Proceedings of the 2012 conference on neural information processing systems (NIPS), Lake Tahoe, NV, USA.

  • Lindsten, F., Douc, R., Moulines, E. (2014). Uniform ergodicity of the particle Gibbs sampler. arXiv, preprint arXiv:1401.0683.

  • Liu, J. S., Chen, R. (1998). Sequential Monte Carlo methods for dynamic systems. Journal of the American Statistical Association, 93(443), 1032–1044.

    Google Scholar 

  • Martin, J. S., Jasra, A., McCoy, E. (2013). Inference for a class of partially observed point process models. Annals of the Institute of Statistical Mathematics, 65(3), 413–437.

    Google Scholar 

  • Neal, R. M. (2001). Annealed importance sampling. Statistics and Computing, 11(2), 125–139.

    Article  MathSciNet  Google Scholar 

  • Pitt, M. K., Shephard, N. (1999). Filtering via simulation: Auxiliary particle filters. Journal of the American Statistical Association, 94(446), 590–599.

    Google Scholar 

  • Poyiadjis, G., Doucet, A., Singh, S. S. (2011). Particle approximations of the score and observed information matrix in state space models with application to parameter estimation. Biometrika, 98(1), 65–80.

    Google Scholar 

  • Rao, V., Teh, Y. W. (2012). Fast MCMC sampling for Markov jump processes and continuous time Bayesian networks. arXiv, preprint arXiv:1202.3760.

  • Van Dyk, D. A., Park, T. (2008). Partially collapsed Gibbs samplers: theory and methods. Journal of the American Statistical Association, 103(482), 790–796.

    Google Scholar 

  • Whiteley, N. (2010). Contribution to the discussion on ‘Particle Markov chain Monte Carlo methods’ by Andrieu, C., Doucet, A., and Holenstein, R. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72(3), 306–307.

  • Whiteley, N., Andrieu, C., Doucet, A. (2010). Efficient Bayesian inference for switching state-space models using discrete particle Markov chain Monte Carlo methods. Bristol University Statistics Research Report, 10, 04.

  • Whiteley, N., Johansen, A. M., Godsill, S. J. (2011). Monte Carlo filtering of piecewise deterministic processes. Journal of Computational and Graphical Statistics, 20(1), 119–139.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK

    Axel Finke, Adam M. Johansen & Dario Spanò

Authors
  1. Axel Finke
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Adam M. Johansen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dario Spanò
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Adam M. Johansen.

Additional information

Axel Finke was supported by Engineering and Physical Sciences Research Council (EPSRC) Doctoral Training Grant EP/J500586/1. Adam M. Johansen was partially supported by EPSRC Grant EP/I017984/1 and is grateful to the Isaac Newton Institute for a visiting fellowship which allowed him time to work on this manuscript during the programme: Inference for Change-Point and Related Processes. Dario Spanò’s research was partially funded by CRiSM, an EPSRC/HEFCE-funded grant.

About this article

Cite this article

Finke, A., Johansen, A.M. & Spanò, D. Static-parameter estimation in piecewise deterministic processes using particle Gibbs samplers. Ann Inst Stat Math 66, 577–609 (2014). https://doi.org/10.1007/s10463-014-0455-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10463-014-0455-z

Keywords

Search

Navigation