Inference for a class of partially observed point process models



This paper presents a simulation-based framework for sequential inference from partially and discretely observed point process models with static parameters. Taking on a Bayesian perspective for the static parameters, we build upon sequential Monte Carlo methods, investigating the problems of performing sequential filtering and smoothing in complex examples, where current methods often fail. We consider various approaches for approximating posterior distributions using SMC. Our approaches, with some theoretical discussion are illustrated on a doubly stochastic point process applied in the context of finance.


Point processes Sequential Monte Carlo Intensity estimation 


  1. Andrieu, C., Jasra, A., Doucet, A., Del Moral, P. (2011). On non-linear Markov chain Monte Carlo. Bernoulli, 17, 987–1014.Google Scholar
  2. Barndorff-Nielsen, O., Shephard, N. (2001). Non-Gaussian Ornstein-Uhlenbeck models and some of their uses in financial economics (with discussion). Journal of the Royal Statistical Society Series B, 63, 167–241.Google Scholar
  3. Beskos, A., Crisan, D., Jasra, A. (2011). On the stability of sequential Monte Carlo methods in high dimensions. Technical Report, Imperial College London, London.Google Scholar
  4. Centanni, S., Minozzo, M. (2006a). A Monte Carlo approach to filtering for a class of marked doubly stochastic Poisson processes. Journal of the American Statistical Association, 101, 1582–1597.Google Scholar
  5. Centanni, S., Minozzo, M. (2006b). Estimation and filtering by reversible jump MCMC for a doubly stochastic Poisson model for ultra-high-frequency financial data. Statistical Modelling, 6, 97–118.Google Scholar
  6. Chopin, N. (2002). A sequential particle filter method for static models. Biometrika, 89, 539–552.Google Scholar
  7. Chopin, N., Jacob, P., Papaspiliopoulos, O. (2012). SMC\(^2\): A sequential Monte Carlo algorithm with particle Markov chain Monte Carlo updates. Journal of the Royal Statistical Society Series B (to appear).Google Scholar
  8. Daley, D. J., Vere-Jones, D. (1988). Introduction to the theory of point processes. New York: Springer.Google Scholar
  9. Del Moral, P. (2004). Feynman-Kac formulae. Genealogical and interacting particle systems. New York: Springer.MATHCrossRefGoogle Scholar
  10. Del Moral, P., Doucet, A., Jasra, A. (2006). Sequential Monte Carlo samplers. Journal of the Royal Statistical Society Series B, 68, 411–32.Google Scholar
  11. Del Moral, P., Doucet, A., Jasra, A. (2007). Sequential Monte Carlo for Bayesian computation (with discussion). In: S. Bayarri, J. O. Berger, J. M. Bernardo, A. P. Dawid, D. Heckerman, A. F. M. Smith, M. West (Eds.), Bayesian statistics (Vol. 8, pp. 115–149). Oxford: OUP.Google Scholar
  12. Del Moral, P., Doucet, A., Jasra, A. (2012). On adaptive resampling procedures for sequential Monte Carlo methods. Bernoulli, 18, 252–278.Google Scholar
  13. Doucet, A., De Freitas, J. F. G., Gordon, N. J. (2001). Sequential Monte Carlo methods in practice. New York: Springer.Google Scholar
  14. Doucet, A., Montesano, L., Jasra, A. (2006). Optimal filtering for partially observed point processes using trans-dimensional sequential Monte Carlo. International Conference on Acoustics, Speech, and Signal Processing, 5, 597–600.Google Scholar
  15. Eberle, A., Marinelli, C. (2012). Quantitative approximations of evolving probability measures and sequential Markov chain Monte Carlo methods. Probability Theory and Related Fields (to appear).Google Scholar
  16. Fearnhead, P. (2004). Exact filtering for partially-observed queues. Statistics and Computing, 14, 261–266.Google Scholar
  17. Glynn, P. W., Meyn, S. P. (1996). A Lyapunov bound for solutions of the Poisson equation. Annals of Probability, 24, 916–931.Google Scholar
  18. Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82, 711–732.Google Scholar
  19. Jasra, A., Stephens, D. A., Holmes, C. C. (2007). On population-based simulation for static inference. Statistics and Computing, 17, 263–279.Google Scholar
  20. Kantas, N., Chopin, N., Doucet, A., Singh, S. S., Maciejowski, J. M. (2011). On particle methods for parameter estimation in general state-space models. Technical Report, Imperial College London, London.Google Scholar
  21. Liu, J. S. (2001). Monte Carlo strategies in scientific computing. New York: Springer.MATHGoogle Scholar
  22. Pitt, M. K., Shephard, N. (1997). Filtering via simulation: Auxiliary particle filters. Journal of the American Statistical Association, 94, 590–599.Google Scholar
  23. Roberts, G. O., Papaspiliopoulos, O., Dellaportas, P. (2004). Bayesian inference for non-Gaussian Ornstein-Uhlenbeck stochastic volatility processes. Journal of the Royal Statistical Society Series B, 66, 369–393.Google Scholar
  24. Rousset, M., Doucet, A. (2006). Discussion of Beskos et al. Journal of the Royal Statistical Society Series B, 68, 374–375.Google Scholar
  25. Rydberg, T. H., Shephard, N. (2000). A modelling framework for the prices and times of trades made on the New York Stock exchange. In W. J. Fitzgerald, R. L. Smith, A. T. Walden, P. C. Young (Eds.), Non-linear and non-stationary signal processing (p. 246). Cambridge: CUP.Google Scholar
  26. Shiryaev, A. (1996). Probability. New York: Springer.Google Scholar
  27. Snyder, D. L. (1972). Filtering and detection for doubly stochastic Poisson processes. IEEE Transactions on Information Theory, 18, 91–102.Google Scholar
  28. Snyder, D. L., Miller, M. I. (1998). Random point processes in space and time. New York: Springer.Google Scholar
  29. Varini, E. (2007). A Monte Carlo method for filtering a marked doubly stochastic Poisson process. Statistical Methods and Applications, 17, 183–193.Google Scholar
  30. Whiteley, N. P., Johansen, A. M., Godsill, S. J. (2011). Monte Carlo filtering of piece-wise deterministic processes. Journal of Computational and Graphical Statistics, 20, 119–139.Google Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2012

Authors and Affiliations

  1. 1.Australian School of BusinessUniversity of New South WalesSydneyAustralia
  2. 2.Department of Statistics and Applied ProbabilityNational University of SingaporeSingaporeSingapore
  3. 3.Department of MathematicsImperial College LondonLondonUK

Personalised recommendations