• Bernoulli
  • Volume 20, Number 4 (2014), 1999-2019.

Optimal filtering and the dual process

Omiros Papaspiliopoulos and Matteo Ruggiero

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We link optimal filtering for hidden Markov models to the notion of duality for Markov processes. We show that when the signal is dual to a process that has two components, one deterministic and one a pure death process, and with respect to functions that define changes of measure conjugate to the emission density, the filtering distributions evolve in the family of finite mixtures of such measures and the filter can be computed at a cost that is polynomial in the number of observations. Special cases of our framework include the Kalman filter, and computable filters for the Cox–Ingersoll–Ross process and the one-dimensional Wright–Fisher process, which have been investigated before. The dual we obtain for the Cox–Ingersoll–Ross process appears to be new in the literature.

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Bernoulli, Volume 20, Number 4 (2014), 1999-2019.

First available in Project Euclid: 19 September 2014

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Bayesian conjugacy Cox–Ingersoll–Ross process finite mixture models hidden Markov model Kalman filter


Papaspiliopoulos, Omiros; Ruggiero, Matteo. Optimal filtering and the dual process. Bernoulli 20 (2014), no. 4, 1999--2019. doi:10.3150/13-BEJ548.

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  • [1] Barbour, A.D., Ethier, S.N. and Griffiths, R.C. (2000). A transition function expansion for a diffusion model with selection. Ann. Appl. Probab. 10 123–162.
  • [2] Bernardo, J.M. and Smith, A.F.M. (1994). Bayesian Theory. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Chichester: Wiley.
  • [3] Cappé, O., Moulines, E. and Rydén, T. (2005). Inference in Hidden Markov Models. Springer Series in Statistics. New York: Springer. With Randal Douc’s contributions to Chapter 9 and Christian P. Robert’s to Chapters 6, 7 and 13, with Chapter 14 by Gersende Fort, Philippe Soulier and Moulines, and Chapter 15 by Stéphane Boucheron and Elisabeth Gassiat.
  • [4] Chaleyat-Maurel, M. and Genon-Catalot, V. (2006). Computable infinite-dimensional filters with applications to discretized diffusion processes. Stochastic Process. Appl. 116 1447–1467.
  • [5] Chaleyat-Maurel, M. and Genon-Catalot, V. (2009). Filtering the Wright–Fisher diffusion. ESAIM Probab. Stat. 13 197–217.
  • [6] Cox, J.C., Ingersoll, J.E. Jr. andRoss, S.A. (1985). A theory of the term structure of interest rates. Econometrica 53 385–407.
  • [7] Dawson, D.A. (1993). Measure-valued Markov processes. In École d’Été de Probabilités de Saint-Flour XXI—1991. Lecture Notes in Math. 1541 1–260. Berlin: Springer.
  • [8] Etheridge, A.M. (2000). An Introduction to Superprocesses. University Lecture Series 20. Providence, RI: Amer. Math. Soc.
  • [9] Ethier, S.N. and Kurtz, T.G. (1981). The infinitely-many-neutral-alleles diffusion model. Adv. in Appl. Probab. 13 429–452.
  • [10] Ethier, S.N. and Kurtz, T.G. (1986). Markov Processes. Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. New York: Wiley.
  • [11] Ethier, S.N. and Kurtz, T.G. (1993). Fleming–Viot processes in population genetics. SIAM J. Control Optim. 31 345–386.
  • [12] Feller, W. (1951). Diffusion processes in genetics. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950 227–246. Berkeley and Los Angeles: Univ. California Press.
  • [13] Genon-Catalot, V. and Kessler, M. (2004). Random scale perturbation of an AR(1) process and its properties as a nonlinear explicit filter. Bernoulli 10 701–720.
  • [14] Griffiths, R.C. (2006). Coalescent lineage distributions. Adv. in Appl. Probab. 38 405–429.
  • [15] Hutzenthaler, M. and Wakolbinger, A. (2007). Ergodic behavior of locally regulated branching populations. Ann. Appl. Probab. 17 474–501.
  • [16] Jansen, S. and Kurt, N. (2013). On the notion(s) of duality for Markov processes. Available at arXiv:1210.7193 [math.PR].
  • [17] Johnson, N.L., Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions. Vol. 1, 2nd ed. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. New York: Wiley.
  • [18] Karlin, S. and Taylor, H.M. (1981). A Second Course in Stochastic Processes. New York: Academic Press.
  • [19] Kawazu, K. and Watanabe, S. (1971). Branching processes with immigration and related limit theorems. Theory Probab. Appl. 16 34–51.
  • [20] Lévy, P. (1948). Processus Stochastiques et Mouvement Brownien. Paris: Gauthier-Villars & Cie.
  • [21] Liggett, T.M. (2005). Interacting Particle Systems. Classics in Mathematics. Berlin: Springer. Reprint of the 1985 original.
  • [22] Sen, A. and Balakrishnan, N. (1999). Convolution of geometrics and a reliability problem. Statist. Probab. Lett. 43 421–426.
  • [23] Tavaré, S. (1984). Line-of-descent and genealogical processes, and their applications in population genetics models. Theoret. Population Biol. 26 119–164.