Hostname: page-component-cb9f654ff-mwwwr Total loading time: 0 Render date: 2025-08-18T00:50:59.571Z Has data issue: false hasContentIssue false
  • English
  • Français

Filtering the Wright-Fisher diffusion

Published online by Cambridge University Press:  11 June 2009

Mireille Chaleyat-Maurel  and
Valentine Genon-Catalot
Mireille Chaleyat-Maurel
Affiliation:
(Corresponding author) Laboratoire MAP5, Université Paris Descartes, UFR de Mathématique et Informatique, CNRS-UMR 8145 and Laboratoire de Probabilités et Modèles Aléatoires (CNRS-UMR 7599), 45 rue des Saints-Pères, 75270 Paris Cedex 06, France; mcm@math-info.univ-paris5.fr
Valentine Genon-Catalot
Affiliation:
Laboratoire MAP5, Université Paris Descartes, UFR de Mathématique et Informatique, CNRS-UMR 8145, 45 rue des Saints-Pères, 75270 Paris Cedex 06, France; genon@math-info.univ-paris5.fr
Get access

Abstract

We consider a Wright-Fisher diffusion (x(t)) whose current state cannot be observed directly. Instead, at times t 1 < t 2 < ..., the observations y(ti) are such that, given the process (x(t)), the random variables (y(ti)) are independent and the conditional distribution of y(ti) only depends on x(ti). When this conditional distribution has a specific form, we prove that the model ((x(ti),y(ti)), i ≥1) is a computable filter in the sense that all distributions involved in filtering, prediction and smoothing are exactly computable. These distributions are expressed as finite mixtures of parametric distributions. Thus, the number of statistics to compute at each iteration is finite, but this number may vary along iterations.

Keywords

Stochastic filteringpartial observationsdiffusion processesdiscrete time observationshidden Markov modelsprior and posterior distributions

Information

Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 13 , July 2009 , pp. 197 - 217
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

O. Cappé, E. Moulines and T. Rydèn, Inference in hidden Markov models. Springer (2005).
Chaleyat-Maurel, M. and Genon-Catalot, V., Computable infinite-dimensional filters with applications to discretized diffusion processes. Stoch. Process. Appl. 116 (2006) 1447–1467. CrossRef
F. Comte, V. Genon-Catalot and M. Kessler, Multiplicative Kalman filtering, Pré-publication 2007-16, MAP5, Laboratoire de Mathématiques Appliquées de Paris Descartes, submitted (2007).
Genon-Catalot, V., A non-linear explicit filter. Statist. Probab. Lett. 61 (2003) 145–154. CrossRef
V. Genon-Catalot and M. Kessler, Random scale perturbation of an AR(1) process and its properties as a nonlinear explicit filter. Bernoulli (10) (2004) 701–720.
S. Karlin and H.M. Taylor, A Second Course in Stochastic Processes. Academic Press (1981).
N.N. Lebedev, Special functions and their applications. Dover publications, Inc. (1972).
A. Nikiforov and V. Ouvarov, Fonctions spéciales de la physique mathématique. Editions Mir, Moscou (1983).
Runggaldier, W. and Spizzichino, F., Sufficient conditions for finite dimensionality of filters in discrete time: A Laplace transform-based approach. Bernoulli 7 (2001) 211–221. CrossRef
Sawitzki, G., Finite dimensional filter systems in discrete time. Stochastics 5 (1981) 107–114.
T. Wai-Yuan, Stochastic models with applications to genetics, cancers, AIDS and other biomedical systems. Series on Concrete and Applicable Mathematics, Vol. 4. World Scientific (2002).
M. West and J. Harrison, Bayesian forecasting and dynamic models. Springer Series in Statistics, second edition. Springer (1997).