Volume 13, January 2009
|Page(s)||197 - 217|
|Published online||11 June 2009|
Filtering the Wright-Fisher diffusion
(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; firstname.lastname@example.org
2 Laboratoire MAP5, Université Paris Descartes, UFR de Mathématique et Informatique, CNRS-UMR 8145, 45 rue des Saints-Pères, 75270 Paris Cedex 06, France; email@example.com
Revised: 13 February 2008
We consider a Wright-Fisher diffusion (x(t)) whose current state cannot be observed directly. Instead, at times t1 < t2 < ..., 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.
Mathematics Subject Classification: Primary 93E11 / 60G35; secondary 62C10
Key words: Stochastic filtering / partial observations / diffusion processes / discrete time observations / hidden Markov models / prior and posterior distributions
© EDP Sciences, SMAI, 2009
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