Filtering the Wright-Fisher diffusion

¹ (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
² 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

Received: 27 April 2007
Revised: 13 February 2008

Abstract

We consider a Wright-Fisher diffusion (x(t)) whose current state cannot be observed directly. Instead, at times t₁ < t₂ < ..., the observations y(t_i) are such that, given the process (x(t)), the random variables (y(t_i)) are independent and the conditional distribution of y(t_i) only depends on x(t_i). When this conditional distribution has a specific form, we prove that the model ((x(t_i),y(t_i)), 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.