# Filtering the Wright-Fisher diffusion

Mireille Chaleyat-Maurel; Valentine Genon-Catalot

ESAIM: Probability and Statistics (2009)

- Volume: 13, page 197-217
- ISSN: 1292-8100

## Access Full Article

top## Abstract

top## How to cite

topChaleyat-Maurel, Mireille, and Genon-Catalot, Valentine. "Filtering the Wright-Fisher diffusion." ESAIM: Probability and Statistics 13 (2009): 197-217. <http://eudml.org/doc/250672>.

@article{Chaleyat2009,

abstract = {
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.
},

author = {Chaleyat-Maurel, Mireille, Genon-Catalot, Valentine},

journal = {ESAIM: Probability and Statistics},

keywords = {Stochastic filtering; partial observations; diffusion processes; discrete time observations; hidden Markov models; prior and posterior distributions; stochastic filtering},

language = {eng},

month = {6},

pages = {197-217},

publisher = {EDP Sciences},

title = {Filtering the Wright-Fisher diffusion},

url = {http://eudml.org/doc/250672},

volume = {13},

year = {2009},

}

TY - JOUR

AU - Chaleyat-Maurel, Mireille

AU - Genon-Catalot, Valentine

TI - Filtering the Wright-Fisher diffusion

JO - ESAIM: Probability and Statistics

DA - 2009/6//

PB - EDP Sciences

VL - 13

SP - 197

EP - 217

AB -
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.

LA - eng

KW - Stochastic filtering; partial observations; diffusion processes; discrete time observations; hidden Markov models; prior and posterior distributions; stochastic filtering

UR - http://eudml.org/doc/250672

ER -

## References

top- O. Cappé, E. Moulines and T. Rydèn, Inference in hidden Markov models. Springer (2005). Zbl1080.62065
- M. Chaleyat-Maurel and V. Genon-Catalot, Computable infinite-dimensional filters with applications to discretized diffusion processes. Stoch. Process. Appl.116 (2006) 1447–1467. Zbl1122.93079
- 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). Zbl1274.62549
- V. Genon-Catalot, A non-linear explicit filter. Statist. Probab. Lett.61 (2003) 145–154. Zbl1041.62079
- 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. Zbl1055.62102
- S. Karlin and H.M. Taylor, A Second Course in Stochastic Processes. Academic Press (1981). Zbl0469.60001
- N.N. Lebedev, Special functions and their applications. Dover publications, Inc. (1972). Zbl0271.33001
- A. Nikiforov and V. Ouvarov, Fonctions spéciales de la physique mathématique. Editions Mir, Moscou (1983).
- W. Runggaldier and F. Spizzichino, Sufficient conditions for finite dimensionality of filters in discrete time: A Laplace transform-based approach. Bernoulli7 (2001) 211–221. Zbl0981.62077
- G. Sawitzki, Finite dimensional filter systems in discrete time. Stochastics5 (1981) 107–114. Zbl0471.60052
- 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). Zbl0991.92001
- M. West and J. Harrison, Bayesian forecasting and dynamic models. Springer Series in Statistics, second edition. Springer (1997). Zbl0871.62026

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.