Utilisation du fitness sharing pour traiter la multimodalité en filtrage particulaire.
(travail commun avec Jean Bérard, Marc Joannides, Éric Fontenas et Olivier François)

Résumé Dans le cas d'une fonction d'observation non injective, le filtre particulaire peut, par manque de particules, perdre un des modes du filtre optimal. Ce problème se rencontre aussi sous le nom de premature convergence dans l'utilisation d'algorithmes génétiques pour l'optimisation. Pour ces algorithmes, le fitness sharing peut être utilisé pour maintenir une certaine diversité. Nous l'utilisons aussi dans le contexte de filtrage.

Stéphane Boucheron (Université Paris-Sud, Orsay)

Identication de l'ordre des HMM à espace d'états fini.
(travail commun avec Élisabeth Gassiat)

Abstract We consider the estimation of the order, that is the number of hidden states, of a discrete-time finite alphabet Hidden Markov Model (HMM). The estimators we investigate are related by BIC estimators: penalized maximum likelihood estimators and penalized versions of the mixture estimator introduced by Liu and Narayan (IEEE-IT 1994). We prove consistency of those estimators without assuming any a priori upper bound on the order and smaller penalties than previous works (Kieffer, IEEE-IT 1993). The difficulty of the problem pertains to the fact that the maximum likelihood estimator in a given model does not have a simple form. The round-about is provided by functional large deviations principles. As a matter of fact the (Bahadur)-efficiency of HMM order estimators is investigated using large deviation principles for the empirical measure of various extended Markov chains. We prove a version of Stein's lemma for HMM order estimation and derive an upper-bound on under-estimation exponents. Then we prove that this upper-bound can be achieved by the penalized maximum likelihood estimators.

Nicolas Chopin (ENSAE, Malakoff)

Filtrage particulaire pour des modèles à espace d'états discret : inférence séquentielle et détermination du nombre d'états.

Abstract We investigate the application of particle filtering methodology to discrete state-space models, that is models which involve a conditioning on an unobserved discrete process. We first focus on the case where the states are Markov (hidden Markov models). In that setting, we show how to marginalize out the states of the considered posterior distribution, in order to reduce the volatility of the particle weights. The corresponding algorithm can be seen as a Monte Carlo generalization of the HMM filter. We then propose a sequential state number determination procedure, in order to detect the number of distinct states that have actually appeared at time t. Within this approach, we show how to make an improper prior modeling possible. Finally we consider to which extent these results apply to hidden semi-Markov models, and to change-point models.

Keywords change-point models, HMM filter, hidden Markov models, hidden semi-Markov models, Metropolis-Hastings, MCMC