Modélisation probabiliste et inférence par l'algorithme Belief Propagation

On s'interesse a la construction et l'estimation - a partir d'observations incompletes - de modeles de variables aleatoires a valeurs reelles sur un graphe. Ces modeles doivent etre adaptes a un probleme de regression non standard ou l'identite des variables observees (et donc celle des variables a predire) varie d'une instance a l'autre. La nature du probleme et des donnees disponibles nous conduit a modeliser le reseau sous la forme d'un champ markovien aleatoire, choix justifie par le principe de maximisation d'entropie de Jaynes. L'outil de prediction choisi dans ces travaux est l'algorithme Belief Propagation - dans sa version classique ou gaussienne - dont la simplicite et l'efficacite permettent son utilisation sur des reseaux de grande taille. Apres avoir fourni un nouveau resultat sur la stabilite locale des points fixes de l'algorithme, on etudie une approche fondee sur un modele d'Ising latent ou les dependances entre variables reelles sont encodees a travers un reseau de variables binaires. Pour cela, on propose une definition de ces variables basee sur les fonctions de repartition des variables reelles associees. Pour l'etape de prediction, il est necessaire de modifier l'algorithme Belief Propagation pour imposer des contraintes de type bayesiennes sur les distributions marginales des variables binaires. L'estimation des parametres du modele peut aisement se faire a partir d'observations de paires. Cette approche est en fait une maniere de resoudre le probleme de regression en travaillant sur les quantiles. D'autre part, on propose un algorithme glouton d'estimation de la structure et des parametres d'un champ markovien gaussien, base sur l'algorithme Iterative Proportional Scaling. Cet algorithme produit a chaque iteration un nouveau modele dont la vraisemblance, ou une approximation de celle-ci dans le cas d'observations incompletes, est superieure a celle du modele precedent. Cet algorithme fonctionnant par perturbation locale, il est possible d'imposer des contraintes spectrales assurant une meilleure compatibilite des modeles obtenus avec la version gaussienne de Belief Propagation. Les performances des differentes approches sont illustrees par des experimentations numeriques sur des donnees synthetiques.

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