Vitesses de concentration pour les procédures bayésiennes empiriques. Applications pour les processus de comptage

Abstract

After some recalls about the setting of nonparametric Bayesian estimation, we consider the issue of tuning hyperparameters of prior models by using empirical Bayes procedures. We provide general conditions on the model and the prior to derive concentration rates for the associated posterior distributions. We aim at providing conditions close to those given in the seminal article by Ghosal and van der Vaart (2007). We then apply our methodology for estimating the intensity of a counting process under the Aalen model by using Dirichlet processes as prior models. We illustrate our results on inhomogeneous Poisson processes or right-censoring models in survival analysis that constitute a major class of Aalen multiplicative intensity models.