Posterior concentration rates for empirical Bayes procedures, with applications to Dirichlet Process mixtures

Sophie Donnet (1, 2) , Vincent Rivoirard (1) , Judith Rousseau (3) , Catia Scricciolo (4)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 MIA-Paris - Mathématiques et Informatique Appliquées
3 CREST - Centre de Recherche en Économie et Statistique
4 Dipartimento di Scienze delle Decision

In this paper we provide general conditions to check on the model and the prior to derive posterior concentration rates for data-dependent priors (or empirical Bayes approaches). We aim at providing conditions that are close to the conditions provided in the seminal paper by Ghosal & van der Vaart (2007). We then apply the general theorem to two different settings: the estimation of a density using Dirichlet process mixtures of Gaussian random variables with base measure depending on some empirical quantities and the estimation of the intensity of a counting process under the Aalen model. A simulation study for inhomogeneous Poisson processes also illustrates our results. In the former case we also derive some results on the estimation of the mixing density and on the deconvolution problem. In the latter, we provide a general theorem on posterior concentration rates for counting processes with Aalen multiplicative intensity with priors not depending on the data.

Mots clés: Empirical Bayes; posterior concentration rates; Dirichlet process mixtures; counting processes; Aalen model

Domaines: Applications [stat.AP]

Sophie Donnet, Vincent Rivoirard, Judith Rousseau, Catia Scricciolo. Posterior concentration rates for empirical Bayes procedures, with applications to Dirichlet Process mixtures. 2014. ⟨hal-01007444⟩