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

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

1 MIA-Paris - Mathématiques et Informatique Appliquées
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 CREST - Centre de Recherche en Économie et Statistique
4 Department of Economics

We provide conditions on the statistical model and the prior probability law to derive contraction rates of posterior distributions corresponding to data-dependent priors in an empirical Bayes approach for selecting prior hyper-parameter values. We aim at giving conditions in the same spirit as those in the seminal article of Ghosal and van der Vaart [23]. We then apply the result to specific statistical settings: density estimation using Dirichlet process mixtures of Gaussian densities with base measure depending on data-driven chosen hyper-parameter values and intensity function estimation of counting processes obeying the Aalen model. In the former setting, we also derive recovery rates for the related inverse problem of density deconvolution. In the latter, a simulation study for inhomogeneous Poisson processes illustrates the results.

Mots clés: Dirichlet process mixtures; empirical Bayes; Aalen model; counting processes; posterior contraction rates

Domaines: Statistiques [math.ST]

Sophie Donnet, Vincent Rivoirard, Judith Rousseau, Catia Scricciolo. Posterior concentration rates for empirical Bayes procedures with applications to Dirichlet process mixtures. Bernoulli, 2018, 24 (1), pp.231-256. ⟨10.3150/16-BEJ872⟩. ⟨hal-01570308v2⟩