Bernstein Von Mises Theorem for linear functionals of the density

Vincent Rivoirard (1) , Judith Rousseau (2, 3)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 CREST - Centre de Recherche en Économie et Statistique

In this paper, we study the asymptotic posterior distribution of linear functionals of the density. In particular, we give general conditions to obtain a semiparametric version of the Bernstein-Von Mises theorem. We then apply this general result to nonparametric priors based on infinite dimensional exponential families. As a byproduct, we also derive adaptive nonparametric rates of concentration of the posterior distributions under these families of priors on the class of Sobolev and Besov spaces.

Mots clés: Bayesian nonparametric; Bernstein Von Mises; rates of convergence; adaptive estimation; wavelet

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Vincent Rivoirard, Judith Rousseau. Bernstein Von Mises Theorem for linear functionals of the density. 2009. ⟨hal-00405849v1⟩