Open AccessBernstein Von Mises Theorem for linear functionals of the density
Author(s) -
Vincent Rivoirard,
Judith Rousseau
Publication year - 2012
Publication title -
hal (le centre pour la communication scientifique directe)
Language(s) - English
DOI - 10.1214/12-aos1004supp
Subject(s) - mathematics , prior probability , nonparametric statistics , sobolev space , class (philosophy) , von mises yield criterion , exponential function , probability density function , mathematical analysis , pure mathematics , econometrics , statistics , computer science , physics , finite element method , thermodynamics , bayesian probability , artificial intelligence
48pagesInternational audienceIn this paper, we study the asymptotic posterior distribution of linear functionals of the density by deriving general conditions to obtain a semi-parametric version of the Bernstein-von Mises theorem. The special case of the cumulative distributive function evaluated at a specific point is widely considered. In particular, we show that for infinite dimensional exponential families, under quite general assumptions, the asymptotic posterior distribution of the functional can be either Gaussian or a mixture of Gaussian distributions with different centering points. This illustrates the positive but also the negative phenomena that can occur for the study of Bernstein-von Mises results