Bernstein Von Mises Theorem for linear functionals of the density

TL;DR

This paper establishes a semiparametric Bernstein-Von Mises theorem for linear functionals of density functions, providing conditions for asymptotic normality and applying it to nonparametric priors with adaptive concentration rates.

Contribution

It introduces general conditions for a semiparametric Bernstein-Von Mises theorem and applies these to infinite-dimensional exponential family priors, deriving adaptive posterior concentration rates.

Findings

Asymptotic normality of linear functionals under certain conditions

Application to nonparametric priors on Sobolev and Besov spaces

Derivation of adaptive posterior concentration rates

Abstract

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.