  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.
