In 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 in the study of Bernstein–von Mises results.
"Bernstein–von Mises theorem for linear functionals of the density." Ann. Statist. 40 (3) 1489 - 1523, June 2012. https://doi.org/10.1214/12-AOS1004