Let $f(x)$ be a $\operatorname{pdf}$ of exponential form with respect to the measure $\mu$. Suppose a prior $\operatorname{pdf}$ $\pi$ has been placed on the natural parameter space $\Omega$, where $\pi$ is a density (with respect to $m$-dimensional Lebesgue measure) which is both positive and continuous at $\tau^\ast$, the true but unknown parameter value. Using basic properties of exponential families and certain associated convex functions, it is shown that the posterior pdf tends to the multivariate normal.
@article{1176343011,
author = {Crain, Bradford R. and Morgan, Ronnie L.},
title = {Asymptotic Normality of the Posterior Distribution for Exponential Models},
journal = {Ann. Statist.},
volume = {3},
number = {1},
year = {1975},
pages = { 223-227},
language = {en},
url = {http://dml.mathdoc.fr/item/1176343011}
}
Crain, Bradford R.; Morgan, Ronnie L. Asymptotic Normality of the Posterior Distribution for Exponential Models. Ann. Statist., Tome 3 (1975) no. 1, pp. 223-227. http://gdmltest.u-ga.fr/item/1176343011/