Asymptotic Normality of the Posterior Distribution for Exponential Models

Abstract

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.