Approximate Behavior of the Posterior Distribution for a Large Observation

Abstract

Let $X$ be a real valued random variable with a family of possible distributions belonging to a one parameter exponential family with the natural parameter $\theta \in(\underline{\theta}, + \infty)$. Let $g$ be a prior probability density for $\theta$ with unbounded support. Under some additional assumptions it is shown that for large values of $x$ the posterior distribution of $\theta$ given $X = x$ is approximately normally distributed about its mode. If $\delta_g$ denotes the Bayes estimator for squared error loss of some function $\gamma(\theta)$ against $g$ then the rate at which $\delta_g(x)$ approaches infinity as $x$ approaches infinity is found. The rate is shown to depend on the behavior of the prior density $g(\theta)$ for large values of $\theta$.