Asymptotic normality with small relative errors of posterior probabilities of half-spaces

Abstract

Let $\Theta$ be a parameter space included in a finite-dimensional Euclidean space and let $A$ be a half-space. Suppose that the maximum likelihood estimate $\theta_n$ of $\theta$ is not in $A$ (otherwise, replace $A$ by its complement) and let $\Delta$ be the maximum log likelihood (at $\theta_n$) minus the maximum log likelihood over the boundary $\partial A$. It is shown that under some conditions, uniformly over all half-spaces $A$, either the posterior probability of $A$ is asymptotic to $\Phi(-\sqrt{2\Delta}\,)$ where $\Phi$ is the standard normal distribution function, or both the posterior probability and its approximant go to 0 exponentially in $n$. Sharper approximations depending on the prior are also defined.