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

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 A be the maximum log likelihood (at theta(n)) minus the maximum log likelihood over the boundary partial derivativeA. It is shown that under some conditions, uniformly over all half-spaces A, either the posterior probability of A is asymptotic to Phi (-root2Delta) 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.