Asymptotic Approximations to the Bayes Posterior Risk

Suppose that, given [formula ommitted] X₁, X₂,… and Y₁, Y₂,… are independent random variables and their respective distribution functions [formula omitted] and [formula omitted] belong to a one parameter exponential family of distributions. We derive approximations to the posterior probabilities of ω lying in closed convex subsets of the parameter space under a general prior density. Using this, we then approximate the Bayes posterior risk for testing the hypotheses [formula omitted] versus [formula omitted] using a zero-one loss function, where Ω₁, and Ω₂ are disjoint closed convex subsets of the parameter space.