  Bayesian and frequentist criteria fundamentally differ, but often posterior
and sampling distributions agree asymptotically (e.g., Gaussian with same
covariance). For the corresponding single-draw experiment, we characterize the
frequentist size of a certain Bayesian hypothesis test of (possibly nonlinear)
inequalities. If the null hypothesis is that the (possibly
infinite-dimensional) parameter lies in a certain half-space, then the Bayesian
test's size is $\alpha$; if the null hypothesis is a subset of a half-space,
then size is above $\alpha$; and in other cases, size may be above, below, or
equal to $\alpha$. Rejection probabilities at certain points in the parameter
space are also characterized. Two examples illustrate our results: translog
cost function curvature and ordinal distribution relationships.
