Frequentist size of Bayesian inequality tests

Bayesian and frequentist criteria are fundamentally different, but often posterior and sampling distributions are asymptotically equivalent (e.g., Gaussian). For the corresponding limit 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 $\alpha$ (sometimes strictly); and in other cases, size may be above, below, or equal to $\alpha$. Two examples illustrate our results: testing stochastic dominance and testing curvature of a translog cost function.