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 tests 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.
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