Frequentist properties of Bayesian inequality tests

Bayesian and frequentist criteria fundamentally differ, but often posterior and sampling distributions agree asymptotically. 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 parameter lies in a specified half-space, then the Bayesian test’s size equals α; if the null hypothesis is a subset of a half-space, then size is above α; otherwise, size may be equal to, above, or below α. Rejection probabilities at certain points are also characterized. Two examples illustrate our results: translog cost function curvature and ordinal distribution relationships.