Bayesian and frequentist nonlinear inequality tests
Bayesian and frequentist criteria are fundamentally different, but often posterior and sampling distributions are asymptotically equivalent (and normal). We compare Bayesian and frequentist inference on nonlinear inequality restrictions in such cases. To quantify the comparison, we examine the (frequentist) size of a Bayesian hypothesis test (based on a comparable loss function). For finite-dimensional parameters, if the null hypothesis is that the parameter vector lies in a certain half-space, then the Bayesian test has size alpha; if the null hypothesis is a subset of a half-space (with strictly smaller volume), then the Bayesian test has size strictly above alpha; and in other cases, the Bayesian test's size may be above or below alpha. For infinite-dimensional parameters, similar results hold. Two examples illustrate our results: inference on stochastic dominance and on curvature of a translog cost function. We hope these results increase awareness of when Bayesian and frequentist inferences may differ significantly in practice, as well as increase intuition about such differences.