Frequentist properties of Bayesian inequality tests

TL;DR

This paper analyzes the frequentist properties of Bayesian inequality tests, showing how their size compares to the nominal level under various hypotheses, with applications to economic models.

Contribution

It provides a characterization of the frequentist size of Bayesian inequality tests, including cases with nonlinear and infinite-dimensional parameters.

Findings

Bayesian tests have size exactly α for half-space nulls

Size can be above, below, or equal to α depending on the hypothesis

Rejection probabilities are characterized for specific points in parameter space

Abstract

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.