Optimal properties of some Bayesian inferences

TL;DR

This paper investigates the optimal properties of relative surprise Bayesian credible regions, demonstrating their minimal false coverage probability, unbiasedness, and maximization of evidence measures, highlighting their theoretical advantages in Bayesian inference.

Contribution

It establishes the optimality and unbiasedness of relative surprise regions in Bayesian inference, revealing their natural emergence through reparameterizations.

Findings

Minimize prior false coverage probability

Maximize Bayes factor and relative belief ratio

Unbiased with respect to false value coverage

Abstract

Relative surprise regions are shown to minimize, among Bayesian credible regions, the prior probability of covering a false value from the prior. Such regions are also shown to be unbiased in the sense that the prior probability of covering a false value is bounded above by the prior probability of covering the true value. Relative surprise regions are shown to maximize both the Bayes factor in favor of the region containing the true value and the relative belief ratio, among all credible regions with the same posterior content. Relative surprise regions emerge naturally when we consider equivalence classes of credible regions generated via reparameterizations.