The E-Posterior

TL;DR

The paper introduces the e-posterior, a new uncertainty representation that offers frequentist-valid risk bounds and safer decision rules compared to Bayesian posteriors, applicable to arbitrary loss functions.

Contribution

It develops the e-posterior framework, providing a robust alternative to Bayesian posteriors with guaranteed frequentist risk bounds and introduces the quasi-conditional paradigm.

Findings

E-posterior provides risk bounds with frequentist validity.

E-posterior minimax decision rules are safer than Bayesian ones.

Re-interpretation of Kiefer-Berger-Brown-Wolpert tests using e-posteriors.

Abstract

We develop a representation of a decision maker's uncertainty based on e-variables. Like the Bayesian posterior, this *e-posterior* allows for making predictions against arbitrary loss functions that may not be specified ex ante. Unlike the Bayesian posterior, it provides risk bounds that have frequentist validity irrespective of prior adequacy: if the e-collection (which plays a role analogous to the Bayesian prior) is chosen badly, the bounds get loose rather than wrong, making *e-posterior minimax* decision rules safer than Bayesian ones. The resulting *quasi-conditional paradigm* is illustrated by re-interpreting a previous influential partial Bayes-frequentist unification, *Kiefer-Berger-Brown-Wolpert conditional frequentist tests*, in terms of e-posteriors.