The E-Posterior

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.