The e-posterior

Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences, vol. 381, pp. 20220146

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. This article is part of the theme issue ‘Bayesian inference: challenges, perspectives, and prospects’.

Authors 1

1. Peter Grünwald corresponding Aachen Mathematical Institute

   Leiden University · Centrum Wiskunde & Informatica

   Affiliation as printed

   Machine Learning Group, CWI, Amsterdam, The Netherlands

   Mathematical Institute, Leiden University, Leiden, The Netherlands