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. 2023 May 15;381(2247):20220146.
doi: 10.1098/rsta.2022.0146. Epub 2023 Mar 27.

The e-posterior

Peter D Grünwald  1   2
Affiliations

The e-posterior

Peter D Grünwald. Philos Trans A Math Phys Eng Sci. .

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'.

Keywords: Bayes-frequentist debate; Savage–Dickey ratio; conditional inference; decision making; e-values; uncertainty quantification.

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