Bayes with No Shame: Admissibility Geometries of Predictive Inference

TL;DR

This paper explores four distinct geometries of admissibility in sequential and distribution-free inference, revealing their differences, relationships, and the criteria certifying optimality within each framework.

Contribution

It introduces a unifying schematic template for these geometries and proves they are pairwise non-nested, highlighting the criterion-relative nature of admissibility.

Findings

Four admissibility geometries are pairwise non-nested.

Each geometry has a unique certificate of optimality.

Martingale coherence relates differently across geometries.

Abstract

Four distinct admissibility geometries govern sequential and distribution-free inference: Blackwell risk dominance over convex risk sets, anytime-valid admissibility within the nonnegative supermartingale cone, marginal coverage validity over exchangeable prediction sets, and Ces\`aro approachability (CAA) admissibility, which reaches the risk-set boundary via approachability-style arguments rather than explicit priors. We prove a criterion separation theorem: the four classes of admissible procedures are pairwise non-nested. Each geometry carries a different certificate of optimality: a supporting-hyperplane prior (Blackwell), a nonnegative supermartingale (anytime-valid), an exchangeability rank (coverage), or a Ces\`aro steering argument (CAA). Martingale coherence is necessary for Blackwell admissibility and necessary and sufficient for anytime-valid admissibility within e-processes, but is not sufficient for Blackwell admissibility and is not necessary for coverage validity or CAA-admissibility. All four criteria can be viewed through a common schematic template (minimize Bayesian risk subject to a feasibility constraint), but the decision spaces, partial orders, and performance metrics differ by criterion, making them geometrically incompatible. Admissibility is irreducibly criterion-relative.