Bayes with No Shame: Admissibility Geometries of Predictive Inference

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 Cesàro 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 Cesàro 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 share a common optimization template (minimize Bayesian risk subject to a feasibility constraint), but the constraint sets operate over different spaces, partial orders, and performance metrics, making them geometrically incompatible. Admissibility is irreducibly criterion-relative.