Bayes with No Shame: Admissibility Geometries of Predictive Inference

2026-03-05

AI Summary

This work clarifies the relationships and boundaries among four admissibility criteria in predictive inference—Blackwell risk dominance, anytime-valid supermartingale cones, marginal coverage validity, and Cesàro approximability—and resolves their geometric incompatibility. By integrating Blackwell’s decision theory, nonnegative supermartingales, exchangeable prediction sets, and Cesàro approximation techniques, the paper constructs a unified multi-space constrained analytical framework. Its central contribution is a discriminative theorem establishing that the four classes of admissible procedures are mutually non-nested, revealing their fundamental incommensurability. The study further characterizes the optimality certificates and necessary and sufficient conditions for each criterion, elucidating the divergent roles of martingale consistency across these frameworks and thereby laying a geometric foundation for predictive inference.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

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 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.

Problem

• admissibility
• predictive inference
• Bayesian risk
• martingale coherence
• optimality criteria

Innovation

• admissibility geometries
• anytime-valid inference
• nonnegative supermartingale
• exchangeable prediction
• Cesàro approachability