Bayes with No Shame: Admissibility Geometries of Predictive Inference

Researchers Nicholas G. Polson and Daniel Zantedeschi have published a groundbreaking statistical theory paper titled 'Bayes with No Shame: Admissibility Geometries of Predictive Inference' that fundamentally challenges how we evaluate AI prediction methods. The paper proves a criterion separation theorem showing that four major approaches to sequential and distribution-free inference - Blackwell risk dominance, anytime-valid admissibility within nonnegative supermartingale cones, marginal coverage validity over exchangeable prediction sets, and Cesàro approachability (CAA) admissibility - are pairwise non-nested, meaning no single method dominates all others. This work establishes that admissibility is irreducibly criterion-relative, with each geometry carrying different certificates of optimality: supporting-hyperplane priors for Blackwell, nonnegative supermartingales for anytime-valid methods, exchangeability ranks for coverage, and Cesàro steering arguments for CAA.

The research demonstrates that while all four criteria share a common optimization template (minimizing Bayesian risk subject to feasibility constraints), their constraint sets operate over fundamentally different mathematical spaces, partial orders, and performance metrics, making them geometrically incompatible. The authors show that martingale coherence is necessary for Blackwell admissibility and necessary/sufficient for anytime-valid admissibility within e-processes, but isn't sufficient for Blackwell admissibility and isn't necessary for coverage validity or CAA-admissibility. This mathematical framework has profound implications for AI system design, suggesting practitioners must choose inference methods based on specific application requirements rather than seeking a universal 'best' approach, potentially explaining why different AI models excel in different prediction tasks despite similar architectures.