aiqbench analysis
This theoretical paper establishes a rigorous framework for predictive inference by mapping the distinct geometric properties of four major criterion classes, including anytime-valid admissibility and choice-based approachability. The authors prove that these paradigms occupy non-nesting spaces within a unified product structure, clarifying the precise conditions under which they diverge or align without implying practical incompatibility. Such foundational work is essential for advancing the mathematical coherence of modern sequential monitoring systems and robust prediction algorithms.
Bayes with No Shame: Admissibility Geometries of Predictive Inference
arXiv:2603.05335v3 Announce Type: replace-cross Abstract: Modern predictive systems combine predictors, sequential monitors, prediction sets, and online strategies, each with a different certificate of optimality. We study four criterion-relative geometries: Blackwell risk dominance, anytime-valid admissibility, fixed-level marginal coverage with expected-length efficiency within a declared rank-indexed family, and choice-based approachability (CApp) boundary-feasibility. We embed the four procedure types in a common product space and prove witness-based pairwise non-nesting: for every ordered pair of criterion classes, an explicit predictive system is active in both relevant coordinates, belongs to one class, and fails the other. The result records non-nesting across different object spaces and partial orders; it does not assert practical incompatibility. We separate three measure-relative coherence notions. In conditionally i.i.d. models, posterior predictive means under a single prior are martingales under the prior predictive law. For a point null, anytime-valid admissibility within e-processes is equivalent to the nonnegative martingale property. Self-consistency under a predictor's own predictive law does not imply Blackwell admissibility, as shown by a Bernoulli log-loss counterexample. Coverage admissibility is certified by exchangeability ranks, while CApp boundary-feasibility uses Cesaro steering. A constrained-Bayes design schema organizes the four paradigms without collapsing their distinct decision spaces, partial orders, or risk functionals. Admissibility is criterion-relative.
Abstract:Modern predictive systems combine predictors, sequential monitors, prediction sets, and online strategies, each with a different certificate of optimality. We study four criterion-relative geometries: Blackwell risk dominance, anytime-valid admissibility, fixed-level marginal coverage with expected-length efficiency within a declared rank-indexed family, and choice-based approachability (CApp) boundary-feasibility. We embed the four procedure types in a common product space and prove witness-based pairwise non-nesting: for every ordered pair of criterion classes, an explicit predictive system is active in both relevant coordinates, belongs to one class, and fails the other. The result records non-nesting across different object spaces and partial orders; it does not assert practical incompatibility. We separate three measure-relative coherence notions. In conditionally i.i.d. models, posterior predictive means under a single prior are martingales under the prior predictive law. For a point null, anytime-valid admissibility within e-processes is equivalent to the nonnegative martingale property. Self-consistency under a predictor's own predictive law does not imply Blackwell admissibility, as shown by a Bernoulli log-loss counterexample. Coverage admissibility is certified by exchangeability ranks, while CApp boundary-feasibility uses Cesaro steering. A constrained-Bayes design schema organizes the four paradigms without collapsing their distinct decision spaces, partial orders, or risk functionals. Admissibility is criterion-relative.
Accepted for publication in Transactions on Machine Learning Research (TMLR). 37 pages, 6 figures, 5 tables. This replacement adds author information, corrects localized scope statements and supporting arguments, and streamlines the exposition. The main separation results and numerical findings are unchanged
Machine Learning (stat.ML); Machine Learning (cs.LG); Statistics Theory (math.ST)
arXiv:2603.05335 [stat.ML]
(or arXiv:2603.05335v3 [stat.ML] for this version)
Full text under the publisher's open licence · 254 words
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