Bayes with No Shame: Admissibility Geometries of Predictive Inference
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| Format: | Preprint |
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2026
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| author | Polson, Nicholas G. Zantedeschi, Daniel |
| author_facet | Polson, Nicholas G. Zantedeschi, Daniel |
| contents | 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 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_05335 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bayes with No Shame: Admissibility Geometries of Predictive Inference Polson, Nicholas G. Zantedeschi, Daniel Machine Learning Statistics Theory 62C15, 62F07, 62L15, 62G15, 91A26, 60G42 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 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. |
| title | Bayes with No Shame: Admissibility Geometries of Predictive Inference |
| topic | Machine Learning Statistics Theory 62C15, 62F07, 62L15, 62G15, 91A26, 60G42 |
| url | https://arxiv.org/abs/2603.05335 |