# Bayes with No Shame: Admissibility Geometries of Predictive Inference

arXiv: https://arxiv.org/abs/2603.05335 · HTML: https://mlsanity.com/paper/2603.05335/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 51 references that are in the map.
- First posted on arXiv 5 Mar 2026.

## Builds on (most cited first, 7 of 7)

- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [Game-theoretic statistics and safe anytime-valid inference](https://mlsanity.com/paper/2210.01948/) (2022)
- [Safe Testing](https://mlsanity.com/paper/1906.07801/) (2019)
- [Admissible anytime-valid sequential inference must rely on nonnegative martingales](https://mlsanity.com/paper/2009.03167/) (2020)
- [Martingale posterior distributions](https://mlsanity.com/paper/2103.15671/) (2021)
- [Combining Evidence Across Filtrations](https://mlsanity.com/paper/2402.09698/) (2024)
- [De Finetti + Sanov = Bayes: Exchangeable Prediction under Moment Constraints](https://mlsanity.com/paper/2509.13283/) (2025)

## Abstract (arXiv)

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 within the declared rank-indexed family, 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.