# Optimal Recovery Meets Bayesian Learning: Where Worst-Case Bounds Pay Off

arXiv: https://arxiv.org/abs/2609.29622 · HTML: https://mlsanity.com/paper/2609.29622/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 35 references that are in the map.
- First posted on arXiv 29 Aug 2026.

## Builds on (most cited first, 8 of 8)

- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [Adaptive Conformal Inference Under Distribution Shift](https://mlsanity.com/paper/2106.00170/) (2021)
- [On Kernelized Multi-armed Bandits](https://mlsanity.com/paper/1704.00445/) (2017)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Optimal Recovery from Inaccurate Data in Hilbert Spaces: Regularize, but what of the Parameter?](https://mlsanity.com/paper/2111.02601/) (2021)
- [S-Procedure Relaxation: a Case of Exactness Involving Chebyshev Centers](https://mlsanity.com/paper/2310.09677/) (2023)
- [Radius of Information for Two Intersected Centered Hyperellipsoids and Implications in Optimal Recovery from Inaccurate Data](https://mlsanity.com/paper/2401.11112/) (2024)

## Abstract (arXiv)

Worst-case Optimal Recovery (OR) and Bayesian learning describe the same Gaussian-quadratic-Hilbert problems in two vocabularies. We sharpen the correspondence - the radius of information equals a nugget-optimized GP posterior variance and is attained by the posterior mean at a closed-form balance nugget - and measure, inside three published Bayesian systems, where the worst-case side pays. The ledger is two-sided: the losses instruct as much as the wins. Morozov calibration tracks a test-access oracle within 1.00-1.19× where σ-blind rules fail, is 4.9-6.3× more reproducible across noise draws (p=0.002-0.004), and is the only deployable rule whose selection survives a change of backend (1.36× against 12-30× for the released weight, ML-II and GCV); tight certificates cover at the information-theoretic floor with no numerical slack. But on exchangeable data split-conformal beats the OR head on interval score, a water-filling prior adds nothing without an oracle noise hint, and under covariate shift the OR band keeps coverage on every dataset yet loses interval score to split-conformal, and to a feature-free constant band, on most cells; what pays is not shift but shift on a learnable target, which a training-free audit statistic predicts before any model is fitted. In Bayesian optimization the certified width is a validity floor whose scalar inflation we prove inert under a checkable margin condition and check at every step. Inertness is graded, not binary, and in the size of the inflation as much as in the objective: kappa=2 is inert wherever kappa=5 is and on more cells besides, while kappa=5 moves half the Ackley seeds and every Griewank seed. Exploration is a shape problem, not a scale one. The design rule: match the guarantee tool to the data regime, and audit the regime first.