# MMD-Balls as Credal Sets: A PAC-Bayesian Framework for Epistemic Uncertainty in Test-Time Adaptation

arXiv: https://arxiv.org/abs/2605.21783 · HTML: https://mlsanity.com/paper/2605.21783/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 22 references that are in the map.
- First posted on arXiv 20 May 2026.

## Builds on (most cited first, 9 of 9)

- [Towards Stable Test-Time Adaptation in Dynamic Wild World](https://mlsanity.com/paper/2302.12400/) (2023)
- [MEMO: Test Time Robustness via Adaptation and Augmentation](https://mlsanity.com/paper/2110.09506/) (2021)
- [Adaptive Conformal Inference Under Distribution Shift](https://mlsanity.com/paper/2106.00170/) (2021)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Robust Test-Time Adaptation in Dynamic Scenarios](https://mlsanity.com/paper/2303.13899/) (2023)
- [Generative Models and Model Criticism via Optimized Maximum Mean Discrepancy](https://mlsanity.com/paper/1611.04488/) (2016)
- [PAC-Bayesian Theory Meets Bayesian Inference](https://mlsanity.com/paper/1605.08636/) (2016)
- [Revisiting Realistic Test-Time Training: Sequential Inference and Adaptation by Anchored Clustering](https://mlsanity.com/paper/2206.02721/) (2022)
- [Uncertainty Quantification for Machine Learning: One Size Does Not Fit All](https://mlsanity.com/paper/2512.12341/) (2025)

## Abstract (arXiv)

Test-time adaptation (TTA) methods improve model performance under distribution shift but lack formal guarantees connecting shift magnitude to prediction reliability. We develop a PAC-Bayesian framework yielding generalization bounds explicitly parameterized by the maximum mean discrepancy (MMD) between source and target distributions. Our principal contribution is interpreting MMD-balls around the source distribution as credal sets in Walley's imprecise probability theory, yielding natural epistemic uncertainty quantification. We establish: (i) a PAC-Bayesian bound with an MMD-dependent shift penalty under an RKHS-Lipschitz loss assumption; (ii) a finite-sample version via MMD concentration; (iii) a uniform worst-case risk bound over all distributions in the credal set, with a lower-upper risk decomposition; and (iv) geodesic preservation bounds explaining why kernel-guided adaptation protects local feature geometry. The credal set interpretation separates epistemic from aleatoric uncertainty and provides a principled decision criterion for when adaptation is warranted.