用MMD球构建可信集，为测试时适应提供不确定性量化新方法
MMD-Balls as Credal Sets: A PAC-Bayesian Framework for Epistemic Uncertainty in Test-Time Adaptation
- 将MMD球视为可信集，结合精确概率理论实现可解释的不确定性估计
- 在核空间利普希茨假设下，给出与分布偏移程度相关的泛化界
- 区分认知与随机不确定性，为是否需要自适应提供决策依据
测试时适应（TTA）方法能缓解分布偏移带来的性能下降，但缺乏将偏移程度与预测可靠性关联的严格保证。本文提出一个基于PAC-Bayesian的框架，其泛化界显式依赖源分布与目标分布之间的最大均值差异（MMD）。核心贡献是将围绕源分布的MMD球解释为瓦利（Walley）不精确概率理论中的可信集，从而自然地实现认知不确定性量化。我们证明了：(i) 在核希尔伯特空间利普希茨损失假设下，存在依赖MMD的偏移惩罚项的PAC-Bayesian界；(ii) 通过MMD集中性推导出有限样本版本；(iii) 在可信集内所有分布上得到统一最坏情况风险界，并实现上下界风险分解；(iv) 建立测地线保持边界，解释为何核引导适应能保护局部特征几何结构。该可信集解释有效分离认知与随机不确定性，为适应必要性提供了原则性判断标准。
原文摘要 · Abstract (English)
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.
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