用随机路径信息揭示马尔可夫鞅的变分恒等式，统一了多种统计不等式。
Information on trajectories: martingales and random times
- 基于路径空间的信息流推导出鞅的精确变分恒等式。
- 在任意随机时刻下，各经典不等式所丢失的松弛量可精确表达。
- 适用于多模型安全检验，提供可解释的“窥探惩罚”机制。
对非负鞅路径空间上的信息流动进行建模，可在任意随机时间点获得其精确的变分恒等式。该方法恢复了从Ville到PAC-Bayes等广泛使用的经典集中不等式，并量化了每种不等式所舍弃的松弛量。尾部界限本身是一个相对熵，可通过链式法则分解为每步的条件散度。三种几何结构下的松弛项有明确形式：Azuma-Hoeffding与PAC-Bayes对应吉布斯倾斜，Ville及合并检验对应交叉过程，$L^p$最大值界则由主导证书表示。该证书的可选停止缺陷可分解为运行最大值的Bregman散度。在路径-时间空间中，同一恒等式引入一个定价前瞻性的因子：任意随机时间携带一个“窥探惩罚”的e过程。分区函数可解释为独立副本的前缀共享概率（共现过程），而测试鞅的几何混合则在多模型安全检验中获得聚合优势。
原文摘要 · Abstract (English)
Accounting for information flow on the path space of trajectories of a nonnegative martingale yields exact variational identities for it, even at arbitrary random times. This recovers the widely used classical concentration inequalities, from Ville to PAC-Bayes, and measures what each one discards. The tail a bound controls is itself a relative entropy, resolved by the chain rule into per-step conditional divergences. The discarded slack has an exact form in each of three geometries: a Gibbs tilt for the Azuma-Hoeffding and PAC-Bayes bounds, the crossing itself for Ville's and for pooled tests, and a dominating certificate for the $L^p$ maximal bound. That certificate's optional-stopping deficit resolves per step into Bregman divergences of the running maximum. On a path-time space, the same identity gains one factor that prices anticipation: an arbitrary random time carries an e-process ``peeking penalty.'' The partition function can be read as a coalescent--a prefix-sharing probability of independent copies--and geometric mixtures of test martingales gain a pooling benefit for multi-model safe testing.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。