AI 中文总结
该研究针对非负鞅轨迹路径空间的信息流，推导了含任意随机时间的精确变分恒等式，揭示经典集中不等式的松弛项形式，引入e过程窥视惩罚，并将配分函数关联溯祖模型，为多模型安全检验提供合并收益。
AI 中文摘要
对非负鞅轨迹的路径空间上的信息流进行核算，可得到其在任意随机时间下的精确变分恒等式，该恒等式涵盖从Ville到PAC-Bayes的广泛使用的经典集中不等式，并衡量各不等式所舍弃的内容。所控制的尾界本身是一种相对熵，通过链式法则可分解为每一步的条件散度；在三种几何结构中，舍弃的松弛项具有精确形式：对应Azuma-Hoeffding和PAC-Bayes界的Gibbs倾斜、对应Ville界和合并检验的交叉项、对应L^p极大界的主导证书，该证书的可选停止缺陷可分解为每一步运行最大值的Bregman散度。在路径-时间空间中，同一恒等式引入了一个定价预期的因子：任意随机时间带有e过程“窥视惩罚”；配分函数可解读为溯祖模型——独立副本的前缀共享概率，而检验鞅的几何混合则为多模型安全检验带来合并收益。
英文摘要
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.
