这篇论文详细探讨了轨迹路径空间上的信息流，揭示了鞅与随机时间之间的关系，对于理解集中不等式和相对熵有重要意义。
本文研究了非负鞅轨迹路径空间上的信息流，得出了其在任意随机时间上的精确变分恒等式。这恢复了从Ville到PAC-Bayes广泛使用的经典集中不等式，并衡量了每个不等式所丢弃的信息。尾端界限控制本身是一个相对熵，通过链式法则分解为每步条件散度。丢弃的松弛在每个几何形状中都有确切形式：Azuma-Hoeffding和PAC-Bayes界限的Gibbs倾斜，Ville的交叉本身以及池化测试的交叉，以及$L^p$最大界限的支配证书。该证书的选停损失分解为每步的运行最大值的Bregman散度。在路径-时间空间上，相同的恒等式获得了一个因素，用于定价预期：任意随机时间携带一个e过程“窥视惩罚”。分函数可以读作一个合并——独立副本的前缀共享概率——测试鞅的几何混合为多模型安全测试获得池化收益。
Information on trajectories: martingales and random times
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.