Given a composite null $ \mathcal P$ and composite alternative $ \mathcal Q$, when and how can we construct a p-value whose distribution is exactly uniform under the null, and stochastically smaller than uniform under the alternative? Similarly, when and how can we construct an e-value whose expectation exactly equals one under the null, but its expected logarithm under the alternative is positive? We answer these basic questions, and other related ones, when $ \mathcal P$ and $ \mathcal Q$ are convex polytopes (in the space of probability measures). We prove that such constructions are possible if and only if $ \mathcal Q$ does not intersect the span of $ \mathcal P$. If the p-value is allowed to be stochastically larger than uniform under $P\in \mathcal P$, and the e-value can have expectation at most one under $P\in \mathcal P$, then it is achievable whenever $ \mathcal P$ and $ \mathcal Q$ are disjoint. More generally, even when $ \mathcal P$ and $ \mathcal Q$ are not polytopes, we characterize the existence of a bounded nontrivial e-variable whose expectation exactly equals one under any $P \in \mathcal P$. The proofs utilize recently developed techniques in simultaneous optimal transport. A key role is played by coarsening the filtration: sometimes, no such p-value or e-value exists in the richest data filtration, but it does exist in some reduced filtration, and our work provides the first general characterization of this phenomenon. We also provide an iterative construction that explicitly constructs such processes, and under certain conditions it finds the one that grows fastest under a specific alternative $Q$. We discuss implications for the construction of composite nonnegative (super)martingales, and end with some conjectures and open problems.

翻译：给定一个复合原假设 $\mathcal P$ 与复合备择假设 $\mathcal Q$，我们何时以及如何能够构造一个在原假设下分布恰好为均匀分布、在备择假设下随机小于均匀分布的 p 值？类似地，我们何时以及如何能够构造一个在原假设下期望值恰好等于一、但在备择假设下其对数期望为正的 e 值？当 $\mathcal P$ 和 $\mathcal Q$ 是凸多面体（在概率测度空间中）时，我们回答了这些基本问题以及其他相关问题。我们证明，这样的构造是可能的，当且仅当 $\mathcal Q$ 不与 $\mathcal P$ 的张成空间相交。如果允许 p 值在 $P\in \mathcal P$ 下随机大于均匀分布，并且 e 值在 $P\in \mathcal P$ 下的期望至多为一，那么只要 $\mathcal P$ 和 $\mathcal Q$ 不相交，该构造就是可实现的。更一般地，即使当 $\mathcal P$ 和 $\mathcal Q$ 不是多面体时，我们也刻画了存在一个有界非平凡 e 变量的条件，该变量在任何 $P \in \mathcal P$ 下的期望值恰好等于一。证明利用了最近在同步最优传输中发展的技术。其中，对滤波的粗化起着关键作用：有时，在最丰富的数据滤波中不存在这样的 p 值或 e 值，但在某些简化的滤波中存在，我们的工作首次提供了这种现象的一般性刻画。我们还提供了一种迭代构造方法，能够显式地构造此类过程，并且在特定条件下，它能找到在给定备择假设 $Q$ 下增长最快的那一个。我们讨论了其对构造复合非负（上）鞅的启示，并以一些猜想和开放性问题作为结尾。