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均匀、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、在备择假设下其对数期望为正的e值？当$\mathcal P$和$\mathcal Q$为凸多面体（在概率测度空间中）时，我们回答了这些基本问题及其他相关问题。我们证明，此类构造是可能的当且仅当$\mathcal Q$不与$\mathcal P$的张成空间相交。若允许p值在$P\in \mathcal P$下随机大于均匀分布，且e值在$P\in \mathcal P$下的期望至多为一，则只要$\mathcal P$与$\mathcal Q$不相交即可实现。更一般地，即使$\mathcal P$和$\mathcal Q$不是多面体，我们也刻画了存在有界非平凡e变量的条件，该变量在任何$P \in \mathcal P$下的期望值恰好为一。证明利用了同步最优输运领域的最新技术。其中关键作用的是对滤波的粗化处理：有时，在数据的最丰富滤波中不存在此类p值或e值，但在某些简化滤波中存在，我们的工作首次给出了这一现象的一般性刻画。我们还提供了一种迭代构造方法，可显式构建此类过程，并在特定条件下找到在给定备择假设$Q$下增长最快的一个。我们讨论了其对构造复合非负（上）鞅的启示，并以若干猜想和开放性问题作结。