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：给定复合零假设$\mathcal P$和复合备择假设$\mathcal Q$，在何种条件下以及如何构造一个p值，使其在零假设下精确服从均匀分布，且在备择假设下随机小于均匀分布？类似地，在何种条件下以及如何构造一个e值，使其在零假设下的期望精确等于1，但在备择假设下的期望对数呈正值？我们针对$\mathcal P$和$\mathcal Q$为凸多面体（位于概率测度空间内）的情形，回答了这些基本问题及其他相关问题。我们证明：当且仅当$\mathcal Q$不与$\mathcal P$的生成空间相交时，此类构造才可能实现。若允许p值在$P\in\mathcal P$下随机大于均匀分布，且e值在$P\in\mathcal P$下的期望不超过1，则当$\mathcal P$和$\mathcal Q$不相交时该构造总是可行的。更一般地，即使$\mathcal P$和$\mathcal Q$并非多面体，我们刻画了存在有界非平凡e变量（其在任意$P\in\mathcal P$下的期望精确等于1）的条件。证明过程利用了同步最优输运领域的最新发展技术。其中，滤子粗化发挥了关键作用：有时在最丰富的数据滤子中不存在此类p值或e值，但在某个约化滤子中却存在，我们的工作首次对此现象进行了通用刻画。我们还提供了一种显式构造此类过程的迭代方法，并在特定条件下找到了在特定备择假设$Q$下增长最快的构造。最后，我们讨论了该方法对构造复合非负（超）鞅的启示，并提出了若干猜想与开放性问题。