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 (the convex hull of) $\mathcal Q$ 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. 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 when or why such a phenomenon occurs. We also provide an iterative construction that explicitly constructs such processes, that under certain conditions finds the one that grows fastest under a specific alternative $\mathcal Q$. 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 $\mathcal Q$，何时以及如何构建一个p值，使其在零假设下分布精确均匀，而在备择假设下随机小于均匀分布？类似地，何时以及如何构建一个e值，使其在零假设下期望精确等于1，而在备择假设下期望对数大于0？我们回答了这些基本问题及其他相关问题，其中 $\mathcal P$ 和 $\mathcal Q$ 是（概率测度空间中的）凸多面体。我们证明，当且仅当 $\mathcal Q$ 的（凸包）不与 $\mathcal P$ 的生成空间相交时，这类构造是可能的。若允许p值在 $P \in \mathcal P$ 下随机大于均匀分布，且e值在 $P \in \mathcal P$ 下期望至多为1，则当 $\mathcal P$ 和 $\mathcal Q$ 不相交时，该构造总是可行的。证明利用了近期发展的同步最优传输技术。一个关键作用来自信息流的粗化：有时，在信息最丰富的数据流中不存在这样的p值或e值，但在某种简化后的信息流中存在，而我们的工作首次给出了此类现象何时或为何发生的通用刻画。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指定备择假设 $\mathcal Q$ 下增长最快的过程。我们讨论了该结果对构造复合非负（超）鞅的意义，并总结了一些猜想与开放问题。