# On the existence of powerful p-values and e-values for composite hypotheses

arXiv: https://arxiv.org/abs/2305.16539 · HTML: https://mlsanity.com/paper/2305.16539/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 May 2023.

## Built on by (most cited first, 1 of 1)

- [A complete characterization of sequential testability and change detectability in i.i.d. models](https://mlsanity.com/paper/2609.05752/) (2026)

## Abstract (arXiv)

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∈ mathcal P, and the e-value can have expectation at most one under P∈ 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 ∈ 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.