When do exact and powerful p-values and e-values exist?
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Given a composite null 𝒫 and composite alternative 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 𝒫 and 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) 𝒬 does not intersect the span of 𝒫. If the p-value is allowed to be stochastically larger than uniform under P ∈𝒫, and the e-value can have expectation at most one under P ∈𝒫, then it is achievable whenever 𝒫 and 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 𝒬. We discuss implications for the construction of composite nonnegative (super)martingales, and end with some conjectures and open problems.
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