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

TL;DR

This work addresses the fundamental problem of constructing exact and powerful p-values and e-values for composite nulls against composite alternatives, focusing on convex polytopes in the probability-measure space. It develops a unified framework based on convex order and simultaneous optimal transport, revealing precise geometric conditions under which exact pivotal p-values and exact e-values exist, and introduces the SHINE construction to explicitly build such statistics. The authors extend results to general finite and infinite composite settings and connect these objects to nonnegative martingales and e-processes, showing how filtration reduction via transport can unlock test power. The findings offer nonasymptotic, constructive tools for composite hypothesis testing with potential impact on sequential testing and the design of robust, distribution-free tests.

Abstract

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.

Figures (6)

• Figure 1: A showcase of simultaneous transport: here the input vector $\boldsymbol{\mu}$ is two-dimensional, as is the output vector $\boldsymbol{\nu}$. The two input distributions are discrete distributions over the same alphabet of size six and are drawn in different colors in the top row, with the height of a bar indicating its mass. The two target distributions are binary, indicated on the bottom row. The simultaneous transport requires that the maps that transport from $\mu_1$ to $\nu_1$ (left) and from $\mu_2$ to $\nu_2$ (right) are identical. This map is achieved by mixing (averaging) the Radon--Nikodym derivatives. Denoting $\bar{\nu}=\nu_1+\nu_2$ and $\bar{\mu} = \mu_1+\mu_2$, we have $\frac{\mathrm{d}\nu_1}{\mathrm{d} \bar{\nu}}(1) = \frac{1}{3} \left( \frac{\mathrm{d}\mu_1}{\mathrm{d} \bar{\mu}}(1) + \frac{\mathrm{d}\mu_1}{\mathrm{d} \bar{\mu}}(3) + \frac{\mathrm{d}\mu_1}{\mathrm{d} \bar{\mu}}(4)\right)$, and analogously for the other coordinate.
• Figure 2: Illustration of Theorem \ref{['theorem:cx']}. The convex set $\Gamma$ is enclosed by the red contour $\partial\Gamma$ on which $\gamma$ (the law of $({\mathrm{d} P_1}/{\mathrm{d} Q},{\mathrm{d} P_2}/{\mathrm{d} Q})$ under $Q$) is supported. The measure $\mu$ is supported on the thick segment on the diagonal $\mathcal{I}$.
• Figure 3: An illustration of Example \ref{['ex:d=2']}: $\gamma$ is supported on the hyperbola $x_1x_2=e^{-1}$, the optimal $\mu$ is supported on the red ray. Dashed arrows indicate the reduction of filtration.
• Figure 4: An illustration of the SHINE construction in dimension $L=2$. Suppose that the measure $\gamma$ is supported on the region enclosed by the red contour, where $\mathrm{bary}(\gamma)=(1,1)$. In the first step of the SHINE construction, we use Proposition \ref{['prop:LdimHx']} to find a line $\ell_1$ through $(1,1)$ that partitions $\gamma$ into two parts $\mu^{(1)}_1$ and $\mu^{(1)}_2$, each of whose barycenters lies on the diagonal. In the second step, we find a line $\ell_2$ through $x^{(1)}_1=\mathrm{bary}(\mu^{(1)}_1)$ that partitions $\mu^{(1)}_1$ into two measures $\mu^{(2)}_1$ and $\mu^{(2)}_2$, each of whose barycenters lies on the diagonal, and similarly a line $\ell_3$. We then proceed iteratively.
• Figure 5: The SHINE construction for Example \ref{['ex:d=2']}.

Theorems & Definitions (109)

• Remark 1.1
• Lemma 2.1
• Definition 2.2
• Proposition 2.3
• proof
• Proposition 2.4
• Lemma 2.5
• proof
• Theorem 3.1
• Lemma 3.2