The E-measure

Abstract: We introduce the E-measure: a measure-like generalization of the E-value to a class of hypotheses. Unlike classical measures, E-measures are closed under infimums instead of addition. They arise from a compatibility axiom with logical implications, that there should be at least as much evidence against more specific hypotheses. We show that E-measures are the only non-dominated such objects, if the hypothesis class is closed under intersections. We propose to use the E-measure to present all the relevant evidence for a problem, where the relevance is captured by the choice of hypothesis class. We showcase this by applying the E-measure to decision making, inducing a hypothesis class from the uncertain consequences of decisions. This results in uniform E-consequence bounds on decisions, which nest high-probability loss bounds. Correcting for multiplicity, we consider 'familywise evidence' and 'false evidence rate' control, generalizing from errors and discoveries to continuous evidence. Remarkably, E-measures control these without multiplicity correction if the hypothesis class is intersection-closed. Moreover, we obtain a 'frequentist' notion of updating from E-prior to E-posterior. Abstracting the notion of a 'hypothesis', we advocate for using E-measures for any unknown quantity, leading to predictive E-measures.

• The paper introduces a novel evidence quantification object, the E-measure, that extends traditional E-values to entire hypothesis classes.
• It rigorously formalizes key axioms—impossibility, implication, and closure—ensuring consistency, uniqueness, and optimality in evidence reporting.
• The framework provides robust solutions for multiple testing and decision theory by enabling uniform, post-hoc valid evidence reporting without additional corrections.

Technical Summary of "The E-measure" (2604.20788)

Introduction and Formalization

The paper provides a unifying extension of the E-value framework by introducing the E-measure—a generalized, measure-like object that quantifies evidence across arbitrary classes of hypotheses. Classical E-values provide evidence against a single hypothesis, while E-measures extend this to simultaneously report evidence across an entire class, governed by a set of axioms motivated by logical consistency with implications and impossibilities.

Unlike classical measures, which are additive, E-measures satisfy an infimum closure property: for any union of hypotheses, the evidence is the infimum of the evidence across the constituent hypotheses. The central Closure axiom ensures that evidence against a hypothesis matches the least evidence against all sub-hypotheses that imply it, aligning with the closure principle in multiple testing. The paper rigorously demonstrates that E-measures are the unique non-dominated evidence reporting objects compatible with these axioms when the hypothesis class is closed under intersection.

Main Axioms and Structural Contributions

Three fundamental axioms structure the E-measure:

• Impossibility: Logically impossible hypotheses receive maximal evidence.
• Implication: If $H$ implies $H'$, evidence against $H$ is at least as large as against $H'$ (antitonicity).
• Closure: Evidence against a hypothesis equals the infimum of evidence against its covers (sub-hypotheses).

Under intersection-closed hypothesis classes—those closed under arbitrary unions and intersections—the paper establishes that E-measures can be fully characterized by their "least" hypotheses, minimal elements under set inclusion containing a given parameter value. This is shown using order-theoretic constructions: intersection-closed classes correspond precisely to hypothesis families generated by preorders.

Core Construction and Uniqueness

Every E-function (a map assigning evidence to each hypothesis, taking maximal value at the empty set) is closed by defining the E-measure as the supremum over all hypothesis covers of the infimum of evidence under the cover. This "closure" operator generates the least E-measure dominating an E-function. In intersection-closed classes, closure simplifies: the E-measure at a hypothesis $H$ is the infimum of the E-measures at the least hypotheses contained in $H$. This ensures every E-capacity (antitone E-function) is dominated by its closed E-measure, guaranteeing uniqueness and optimality among non-dominated evidence reports.

Generalization, Validity, and Multiplicity

A crucial operational implication is validity: valid E-measure kernels are those for which the expected evidence under any true distribution is at most one. In the context of multiplicity, the E-measure's structure provides familywise evidence rate and false evidence rate (FER) control analogues of FWER and FDR, but for continuous evidence rather than binary rejections. Key claim: In intersection-closed classes, hypothesis-wise validity simultaneously ensures strong familywise and uniform false evidence rate control, obviating additional multiplicity corrections typically required in classical multiple testing settings.

The E-measure allows for uniform, post-hoc valid evidence reporting across data-driven selection rules—an extension of post-selection inference frameworks—without the need for correction when the hypothesis family is intersection-closed.

Applications to Decision Theory

Via the E-principle, which generalizes the likelihood principle, the E-measure is advocated as the sole object capturing all relevant evidence regarding a statistical question, where relevance is imposed by the choice of the hypothesis class. In decision-theory contexts, this induces a natural hypothesis structure reflecting lower bounds on consequences, enabling the construction of uniform E-consequence bounds on decisions. These bounds generalize recent E-value and probabilistic approaches to high-probability loss bounds, such as the E-posterior minimax risk framework ([Grünwald, 2023] [grunwald2023posterior]).

The E-measure's aggregated consequence bound nests these results, and the E-integration mechanism (mirroring the Shilkret/maxitive integral) provides an operational means to aggregate consequences in evidence-driven fashion. The resulting minimization of evidence-weighted loss extends optimal decision selection into preorder/consequence spaces, not limited to real-valued losses.

Bayesian-Like Updating and Sequential Settings

The E-measure formalism supports a frequentist updating rule analogous to Bayesian updating: products of prior E-measures with valid data-driven E-measure kernels yield valid posterior E-measures under closure. This produces a "frequentist" E-posterior, remaining valid under the E-measure axiomatization, and yielding a generalized notion of martingales and anytime-valid E-measure processes. This deepens the connection between E-value-based sequential inference and the law of (E-)measure processes, extending the result that admissible (sequential) evidence reporting for composite hypotheses must be constructed as the infimum over admissible processes for point, or least, hypotheses ([Ramdas et al., 2022] [ramdas2022admissible]).

Abstraction: Generalized Hypotheses and Predictive Inference

The E-measure approach is fully abstract over the notion of hypothesis. "Hypotheses" need not correspond to collections of probability distributions; they can be arbitrary measurable objects, e.g., predictive sets. In predictive inference, the paper illustrates that E-measure kernels encode fuzzified prediction sets, and valid E-measure kernels reduce predictive validity to validity on least hypotheses (singletons), paralleling conformal inference guarantees ([Koning & van Meer, 2025] [koning2025fuzzy]). The framework canonically extends E-measures via pushforwards under measurable maps to arbitrary parameter or decision spaces.

Implications and Theoretical Perspective

This work provides a fully rigorous, order-theoretic foundation for evidence quantification in statistics, unifying and generalizing existing E-value methodologies, and offering a maximalist alternative to classical p-value or confidence-set thinking. The formal equivalence between E-measures and maxitive measures (Shilkret integrals) underlines a deep connection to non-additive probability theory, but rooted in operational evidence.

The results demonstrate that when the class of hypotheses respects logical closure (intersections), evidence quantification for multiple testing, selective inference, and decision theory can be rendered multiplicity-robust without conservativeness or the loss of power associated with classical error-rate corrections. The E-measure formalism is agnostic to the semantic interpretation of hypotheses, enabling its application in parametric, nonparametric, predictive, and causal inference settings.

Conclusion

The E-measure constitutes a unified, logically principled, and operationally sound structure for reporting evidence across hypothesis classes, generalizing and subsuming numerous modern statistical methods that use E-values, confidence sets, and evidence post-processing. Its closure properties guarantee both uniqueness and optimality within its axiomatic system, and its structure addresses multiplicity and post-selection inference intrinsically when logical closure is respected. Practically, E-measures open a path toward robust, interpretable, and simultaneously valid evidence reporting across statistical decision-making, multiple testing, and predictive inference domains.

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References:

• Grünwald, P. D., "The e-posterior" [grunwald2023posterior]
• Ramdas, A., et al., "Admissible anytime-valid sequential inference must rely on nonnegative martingales" [ramdas2022admissible]
• Koning, N., van Meer, S., "Fuzzy prediction sets: Conformal prediction with e-values" [koning2025fuzzy]
• Xu, Z., et al., "Bringing closure to false discovery rate control: A general principle for multiple testing" (Xu et al., 2 Sep 2025)
• Wang, R., Ramdas, A., "False Discovery Rate Control with E-values" [wang2022false]
• Shilkret, N., "Maxitive measure and integration," [shilkret1971maxitive]