E-values as statistical evidence: A comparison to Bayes factors, likelihoods, and p-values

2026-03-25

AI Summary

This work proposes and systematically evaluates e-values and their sequential counterpart—e-processes—as a unified measure of statistical evidence, addressing key limitations of existing metrics such as p-values, likelihood ratios, and Bayes factors in settings involving composite hypotheses, optional stopping, and evidence aggregation. Framed through a game-theoretic lens, e-values are shown to uniquely reconcile frequentist error guarantees, Bayesian flexibility, and likelihood-based interpretability. Theoretical analysis and empirical results demonstrate that e-values excel in controlling long-run error rates, handling composite hypotheses, accommodating arbitrary stopping rules, and enabling coherent combination of evidence across studies, thereby offering strong potential as a foundational tool for statistical inference.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Calibration & Uncertainty Quantification
• Game Theory and Economic Paradigms: Social Choice / Voting

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

A recurring debate in the philosophy of statistics concerns what, exactly, should count as a measure of evidence for or against a given hypothesis. P-values, likelihood ratios, and Bayes factors all have their defenders. In this paper we add two additional candidates to this list: the e-value and its sequential analogue, the e-process. E-values enjoy several desirable properties as measures of evidence: they combine naturally across studies, handle composite hypotheses, provide long-run error rates, and admit a useful interpretation as the wealth accrued by a bettor in a game against the null distribution. E-processes additionally handle optional stopping and optional continuation. This work examines the extent to which e-values and e-processes satisfy the evidential desiderata of different statistical traditions, concluding that they combine attractive features of p-values, likelihood ratios, and Bayes factors, and merit serious consideration as interpretable and intuitive measures of statistical evidence.

Problem

• e-values
• statistical evidence
• Bayes factors
• likelihood ratios
• p-values

Innovation

• e-values
• e-processes
• statistical evidence
• optional stopping
• composite hypotheses