Dynamic $e$-closure for online hypotheses with any-time-valid evidence: closure principles and projective mergers

2026-08-10

AI Summary

This work addresses the challenge of controlling multiple testing errors in online hypothesis testing, where hypotheses arrive dynamically and evidence becomes observable at arbitrary times. The authors propose a dynamic e-closure method that integrates e-values, dynamic closure principles, and cross-time consistency constraints to establish the first theoretical framework capable of effectively controlling the supremum false discovery rate (SupFDR) under simultaneous stopping and ensuring the persistence of rejection sets. Key contributions include establishing universal guarantees under a canonical normalized loss process, revealing structural limitations inherent to pointwise merging procedures, and constructing a globally shared control procedure via projection merging, alongside a counterexample demonstrating the failure of consistency under certain conditions.

Abstract

Many modern testing problems are sequential along two axes: new hypotheses may arrive over time, while evidence for hypotheses already under consideration continues to evolve and may be inspected at arbitrary stopping times. We develop dynamic $e$-closure for this setting. At a global stopping time the active true-null intersection is random. Future-extension coherence allows its certificate to be compared with that of a fixed terminal intersection, yielding simultaneous stopped-FDR control. If the certificates are also time-monotone, the resulting closure controls simultaneous SupFDR and is setwise persistent. Conversely, every procedure satisfying either criterion is contained in a dynamic closure generated by canonical normalized-loss processes. For pointwise mergers, fixed-dimensional admissibility is equivalent to ordinary arbitrary-dependence $e$-merging. Coherence across horizons then forces a single globally summable weight sequence and exact neutrality under padding by the $e$-value one; on a countably infinite hypothesis universe, this rules out nontrivial symmetric mergers in the admissible pointwise class. The theory extends from FDP to bounded losses that are monotone in the possible true-null configuration and local in the reported action. We also give a coherence counterexample, persistent constructions, and a globally valid shared-control model.

Problem

• sequential testing
• multiple hypothesis testing
• false discovery rate
• e-values
• any-time valid inference

Innovation

• dynamic e-closure
• any-time-valid inference
• stopped-FDR control
• future-extension coherence
• e-value merging