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

Rianne de Heide

What is this paper about?

This preprint develops a closure principle for the genuinely online setting in which new hypotheses arrive over time while the evidence for existing hypotheses keeps evolving. It studies what coherence across time and across expanding hypothesis families requires, and derives simultaneous stopped-FDR and stronger persistent guarantees.

Summary

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. The theory extends from FDP to bounded losses that are monotone in the possible true-null configuration and local in the reported action.

Topics

E-values and anytime-valid inference · Multiple testing

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