Time-Sensitive Anytime-Valid Testing
A time-sensitive testing-by-betting framework favors early rejection via time-weighted rewards, yielding Bellman-optimal e-processes and an exponential-decay-optimal criterion recovering classical growth-rate optimality at large scales.
Published 2026Sydney Poster Session 6 · Thu, Dec 10, 5:00 PM–8:00 PM local time · Hall 1-4arXiv ↗OpenReview ↗
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The paper elegantly extends anytime-valid testing with a time-sensitive betting framework and sharp Bellman reductions, but it remains purely theoretical without empirical baselines or concrete applications for hard deadlines.
Abstract
Anytime-valid tests allow evidence to be checked during data collection: one can either continue testing or stop and reject the null while still controlling type-I error. Yet, in many applications rejection is useful only if it comes soon enough. We introduce a time-sensitive testing-by-betting framework that favours early rejection by assigning rewards to rejection times and maximising their expected value under a given alternative. This encompasses hard deadlines and softer time preferences. The resulting optimal control problem admits a Bellman representation in terms only of time and evidence against the null, rather than the full history. For hard deadlines, the simple-vs-simple case reduces to a finite-horizon Neyman--Pearson problem and identify the corresponding optimal e-process. Furthermore, we show that exponentially decaying rewards admit a stationary approximation, yielding the exponential-decay-optimal (EDO) criterion: a finite-time-scale counterpart to the classical growth-rate-optimal (GRO) viewpoint in anytime-valid statistics, with the GRO criterion recovered in the large-time-scale limit.