# Time-sensitive anytime-valid testing

arXiv: https://arxiv.org/abs/2605.06521 · HTML: https://mlsanity.com/paper/2605.06521/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 May 2026.

## Built on by (most cited first, 1 of 1)

- [Confidence Horizons](https://mlsanity.com/paper/2608.03889/) (2026)

## Abstract (arXiv)

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.