# Anytime Validity is Free: Inducing Sequential Tests

arXiv: https://arxiv.org/abs/2501.03982 · HTML: https://mlsanity.com/paper/2501.03982/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jan 2025.

## Built on by (most cited first, 4 of 4)

- [Fuzzy Prediction Sets: Conformal Prediction with E-values](https://mlsanity.com/paper/2509.13130/) (2025)
- [Learning to Bet for Horizon-Aware Anytime-Valid Testing](https://mlsanity.com/paper/2603.19551/) (2026)
- [Detect Early, Escalate Rarely: Anytime Detection of AI-Generated Video from the Compressed Bitstream](https://mlsanity.com/paper/2607.19476/) (2026)
- [Two Conformal Constructions for Adaptive Within-Document AI-Text Screening](https://mlsanity.com/paper/2609.31547/) (2026)

## Abstract (arXiv)

Anytime valid sequential tests permit us to stop testing based on the current data, without invalidating the inference. Given a maximum number of observations N, one may believe this must come at the cost of power when compared to a conventional test that waits until all N observations have arrived. Our first contribution is to show that this is false: for any valid test based on N observations, we show how to construct an anytime valid sequential test that matches it after N observations. Our second contribution is that we may continue testing by using the outcome of a [0, 1]-valued test as a conditional significance level in subsequent testing, leading to an overall procedure that is valid at the original significance level. This shows that anytime validity and optional continuation are readily available in traditional testing, without requiring explicit use of e-values. We illustrate this by deriving the anytime valid sequentialized z-test and t-test, which at time N coincide with the traditional z-test and t-test. Finally, we characterize the SPRT by invariance under test induction, and also show under an i.i.d. assumption that the SPRT is induced by the Neyman-Pearson test for a tiny significance level and huge N.