# Confidence Horizons

arXiv: https://arxiv.org/abs/2608.03889 · HTML: https://mlsanity.com/paper/2608.03889/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 79 references that are in the map.
- First posted on arXiv 4 Aug 2026.

## Builds on (most cited first, 14 of 14)

- [Admissible anytime-valid sequential inference must rely on nonnegative martingales](https://mlsanity.com/paper/2009.03167/) (2020)
- [Efficient Adaptive Experimental Design for Average Treatment Effect Estimation](https://mlsanity.com/paper/2002.05308/) (2020)
- [Instance-Log-Optimality of Portfolio-Based E-Processes and their Sequential Hypothesis Tests](https://mlsanity.com/paper/2504.02818/) (2025)
- [Near-Optimal Non-Parametric Sequential Tests and Confidence Sequences with Possibly Dependent Observations](https://mlsanity.com/paper/2212.14411/) (2022)
- [An Experimental Design for Anytime-Valid Causal Inference on Multi-Armed Bandits](https://mlsanity.com/paper/2311.05794/) (2023)
- [On Stopping Times of Power-one Sequential Tests: Tight Lower and Upper Bounds](https://mlsanity.com/paper/2504.19952/) (2025)
- [Logarithmic Neyman Regret for Adaptive Estimation of the Average Treatment Effect](https://mlsanity.com/paper/2411.14341/) (2024)
- [Post-Hoc Large-Sample Statistical Inference](https://mlsanity.com/paper/2603.08002/) (2026)
- [Demonstration Experiments](https://mlsanity.com/paper/2603.06941/) (2026)
- [Distribution-uniform anytime-valid sequential inference and the Robbins-Siegmund distributions](https://mlsanity.com/paper/2311.03343/) (2023)
- [Confidence sequences with informative, bounded-influence priors](https://mlsanity.com/paper/2506.22925/) (2025)
- [Learning to Bet for Horizon-Aware Anytime-Valid Testing](https://mlsanity.com/paper/2603.19551/) (2026)
- [Asymptotic e-processes](https://mlsanity.com/paper/2604.19353/) (2026)
- [Time-sensitive anytime-valid testing](https://mlsanity.com/paper/2605.06521/) (2026)

## Abstract (arXiv)

Anytime-valid inference enables analysts to continuously monitor their data and stop experiments early. However, the majority of these methods incur a certain conservativeness by remaining valid on infinite time horizons. In practice, a bound on the horizon may be imposed due to budgetary, practical, or ethical constraints. In this paper, we ask the question: "Is it possible to obtain sharper large-sample anytime-valid inference by forgoing validity beyond some finite time horizon?". We provide a positive answer to this question by proposing a family of statistical objects that we call "confidence horizons". These objects can be viewed as large-sample confidence sequences on bounded time horizons, or alternatively as group sequential repeated confidence intervals with a maximal number of interim peeking times. We make explicit connections to the group sequential boundaries of Pocock [1977], O'Brien--Fleming [1979], and Wang--Tsiatis [1987]. We derive closed-form distribution functions of certain statistics which can be used to calculate the asymptotic quantiles of confidence horizons exactly, sidestepping the repeated integration typically employed in group sequential methods. We illustrate the use of confidence horizons for treatment effect estimation in sequentially randomized experiments under adaptive Neyman allocation.