# A (tight) upper bound for the length of confidence intervals with conditional coverage

arXiv: https://arxiv.org/abs/2007.12448 · HTML: https://mlsanity.com/paper/2007.12448/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jul 2020.

## Built on by (most cited first, 5 of 5)

- [Confidence on the Focal: Conformal Prediction with Selection-Conditional Coverage](https://mlsanity.com/paper/2403.03868/) (2024)
- [Exact Selective Inference with Randomization](https://mlsanity.com/paper/2212.12940/) (2022)
- [Black-box Selective Inference via Bootstrapping](https://mlsanity.com/paper/2203.14504/) (2022)
- [A parametric distribution for exact post-selection inference with data carving](https://mlsanity.com/paper/2305.12581/) (2023)
- [Hierarchical Clustering With Confidence](https://mlsanity.com/paper/2512.06522/) (2025)

## Abstract (arXiv)

We show that two popular selective inference procedures, namely data carving (Fithian et al., 2017) and selection with a randomized response (Tian et al., 2018b), when combined with the polyhedral method (Lee et al., 2016), result in confidence intervals whose length is bounded. This contrasts results for confidence intervals based on the polyhedral method alone, whose expected length is typically infinite (Kivaranovic and Leeb, 2020). Moreover, we show that these two procedures always dominate corresponding sample-splitting methods in terms of interval length.