# Fuzzy Prediction Sets: Conformal Prediction with E-values

arXiv: https://arxiv.org/abs/2509.13130 · HTML: https://mlsanity.com/paper/2509.13130/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 32 references that are in the map.
- First posted on arXiv 16 Sep 2025.

## Builds on (most cited first, 19 of 19)

- [LEAF: A Benchmark for Federated Settings](https://mlsanity.com/paper/1812.01097/) (2018)
- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [Learning by Transduction](https://mlsanity.com/paper/1301.7375/) (2013)
- [Safe Testing](https://mlsanity.com/paper/1906.07801/) (2019)
- [There are natural scores: Full comment on Shafer, "Testing by betting: A strategy for statistical and scientific communication"](https://mlsanity.com/paper/2102.05569/) (2021)
- [Decision Theoretic Foundations for Conformal Prediction: Optimal Uncertainty Quantification for Risk-Averse Agents](https://mlsanity.com/paper/2502.02561/) (2025)
- [E-Values Expand the Scope of Conformal Prediction](https://mlsanity.com/paper/2503.13050/) (2025)
- [Conformal e-prediction](https://mlsanity.com/paper/2001.05989/) (2020)
- [Enhancing Conformal Prediction Using E-Test Statistics](https://mlsanity.com/paper/2403.19082/) (2024)
- [Beyond Neyman-Pearson: e-values enable hypothesis testing with a data-driven alpha](https://mlsanity.com/paper/2205.00901/) (2022)
- [The numeraire e-variable and reverse information projection](https://mlsanity.com/paper/2402.18810/) (2024)
- [Measuring Evidence against Exchangeability and Group Invariance with E-values](https://mlsanity.com/paper/2310.01153/) (2023)
- [E-Statistics, Group Invariance and Anytime Valid Testing](https://mlsanity.com/paper/2208.07610/) (2022)
- [Selection from Hierarchical Data with Conformal e-values](https://mlsanity.com/paper/2501.02514/) (2025)
- [Diversifying Conformal Selections](https://mlsanity.com/paper/2506.16229/) (2025)
- [The E-Posterior](https://mlsanity.com/paper/2301.01335/) (2023)
- [Anytime Validity is Free: Inducing Sequential Tests](https://mlsanity.com/paper/2501.03982/) (2025)
- [Boosting e-BH via conditional calibration](https://mlsanity.com/paper/2404.17562/) (2024)
- [Full-conformal novelty detection](https://mlsanity.com/paper/2501.02703/) (2025)

## Built on by (most cited first, 3 of 3)

- [Conformal Selective Prediction with General Risk Control](https://mlsanity.com/paper/2603.24704/) (2026)
- [Elements of Conformal Prediction](https://mlsanity.com/paper/2603.23923/) (2026)
- [Beyond Fixed False Discovery Rates: Post-Hoc Conformal Selection with E-Variables](https://mlsanity.com/paper/2604.11305/) (2026)

## Abstract (arXiv)

Prediction sets offer a binary inclusion/exclusion for each element at the same fixed confidence level. We generalize to fuzzy prediction sets, which exclude elements at their own data-driven confidence level. Our key insight is that a fuzzy prediction set is an e-value, capturing precisely what e-values bring to predictive inference. Fuzzy prediction sets inherit the merging properties of their e-value, offer richer guarantees to decision-makers. We also show in what sense optimal e-values give rise to optimal (fuzzy) prediction sets. We apply our results to conformal prediction, deriving optimal fuzzy conformal prediction sets, and characterizing in what sense classical conformal prediction is optimal.