# Distribution-free binary classification: prediction sets, confidence intervals and calibration

arXiv: https://arxiv.org/abs/2006.10564 · HTML: https://mlsanity.com/paper/2006.10564/

## Key facts

- Cited by 79 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jun 2020.

## Built on by (most cited first, 30 of 79)

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- [Online Multivalid Learning: Means, Moments, and Prediction Intervals](https://mlsanity.com/paper/2101.01739/) (2021)
- [Top-label calibration and multiclass-to-binary reductions](https://mlsanity.com/paper/2107.08353/) (2021)
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- [Tracking the risk of a deployed model and detecting harmful distribution shifts](https://mlsanity.com/paper/2110.06177/) (2021)
- [Don't Just Blame Over-parametrization for Over-confidence: Theoretical Analysis of Calibration in Binary Classification](https://mlsanity.com/paper/2102.07856/) (2021)
- [Calibrated Selective Classification](https://mlsanity.com/paper/2208.12084/) (2022)
- [Conformal Prediction Sets with Limited False Positives](https://mlsanity.com/paper/2202.07650/) (2022)
- [Restoring Calibration for Aligned Large Language Models: A Calibration-Aware Fine-Tuning Approach](https://mlsanity.com/paper/2505.01997/) (2025)
- [T-Cal: An optimal test for the calibration of predictive models](https://mlsanity.com/paper/2203.01850/) (2022)
- [Efficiently Controlling Multiple Risks with Pareto Testing](https://mlsanity.com/paper/2210.07913/) (2022)
- [An Information Theoretic Perspective on Conformal Prediction](https://mlsanity.com/paper/2405.02140/) (2024)
- [Conformal Semantic Image Segmentation: Post-hoc Quantification of Predictive Uncertainty](https://mlsanity.com/paper/2405.05145/) (2024)
- [Theoretical characterization of uncertainty in high-dimensional linear classification](https://mlsanity.com/paper/2202.03295/) (2022)
- [Improving Screening Processes via Calibrated Subset Selection](https://mlsanity.com/paper/2202.01147/) (2022)
- [Causal isotonic calibration for heterogeneous treatment effects](https://mlsanity.com/paper/2302.14011/) (2023)
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- [Modular Conformal Calibration](https://mlsanity.com/paper/2206.11468/) (2022)
- [On Temperature Scaling and Conformal Prediction of Deep Classifiers](https://mlsanity.com/paper/2402.05806/) (2024)
- [QA-Calibration of Language Model Confidence Scores](https://mlsanity.com/paper/2410.06615/) (2024)
- [Distributionally Robust Statistical Verification with Imprecise Neural Networks](https://mlsanity.com/paper/2308.14815/) (2023)
- [Regression Trees for Fast and Adaptive Prediction Intervals](https://mlsanity.com/paper/2402.07357/) (2024)
- [Can a calibration metric be both testable and actionable?](https://mlsanity.com/paper/2502.19851/) (2025)
- [MBCT: Tree-Based Feature-Aware Binning for Individual Uncertainty Calibration](https://mlsanity.com/paper/2202.04348/) (2022)
- [Understanding the Under-Coverage Bias in Uncertainty Estimation](https://mlsanity.com/paper/2106.05515/) (2021)
- [Uncertainty Quantification for Fairness in Two-Stage Recommender Systems](https://mlsanity.com/paper/2205.15436/) (2022)

## Abstract (arXiv)

We study three notions of uncertainty quantification -- calibration, confidence intervals and prediction sets -- for binary classification in the distribution-free setting, that is without making any distributional assumptions on the data. With a focus towards calibration, we establish a 'tripod' of theorems that connect these three notions for score-based classifiers. A direct implication is that distribution-free calibration is only possible, even asymptotically, using a scoring function whose level sets partition the feature space into at most countably many sets. Parametric calibration schemes such as variants of Platt scaling do not satisfy this requirement, while nonparametric schemes based on binning do. To close the loop, we derive distribution-free confidence intervals for binned probabilities for both fixed-width and uniform-mass binning. As a consequence of our 'tripod' theorems, these confidence intervals for binned probabilities lead to distribution-free calibration. We also derive extensions to settings with streaming data and covariate shift.