# An analysis of binary isotonic regression: degrees of freedom and implications for calibration

arXiv: https://arxiv.org/abs/2607.27301 · HTML: https://mlsanity.com/paper/2607.27301/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 21 references that are in the map.
- First posted on arXiv 29 Jul 2026.

## Builds on (most cited first, 8 of 8)

- [Distribution-free binary classification: prediction sets, confidence intervals and calibration](https://mlsanity.com/paper/2006.10564/) (2020)
- [A Unifying Theory of Distance from Calibration](https://mlsanity.com/paper/2211.16886/) (2022)
- [U-Calibration: Forecasting for an Unknown Agent](https://mlsanity.com/paper/2307.00168/) (2023)
- [Classifier Calibration with ROC-Regularized Isotonic Regression](https://mlsanity.com/paper/2311.12436/) (2023)
- [Causal isotonic calibration for heterogeneous treatment effects](https://mlsanity.com/paper/2302.14011/) (2023)
- [Can a calibration metric be both testable and actionable?](https://mlsanity.com/paper/2502.19851/) (2025)
- [In-sample calibration yields conformal calibration guarantees](https://mlsanity.com/paper/2503.03841/) (2025)
- [Honest calibration assessment for binary outcome predictions](https://mlsanity.com/paper/2203.04065/) (2022)

## Abstract (arXiv)

Isotonic regression is a canonical tool for estimating monotone functions and calibrating probabilistic predictors. We provide a fully sharp finite-sample characterization of its worst-case degrees of freedom on binary samples. Specifically, we identify the binary sequences that maximize the number of distinct fitted values produced by isotonic regression. We develop a sharp bound on the degrees of freedom with a leading term of frac3(4π2)1/3 n2/3 using analytic number theory, improving on previous bounds. We then apply this result to calibration. Calibration is a central requirement for probabilistic prediction, and isotonic regression is a widely used post-processing method for improving calibration. Building on deterministic degrees-of-freedom bounds, we derive, to our knowledge, the first nontrivial distribution-free guarantee on the Expected Calibration Error (ECE) of isotonic regression. This ECE bound is fully model-free and distribution-free, only assuming Y ∈ 0,1.