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

A point prediction that is well calibrated on average can still be systematically biased conditional on its own value, undermining its use in downstream decision-making. We consider two objectives for reliable uncertainty quantification: self-calibration, requiring a point prediction to be unbiased conditional on its own value, and prediction-conditional validity, requiring a prediction interval to attain nominal coverage conditional on the prediction. Self-Calibrating Conformal Prediction (SC-CP) attains both objectives exactly in finite samples, but requires refitting its calibrator for every candidate outcome, which is computationally prohibitive for continuous outcomes. We propose Isotonic Conformal Prediction (ICP), a framework that decouples calibration from prediction-set construction by fitting a single isotonic recalibration map and constructing prediction intervals within strata of similar recalibrated predictions. Within this framework we develop two procedures. Split Isotonic Conformal Prediction (SICP) attains prediction-conditional validity in finite samples and self-calibration asymptotically, at the computational cost of split conformal prediction. Transductive Isotonic Conformal Prediction (TICP) attains both objectives exactly in finite samples through a per-test-point inner loop that avoids refitting the isotonic calibrator. On synthetic heteroscedastic regression problems and a real-world healthcare-utilization dataset, both procedures match the coverage of SC-CP at substantially lower computational cost.

We study the online binary sequential calibration problem. A recent breakthrough by \citet{dagan2024breaking} overcomes the classical $T^{2/3}$ barrier for calibration error. Building on this result, we present an efficient randomized forecaster that achieves an expected calibration error $O(T^{2/3-\varepsilon})$ for some constant $\varepsilon>0$. Our forecaster combines the \textsc{SPR-Calibration} procedure \citep{dagan2024breaking} with an outer Blackwell-style correction layer. The \textsc{SPR-Calibration} procedure controls calibration with respect to a surrogate sequence of conditional-mean estimates, while the correction layer controls the additional error incurred when these surrogates are used to approximate the true outcomes. The analysis decomposes the total calibration error into the surrogate calibration error and the residual discrepancy between the surrogate sequence and the true outcomes. The former is bounded by the \textsc{SPR-Calibration} guarantee in \citet{dagan2024breaking}, and the latter is controlled using a quadratic potential argument together with the sparsity of the \textsc{SPR-Calibration} forecaster.

When providing forecasted probabilities with a predictive model, the ideal model offers perfect calibration: the true probability of the outcome (i.e., the probability that $Y=1$) exactly matches the forecasted probability $f(X)$. In practice, models inevitably exhibit calibration error, and it is therefore important to be able to measure this miscalibration to assess a model's reliability. The Expected Calibration Error (ECE) is the most widely used measure of miscalibration, but is known to be impossible to estimate the ECE with guaranteed accuracy in an assumption-free setting. In this work, we propose an alternative measure, the rankECE, that is based on comparing points with neighboring values of the predicted probability $f(X)$. Our theoretical guarantees and empirical results establish that rankECE provides a better proxy for ECE as compared to binned approximations to ECE, which are the most commonly-used approximations in practice.