A Perfectly Truthful Calibration Measure

Calibration requires that predictions are conditionally unbiased and, therefore, reliably interpretable as probabilities. Calibration measures quantify how far a predictor is from perfect calibration. As introduced by Haghtalab et al. (2024), a calibration measure is truthful if it is minimized in expectation when a predictor outputs the ground-truth probabilities. Although predicting the true probabilities guarantees perfect calibration, in reality, when calibration is evaluated on a finite sample, predicting the truth is not guaranteed to minimize any known calibration measure. All known calibration measures incentivize predictors to lie in order to appear more calibrated on a finite sample. Such lack of truthfulness motivated Haghtalab et al. (2024) and Qiao and Zhao (2025) to construct approximately truthful calibration measures in the sequential prediction setting, but no perfectly truthful calibration measure was known to exist even in the more basic batch setting.