Distribution-free calibration guarantees for histogram binning without sample splitting

In classification, the goal is to learn a model that uses observed feature measurements to make a class prediction on the categorical outcome. However, for safety-critical areas such as medicine and finance, a single class prediction might be insufficient and reliable measures of confidence or certainty may be desired. Such uncertainty quantification is often provided by predictors that produce not just a class label, but a probability distribution over the labels. If the predicted probability distribution is consistent with observed empirical frequencies of labels, the predictor is said to be calibrated [Dawid, 1982]. In this paper we study the problem of calibration for binary classification; let X and Y " t0, 1u denote the feature and label spaces. We focus on the recalibration or post-hoc calibration setting, a standard statistical setting where the goal is to recalibrate existing ('pre-learnt') classifiers that are powerful and (statistically) efficient for classification accuracy, but do not satisfy calibration properties out-of-the-box. This setup is popular for recalibrating pre-trained deep nets. For example, Guo et al. [2017, Figure 4] demonstrated that a pre-learnt ResNet is initially miscalibrated, but can be effectively post-hoc calibrated. In the case of binary classification, the pre-learnt model can be any arbitrary function that provides a classification'score' g: X Ñ r0, 1s.