Distribution

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 distributionfree 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.