Distribution-free binary classification: prediction sets, confidence intervals and calibration
This work addresses uncertainty quantification for binary classification in distribution-free settings, providing theoretical foundations and practical methods, but it is incremental as it builds on existing notions like calibration and binning.
The paper tackles the problem of uncertainty quantification in binary classification without distributional assumptions, establishing connections between calibration, confidence intervals, and prediction sets, and showing that distribution-free calibration is only possible with scoring functions having at most countably many level sets, leading to confidence intervals for binned probabilities.
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 distribution-free 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. To close the loop, we derive distribution-free confidence intervals for binned probabilities for both fixed-width and uniform-mass binning. As a consequence of our 'tripod' theorems, these confidence intervals for binned probabilities lead to distribution-free calibration. We also derive extensions to settings with streaming data and covariate shift.