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Multi-class classification methods that produce sets of probabilisticclassifiers, such as ensemble learning methods, are able to model aleatoric andepistemic uncertainty. Aleatoric uncertainty is then typically quantified viathe Bayes error, and epistemic uncertainty via the size of the set. In thispaper, we extend the notion of calibration, which is commonly used to evaluatethe validity of the aleatoric uncertainty representation of a singleprobabilistic classifier, to assess the validity of an epistemic uncertaintyrepresentation obtained by sets of probabilistic classifiers. Broadly speaking,we call a set of probabilistic classifiers calibrated if one can find acalibrated convex combination of these classifiers. To evaluate this notion ofcalibration, we propose a novel nonparametric calibration test that generalizesan existing test for single probabilistic classifiers to the case of sets ofprobabilistic classifiers. Making use of this test, we empirically show thatensembles of deep neural networks are often not well calibrated.

On the Calibration of Probabilistic Classifier Sets