Optimal strategies for reject option classifiers

In classification with a reject option, the classifier is allowed in uncertain cases to abstain from prediction. The classical cost-based model of a reject option classifier requires the cost of rejection to be defined explicitly. An alternative bounded-improvement model, avoiding the notion of the reject cost, seeks for a classifier with a guaranteed selective risk and maximal cover. We coin a symmetric definition, the bounded-coverage model, which seeks for a classifier with minimal selective risk and guaranteed coverage. We prove that despite their different formulations the three rejection models lead to the same prediction strategy: a Bayes classifier endowed with a randomized Bayes selection function. We define a notion of a proper uncertainty score as a scalar summary of prediction uncertainty sufficient to construct the randomized Bayes selection function. We propose two algorithms to learn the proper uncertainty score from examples for an arbitrary black-box classifier. We prove that both algorithms provide Fisher consistent estimates of the proper uncertainty score and we demonstrate their efficiency on different prediction problems including classification, ordinal regression and structured output classification.