The Foundations of Cost-Sensitive Learning

Abstract

This paper revisits the problem of optimal learning and decision-making when different misclassification errors incur different penalties. We characterize precisely but intuitively when a cost matrix is reasonable, and we show how to avoid the mistake of defining a cost matrix that is economically incoherent. For the two-class case, we prove a theorem that shows how to change the proportion of negative examples in a training set in order to make optimal cost-sensitive classification decisions using a classifier learned by a standard non-costsensitive learning method. However, we then argue that changing the balance of negative and positive training examples has little effect on the classifiers produced by standard Bayesian and decision tree learning methods. Accordingly, the recommended way of applying one of these methods in a domain with differing misclassification costs is to learn a classifier from the training set as given, and then to compute optimal decisions explicitly using the probability estimates given by the classifier. 1 Making decisions based on a cost matrix Given a specification of costs for correct and incorrect predictions, an example should be predicted to have the class that leads to the lowest expected cost, where the expectation is computed using the conditional probability of each class given the example. Mathematically, let the entry in a cost matrix be the cost of predicting class when the true class is. If then the prediction is correct, while if the prediction is incorrect. The optimal prediction for an example is the class that minimizes