Likelihood ratio equivalence and imbalanced binary classification

Abstract

This contribution proves that neutral re-balancing mechanisms, that do not alter the likelihood ratio, and training discriminative machines using Bregman divergences as surrogate costs are necessary and sufficient conditions to estimate the likelihood ratio of imbalanced binary classification problems in a consistent manner. These two conditions permit the estimation of the theoretical Neyman–Pearson operating characteristic corresponding to the problem under study. In practice, a classifier operates at a certain working point corresponding to, for example, a given false positive rate. This perspective allows the introduction of an additional principled procedure to improve classification performance by means of a second design step in which more weight is assigned to the appropriate training samples. The paper includes a number of examples that demonstrate the performance capabilities of the methods presented, and concludes with a discussion of relevant research directions and open problems in the area.