A unified view of performance metrics: translating threshold choice into expected classification loss

Many performance metrics have been introduced in the literature for the evaluation of classification performance, each of them with different origins and areas of application. These metrics include accuracy, unweighted accuracy, the area under the ROC curve or the ROC convex hull, the mean absolute error and the Brier score or mean squared error (with its decomposition into refinement and calibration). One way of understanding the relations among these metrics is by means of variable operating conditions (in the form of misclassification costs and/or class distributions). Thus, a metric may correspond to some expected loss over different operating conditions. One dimension for the analysis has been the distribution for this range of operating conditions, leading to some important connections in the area of proper scoring rules. We demonstrate in this paper that there is an equally important dimension which has so far received much less attention in the analysis of performance metrics. This dimension is given by the decision rule, which is typically implemented as a threshold choice method when using scoring models. In this paper, we explore many old and new threshold choice methods: fixed, score-uniform, score-driven, rate-driven and optimal, among others. By calculating the expected loss obtained with these threshold choice methods for a uniform range of operating conditions we give clear interpretations of the 0-1 loss, the absolute error, the Brier score, the AUC and the refinement loss respectively. Our analysis provides a comprehensive view of performance metrics as well as a systematic approach to loss minimisation which can be summarised as follows: given a model, apply the threshold choice methods that correspond with the available information about the operating condition, and compare their expected losses. In order to assist in this procedure we also derive several connections between the aforementioned performance metrics, and we highlight the role of calibration in choosing the threshold choice method.