Threshold Choice Methods: the Missing Link
  Many performance metrics have been introduced for the evaluation of
classification performance, with different origins and niches of application:
accuracy, macro-accuracy, area under the ROC curve, the ROC convex hull, the
absolute error, and the Brier score (with its decomposition into refinement and
calibration). One way of understanding the relation among some of these metrics
is the use of variable operating conditions (either in the form of
misclassification costs or class proportions). Thus, a metric may correspond to
some expected loss over a range of operating conditions. One dimension for the
analysis has been precisely the distribution we take for this range of
operating conditions, leading to some important connections in the area of
proper scoring rules. However, we show that there is another dimension which
has not received attention in the analysis of performance metrics. This new
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 loss of these methods
for a uniform range of operating conditions we get the 0-1 loss, the absolute
error, the Brier score (mean squared error), the AUC and the refinement loss
respectively. This provides a comprehensive view of performance metrics as well
as a systematic approach to loss minimisation, namely: take a model, apply
several threshold choice methods consistent with the information which is (and
will be) available 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.
