Evaluating probabilistic classifiers: The triptych

Probability forecasts for binary outcomes, often referred to as probabilistic classifiers or confidence scores, are ubiquitous in science and society, and methods for evaluating and comparing them are in great demand. We propose and study a triptych of diagnostic graphics focusing on distinct and complementary aspects of forecast performance: Reliability curves address calibration, receiver operating characteristic (ROC) curves diagnose discrimination ability, and Murphy curves visualize overall predictive performance and value. A Murphy curve shows a forecast’s mean elementary scores, including the widely used misclassification rate, and the area under a Murphy curve equals the mean Brier score. For a calibrated forecast, the reliability curve lies on the diagonal, and for competing calibrated forecasts, the ROC and Murphy curves share the same number of crossing points. We invoke the recently developed CORP (Consistent, Optimally binned, Reproducible, and Pool-Adjacent-Violators (PAV) algorithm-based) approach to craft reliability curves and decompose a mean score into miscalibration (MCB), discrimination (DSC), and uncertainty (UNC) components. Plots of the $\text{DSC}$ measure of discrimination ability versus the calibration metric $\text{MCB}$ visualize classifier performance across multiple competitors. The proposed tools are illustrated in empirical examples from astrophysics, economics, and social science.