Performance metrics in machine learning: a scenario-based guide from the confusion matrix to calibration, regression and uncertainty

Discrimination, calibration, and ranking are the three pillars of classifier evaluation, yet no single algebraic language spans all three. A recent indicator-matrix account unifies discrimination measures (precision, recall, F1, Matthews correlation) across binary, multiclass, multilabel and related settings, using Boolean cell operations on indicator label matrices, generalised to soft labels by t-norms. Calibration measures (Brier score, log-loss) and ranking measures (nDCG, AUC) fall outside it. We close the gap with two further classes of cell operators: deviation operators phi(y - yhat), which give a Calibration Framework, and rank- and slice-dependent operators (positional gains for DCG, column concordance cells for AUC), which give a Ranking Framework. Both inherit the micro/macro/exemplar aggregation architecture. For calibration, the three Brier aggregations collapse to one scalar; the Murphy decomposition holds in matrix form (unbinned, binned and isotonic); and, within the Brier-type class-indexed family, a measure is skew-invariant if and only if it depends only on the class-conditional errors alpha_j and beta_j, the calibration analogues of recall and specificity. For ranking, raw DCG collapses across aggregations while nDCG does not; per-class nDCG and per-class AUC are related by a row-column duality; micro-AUC differs from pooled ROC-AUC; raw positional scores are never strictly proper, and nDCG and AUC need not be proper. The skew-invariant per-class building blocks share one class-conditional expectation template. Experiments on two multilabel benchmarks illustrate the distinctions. Many of the results are machine-checked in Lean 4; the proofs, the experiments and the aggmetrics.scores package are public: https://github.com/rcprati/discrimination-calibration-ranking

Fairness evaluation of clinical prediction models for binary outcomes often involves fairness metrics that compare the area under the receiver operating characteristic curve (AUC, or c-statistic) across subgroups. However, the AUC has been shown to be a non-collapsible metric, meaning that its subgroup specific and overall values are not directly comparable as often implicitly assumed. This raises important questions about how existing AUC-based fairness metrics should be used and interpreted. We examine eight AUC-based fairness metrics proposed in the literature, including: within-group AUC parity; cross-group AUC (xAUC) parity; gap-to-overall AUC; equity-scaled AUC; minimum-based metrics; $\mathrm{xAUC}^1$; $\mathrm{xAUC}^0$; and $\mathrm{xAUC}^1$ or $\mathrm{xAUC}^0$ parity. We characterise their estimands, relate them to the decomposition of overall AUC (into within- and cross-group components), and critically appraise them. We then illustrate their behaviour using simulations based on the EuroSCORE II, a clinical prediction model for in-hospital mortality. We recommend reporting an AUC matrix, containing the overall AUC and all within- and cross-group AUC components, alongside population characteristics. When fairness metrics are needed, we recommend reporting the within-group AUC parity, with cross-group AUC parity providing supplementary information but requiring caution due to its potential conflict with calibration. Metrics involving $\mathrm{xAUC}^1$ and $\mathrm{xAUC}^0$ have merit, summarising within- and cross-group AUCs into single values, but their interpretation is non-trivial, and they should be used with caution. Gap-to-overall AUC, equity-scaled AUC, and minimum-based metrics compare quantities that should not be compared and it is unclear how to operationalise them, making them not fit for purpose and generally not recommended.

Reliable probability estimates are critical in many machine learning applications, yet modern classifiers are often poorly calibrated. Post-hoc calibration provides a simple and widely used solution, but the large number of proposed methods, combined with small-scale and inconsistent evaluations, makes it difficult to determine which approaches are truly effective in practice. We introduce a large-scale, standardized benchmark for post-hoc calibration, covering nearly 2000 experiments across tabular and computer vision tasks, including binary, multiclass, and large-scale classification settings. Our benchmark aggregates predictions from a diverse set of classical models, modern deep learning architectures, and foundation models, and provides unified, reproducible implementations of dozens of calibration methods within a common evaluation framework. We argue that Post-Hoc Improvement (PHI) in proper scoring rules offers a principled alternative to traditional calibration error estimators for comparing post-hoc methods, capturing both calibration quality and potential degradation to the model's predictive performance. Using this framework, we conduct the most comprehensive empirical study of post-hoc calibration to date. Our results reveal consistent patterns across domains: smooth calibration functions outperform binning-based approaches, dedicated multiclass methods are essential in high-dimensional settings, and generic machine learning models are not competitive without calibration-specific design. To facilitate future research, we release all data, code, and evaluation tools, providing a plug-and-play benchmark for developing and comparing calibration methods.