Discrimination, Calibration, and Ranking: A Unified Matrix Framework
Ronaldo Cristiano Prati, Farangis Johari
Abstract
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