A Theoretical and Empirical Taxonomy of Class Imbalance in Binary Classification

Class imbalance significantly degrades classification performance, yet its effects are predominantly understood through empirical heuristics rather than first-principles analysis. In this work, we propose a principled theoretical framework that characterizes imbalance through three fundamental scales: the imbalance coefficient $\eta$, the sample dimension ratio $\kappa$, and the effective class separability $\Delta$. Grounded in Bayes decision theory, we derive closed-form expression for the Bayes errors and show how imbalance shifts the discriminant boundary, yielding a deterioration slope that characterizes four regimes: Normal, Mild, Extreme, and Catastrophic. Using a balanced high-dimensional dataset, we vary only $\eta$ while keeping $\kappa$ and $\Delta$ fixed. Across parametric and non-parametric models, empirical degradation closely follows theoretical predictions: minority Recall collapses once $\log(\eta)$ exceeds $\Delta\sqrt{\kappa}$, Precision increases asymmetrically, and F1-score and PR-AUC decline in line with the predicted regimes. These results show that the triplet $(\eta,\kappa,\Delta)$ provides a model-agnostic, geometrically grounded explanation of imbalance-induced deterioration, and clarifies when learning under severe imbalance becomes fundamentally unreliable.}