The Manokhin Probability Matrix: A Diagnostic Framework for Classifier Probability Quality

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.

Post-hoc calibration is widely adopted to correct probability estimates from trained classifiers, yet most evaluations report aggregate performance without testing whether that performance holds across distinct operating conditions within a single dataset. We present a pre-registered, condition-stratified robustness analysis comparing temperature scaling (TEMP) and isotonic regression (ISO) across four controlled conditions (C1--C4). Four hypothesis groups are evaluated: discrimination deltas with Holm-corrected multiplicity control (H1), Brier score differences (H2), calibration slope outcomes (H3), and AUROC differences under best-condition setups (H4). TEMP-minus-ISO discrimination deltas remain small across all conditions (-0.0155 to 0.0139), with Holm-adjusted p-values of 0.9895 everywhere. TEMP Brier differences are consistently negative (C1: -0.0002 through C4: -0.0074), while ISO shows sign reversals. TEMP calibration slopes stay closer to unity in every condition (range 0.7597--0.9493) than ISO slopes (0.1364--0.2726). AUROC differences shift from near zero in C1 (-0.0004) to positive in C4 (0.0264). These results establish that in-dataset robustness is condition-dependent and metric-specific. No claim of external transportability is made.

Calibration, the alignment of predicted probabilities with true outcome frequencies, is essential for reliable decision-making. While extensively studied for classification and regression, calibration has not been formally addressed for probabilistic label ranking, where the goal is to predict a distribution over orderings of a label set. Naively treating rankings as classes ignores their structure and fails to capture important modalities such as pairwise and top-k predictions. We formalize calibration for label ranking and develop a hierarchy of notions covering full rankings, sub-rankings, and top-k rankings. We prove that full-rank calibration implies the others but not conversely, and sub-ranking and top-k calibration are incomparable. Empirically, we find popular label ranking models are often poorly calibrated, with substantial differences between sub-ranking and top-k metrics. Applying our framework to RLHF reward models, we find that calibration correlates strongly but not perfectly with benchmark accuracy, suggesting it captures a meaningful quality dimension beyond top-1 accuracy. These findings motivate future work on understanding the downstream effects of miscalibration and developing methods to correct it.