Bayes-Optimal BER and AUC: Estimation and Evaluation of Estimators

A fundamental quantity in machine learning is the optimal performance achievable by any model on a given task. Estimating this quantity allows us to distinguish the irreducible part of the error from a deficiency of the model, telling us how much room for improvement remains. Recent work has shown that the Bayes error, or equivalently the optimal accuracy, can be estimated from soft labels in binary classification. However, accuracy is often a poor summary of performance in settings with severe class imbalance or noisy annotations, where metrics such as the balanced error rate (BER) and the area under the ROC curve (AUC) are more appropriate. We address this gap with two complementary contributions. (i) Estimation. We propose soft-label-based estimators for the optimal BER and AUC. We first consider the clean setting in which true soft labels and the class prior are known, and then extend the estimators to a more realistic setting in which the class prior is unknown and the observed soft labels are corrupted by an unknown order-preserving transformation, possibly followed by additive noise. In the latter setting, we approximately recover the clean soft labels via isotonic regression with auxiliary hard labels, estimate the class prior with a clipped mean of the hard labels, and derive finite-sample error bounds for the resulting plug-in estimators. (ii) Evaluation. Since the optimum is unobservable on real datasets, evaluating any such estimator is itself nontrivial. We extend the FeeBee framework, originally proposed for evaluating Bayes-error estimators, to the optimal BER and AUC. The resulting procedure provides practical evaluation scores without requiring knowledge of the optimum, and applies to any estimator of the optimal BER or AUC, not only our proposed ones. Experiments on synthetic and real-world datasets validate both the estimators and the evaluation procedure.

Collapsibility of Performance Metrics in Clinical Predictive AI

Background: Population level assessments of predictive artificial intelligence (AI) can conceal performance disparities across subgroups. Fairness evaluations commonly rely on performance analyses across subgroups. However, some performance metrics are non-collapsible, meaning that the overall population performance value does not equal the weighted average of subgroup specific values. Objective: To examine the collapsibility properties of commonly reported performance metrics in predictive AI, with a focus on the area under the receiver operating characteristic curve (AUC, also known as c-statistic). Methods: We investigate the collapsibility of 15 performance metrics, either by expressing each metric as a linear combination of its stratum specific values or, where non-collapsible, by providing a counterexample inspired by Simpson's paradox as a formal disproof. Results: Five performance metrics (AUC, calibration intercept, calibration slope, expected calibration error, and Nagelkerke R^2) are shown to be non-collapsible, and ten (O:E ratio, logloss, Brier score, accuracy, F1-score, true positive rate, true negative rate, positive predictive value, negative predictive value, and net benefit) are shown to be collapsible. The AUC is shown to be non-collapsible because it decomposes into within- and cross-group AUC terms when subpopulations coexist, such that its overall value may fall outside the range of subgroup specific AUCs. Conclusions: Non-collapsibility of performance metrics has important consequences for reporting, model appraisal, and fairness evaluation. It can generate spurious differences between subgroup and overall performance, which may mislead fairness evaluations. Explicitly acknowledging and reporting the collapsibility properties of performance metrics improves both the interpretability and transparency of fairness assessments.

Inferential Evaluation of Surrogate-Derived Models under Covariate Shift

In transfer-learning settings, a model derived from abundant surrogate labels may be deployed in a target population where gold-standard outcomes are unobserved. Evaluating its target performance is essential for determining whether decisions based on the model remain reliable, yet it is difficult when gold labels are scarce, and covariate distributions differ across data sources. We study a three-sample setting with a small gold-labeled source, a larger surrogate-labeled source, and an unlabeled target. Under conditional transportability, we evaluate the surrogate-derived model against the latent gold-standard outcome in the target population. We propose cross-fitted estimators that transport information from the two labeled sources through source-specific density ratios. We also combine outcome-regression augmentation with a kernel correction for estimating the model near a threshold, accounting for uncertainty from all three samples. We establish asymptotically linear inference for TPR and FPR, consistency and pointwise inference for the ROC curve, and asymptotically normal inference for AUC. Simulations assess bias, coverage, and sensitivity to bandwidth and relative sample sizes. A retrospective temporal validation on Chatbot Arena and a semi-synthetic ACS-Income study provide validation in real-world AI applications.

When AUC Misleads: Polarization-Aware Evaluation of Deepfake Detectors under Domain Shift

Recent advances in generative AI, such as diffusion models and face-swapping tools, have enabled the creation of highly realistic deepfakes, leading to real-world harms including financial fraud and non-consensual explicit content. In response, deepfake detection has become an active research area, with recent methods increasingly focusing on improving generalization to unseen manipulations. This is typically evaluated using the Area Under the ROC Curve (AUC) measured separately across multiple datasets. However, such an evaluation fails to reflect real-world scenarios where detectors face a mixture of data sources and varying artifact types. To address this limitation, we introduce a novel metric, Cross-dataset AUC (Cross-AUC) that averages per-domain AUCs with a measure of prediction polarization for taking into account the robustness to domain shift. The polarization extent is quantified by the Wasserstein Distance between class score distributions. Cross-AUC not only assesses the generalization capabilities of deepfake detectors under domain shifts more realistically, but it is also interpretable as it better explains the reason behind a drop in performance. Experiments performed on seven benchmark datasets demonstrate its practical relevance.