The team trained and evaluated all four models on the Wisconsin Diagnostic Breast Cancer dataset, drawn from the UCI Machine Learning Repository. The dataset contains 569 instances described by 30 numerical features of breast tissue, such as radius, texture, perimeter, area, and concavity, each measured across mean, standard error, and worst-case values. Every instance is labeled benign or malignant, with 357 benign cases (62.7 percent) and 212 malignant cases (37.3 percent)—a moderate imbalance the researchers handled with stratified splitting, preserving the same class ratio in training and test sets. The data were divided 80/20 with a fixed random seed, identifiers were dropped, and all features were standardized before modeling. No missing-value imputation was needed. The researchers also note that the receiver operating characteristic curve is independent of the underlying class imbalance, because its axes consider the positive and negative classes separately, so the headline ROC-AUC comparisons were not distorted by the class distribution.

The Bayesian neural network at the heart of the study departs fundamentally from conventional deep learning. Instead of learning a single fixed value for every weight, the variational BNN models each weight as a Gaussian distribution with a learnable mean and log-variance. The architecture comprises three Bayesian linear layers of 64 hidden units with ReLU activations and a sigmoid output, taking the 30 standardized features through a 30-to-64-to-64-to-1 pathway with no bias terms, under a standard normal prior on every weight. Because computing the exact posterior of the weights is intractable, the researchers approximated it with an independent mean-field Gaussian and trained the network by minimizing the variational free energy, the negative evidence lower bound. This loss combines the KL divergence between the approximate posterior and the prior, which restrains model complexity, with the binary cross-entropy of the data, which rewards good fit. The reparameterization trick made weight sampling differentiable, so standard backpropagation could update both the mean and log-variance of every weight using the Adam optimizer over 1000 epochs.

After training, the network’s uncertainty was quantified through Monte Carlo inference: 100 stochastic forward passes per test sample, each pass drawing a fresh set of weights from the learned posteriors. Rather than a single point estimate, every patient receives a distribution of predicted probabilities. Following a theoretically grounded decomposition framework, the total predictive variance splits into an epistemic component, reflecting limited model knowledge, and an aleatoric component, reflecting irreducible randomness in the data. Because the diagnostic outcome is binary, the conditional predictive distribution is Bernoulli, so the aleatoric term reduces to the averaged variance p(1 − p), which peaks when the model hovers near a 50/50 verdict. The researchers checked how many forward passes were actually needed by monitoring stability across 10, 25, 50, 100, 200, and 500 samples. The standard deviation of the mean epistemic uncertainty fell from 0.00360 at 10 passes to 0.00148 at 100 and 0.00090 at 500, while accuracy stabilized—making 100 passes a practical compromise between computational cost and estimate reliability.

The resulting uncertainty profile is striking. Across the test set, mean epistemic uncertainty was 0.0272 with a maximum of 0.2329, while mean aleatoric uncertainty was substantially lower at 0.0042 with a maximum of 0.0436, yielding a mean total uncertainty of 0.0314. The dominance of epistemic over aleatoric uncertainty indicates that most predictive doubt in this dataset stems from limited model confidence on atypical or boundary cases rather than noise baked into the measurements themselves. Most patients showed uncertainty values close to zero, with only a small number of cases presenting elevated values, and the two uncertainty types rose together in the hardest cases. Classification accuracy remained stable across repeated Monte Carlo runs despite minor run-to-run variation in the uncertainty estimates themselves. In other words, the model knows when it knows: the vast majority of tumors receive confident verdicts, while a distinct minority of diagnostically awkward cases is flagged automatically for scrutiny.

To test whether these uncertainty scores carry clinically useful information, the researchers conducted a selective prediction analysis, dividing test samples into certain and uncertain groups using percentile thresholds on total uncertainty. Five thresholds were examined—the 75th, 80th, 85th, 90th, and 95th percentiles—and at every one, the certain samples outperformed the uncertain ones. The 85th percentile was chosen as the operating point to balance coverage and confidence. At this threshold, 97 of the 114 test samples were classified as certain, achieving 98.97 percent accuracy, while the remaining 17 uncertain samples reached only 82.35 percent. The overall test accuracy stood at 96.49 percent. The gap is the study’s central finding: the model’s own uncertainty estimates reliably separated predictions that could be trusted from those warranting further review, a capability conventional classifiers simply do not offer.

On the held-out test set, the hybrid random forest–SVM model posted the strongest raw numbers: 98.25 percent accuracy, an F1-score of 98.61 percent, and a ROC-AUC of 0.9954. The BNN followed with 96.49 percent accuracy and a ROC-AUC of 0.9937, essentially tied with the multilayer perceptron at 95.61 percent, while XGBoost recorded 94.74 percent. The hybrid works by training a random forest on the training set, extracting its out-of-bag positive-class probability as an additional model-derived feature, and feeding this 31-dimensional feature set into an RBF-kernel support vector machine. Calibration analysis added another dimension to the comparison. Measured by the Brier score, which penalizes overconfident probabilistic predictions, the RF-SVM hybrid achieved the best reliability at 0.0226, followed by the MLP at 0.0257 and the BNN at 0.0284, with XGBoost trailing at 0.0362. The BNN’s calibration curve showed broad compatibility between predicted probabilities and observed outcomes, with some deviations from the ideal line.

Interpretability was addressed through SHAP analysis, which quantifies each feature’s contribution to individual predictions using Shapley values. Because the BNN generates predictions from a posterior distribution of weights rather than fixed parameters, and because KernelSHAP approximations on patient datasets are prone to sampling noise and computational instability, the researchers applied SHAP to the XGBoost model, whose performance closely mirrors the BNN’s. The analysis revealed that tumor morphology dominates the model’s reasoning. Perimeter worst carried the highest overall impact with a mean absolute SHAP value of approximately 1.18, followed by concave points worst and texture worst at roughly 0.80 each, then area standard error and radius worst at about 0.7 and 0.6. High feature values in these measures pushed predictions toward malignancy, while low values pushed toward benign. Remaining contributors, including concavity worst, smoothness mean, and concave points mean, added more modest impacts between 0.25 and 0.45, mapping the model’s primary diagnostic drivers in a form clinicians can inspect.

Five-fold stratified cross-validation, with the BNN retrained from scratch in each fold and the 85th-percentile threshold reapplied within each fold’s training data to avoid leakage, confirmed the pattern’s robustness. The BNN achieved the highest mean cross-validation accuracy of 98.07 percent with a mean ROC-AUC of 0.9943, ahead of the RF-SVM hybrid at 97.72 percent, the MLP at 97.37 percent, and XGBoost at 96.13 percent. Crucially, in four of the five folds, accuracy for uncertain samples was lower than for certain ones, suggesting the confident-versus-uncertain gap is stable across data partitions rather than an artifact of a single split. The authors are careful about limits: the evaluation used a single dataset with relatively few samples, no comparable external dataset was identified for independent validation, and the uncertainty threshold was selected after viewing test results, so accuracy for certain cases may be optimistic for new patients. No clinical validation has been performed, and the clinical meaning of the aleatoric uncertainty remains unconfirmed. The findings should be treated as a rigorous benchmark for uncertainty-aware diagnostic AI, not as a ready clinical tool.

Subject of Research: Uncertainty-aware Bayesian deep learning for breast cancer classification

Article Title: Evaluation of comparative performances of uncertainty-aware Bayesian Neural Network with machine and deep learning models for breast cancer classification

Article References: Farid, T. B., Islam, M. A., & Banik, S. C. (2026). Evaluation of comparative performances of uncertainty-aware Bayesian Neural Network with machine and deep learning models for breast cancer classification. Results in Engineering, 32, Article 113610. https://doi.org/10.1016/j.rineng.2026.113610

Image Credits: AI Generated

DOI: 10.1016/j.rineng.2026.113610

Keywords: Bayesian neural network, breast cancer, uncertainty quantification, machine learning, deep learning, Monte Carlo inference, SHAP, XGBoost, calibration, WDBC dataset, explainable AI, selective prediction