Explainable Bayesian Framework for Medical Diagnosis: Asymmetric Decisions, Asymptotic Calibration, and Scalable Inference

Abstract

Automated medical diagnosis requires predictive models that not only achieve high accuracy but also provide principled and interpretable uncertainty quantification aligned with clinical decision-making. In this work, we develop a comprehensive mathematical framework for binary medical classification grounded in Bayesian inference, integrating clinical decision theory under asymmetric loss, strong asymptotic guarantees, and computational analysis. We show that the Bayesian predictor minimizes expected clinical risk under realistic loss functions, is asymptotically calibrated, and satisfies the Bernstein–von Mises theorem under regularity conditions. We introduce the Clinical Uncertainty Index (CI), a theoretically grounded metric that decomposes predictive uncertainty into epistemic and aleatory components. The CI is proven to converge to zero with increasing sample size under correct model specification and to act as a diagnostic indicator of model misspecification. Extending the analysis to approximate inference, we formally demonstrate that mean-field variational approximations systematically underestimate posterior uncertainty, potentially inducing diagnostic overconfidence. These findings highlight the critical role of faithful uncertainty estimation in high-stakes clinical applications. Overall, this study provides rigorous mathematical foundations for the development of explainable and clinically reliable medical AI systems with explicit probabilistic guarantees.