Geometric Calibration and Neutral Zones for Uncertainty-Aware Multi-Class Classification

Modern artificial intelligence systems make critical decisions yet often fail silently when uncertain. We develop a geometric framework for post-hoc calibration of neural network probability outputs, treating probability vectors as points on the $(c-1)$-dimensional probability simplex equipped with the Fisher--Rao metric. Our approach yields Additive Log-Ratio (ALR) calibration maps that reduce exactly to Platt scaling for binary problems (Proposition~1) while extending naturally to multi-class settings -- providing a principled generalization that existing methods lack. Complementing calibration, we define geometric reliability scores based on Fisher--Rao distance and construct neutral zones for principled deferral of uncertain predictions. Theoretical contributions include: (i) consistency of the calibration estimator at rate $O_p(n^{-1/2})$ via M-estimation theory (Theorem~1), and (ii) tight concentration bounds for reliability scores with explicit sub-Gaussian parameters enabling sample size calculations for validation set design (Theorem~2). We conjecture Neyman--Pearson optimality of our neutral zone construction based on connections to Bhattacharyya coefficients. Empirical validation on Adeno-Associated Virus classification demonstrates that the two-stage framework (calibration followed by reliability-based deferral) captures 72.5\% of errors while deferring 34.5\% of samples. Notably, this operational gain is achievable with any well-calibrated probability output; the contribution of geometric calibration lies in its theoretical foundations rather than empirical superiority over simpler alternatives. This work bridges information geometry and statistical learning, offering formal guarantees relevant to applications requiring rigorous validation.