Anchoring Aleatoric Uncertainty: A Four-Term Decomposition of Predictive Risk at the Bayes-Optimal Predictor

Existing uncertainty quantification methods in Bayesian deep learning commonly decompose total predictive variance into an aleatoric component, captured by the expected conditional variance $E[Var(t \mid x, \mu)]$, and an epistemic component, captured by the variance of the conditional mean $Var[E(t \mid x, \mu)]$. However, both quantities are expressed purely in terms of model outputs, conflating the irreducible randomness inherent in the data with uncertainty arising from model misspecification. We propose a finer decomposition that traces aleatoric uncertainty to the discrepancy between the theoretical optimal predictor and the observed label, and further splits the expected risk into four interpretable components: Model Variance, Model Bias, Optimal Spread, and Bayes Risk.