Uncertainty quantification in predictive modeling often relies on ad hoc methods as there is no universally accepted formal framework for that. This paper introduces a theoretical approach to understanding uncertainty through statistical risks, distinguishing between aleatoric (data-related) and epistemic (model-related) uncertainties. We explain how to split pointwise risk into Bayes risk and excess risk. In particular, we show that excess risk, related to epistemic uncertainty, aligns with Bregman divergences. To turn considered risk measures into actual uncertainty estimates, we suggest using the Bayesian approach by approximating the risks with the help of posterior distributions. We tested our method on image datasets, evaluating its performance in detecting out-of-distribution and misclassified data using the AUROC metric. Our results confirm the effectiveness of the considered approach and offer practical guidance for estimating uncertainty in real-world applications.
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