  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.
