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.

翻译：预测建模中的不确定性量化通常依赖于临时方法，因为目前尚无普遍接受的形式化框架。本文提出了一种通过统计风险理解不确定性的理论方法，区分了偶然性（与数据相关）和认知性（与模型相关）的不确定性。我们阐述了如何将逐点风险分解为贝叶斯风险和超额风险。特别地，我们证明了与认知不确定性相关的超额风险与Bregman散度相一致。为了将所考虑的风险度量转化为实际的不确定性估计，我们建议采用贝叶斯方法，借助后验分布来近似这些风险。我们在图像数据集上测试了该方法，使用AUROC指标评估了其在检测分布外数据和误分类数据方面的性能。我们的结果证实了所考虑方法的有效性，并为实际应用中的不确定性估计提供了实用指导。