Distinguishing sources of predictive uncertainty is of crucial importance in the application of forecasting models across various domains. Despite the presence of a great variety of proposed uncertainty measures, there are no strict definitions to disentangle them. Furthermore, the relationship between different measures of uncertainty quantification remains somewhat unclear. In this work, we introduce a general framework, rooted in statistical reasoning, which not only allows the creation of new uncertainty measures but also clarifies their interrelations. Our approach leverages statistical risk to distinguish aleatoric and epistemic uncertainty components and utilizes proper scoring rules to quantify them. To make it practically tractable, we propose an idea to incorporate Bayesian reasoning into this framework and discuss the properties of the proposed approximation.

Despite the huge success of deep neural networks (NNs), finding good mechanisms for quantifying their prediction uncertainty is still an open problem. Bayesian neural networks are one of the most popular approaches to uncertainty quantification. On the other hand, it was recently shown that ensembles of NNs, which belong to the class of mixture models, can be used to quantify prediction uncertainty. In this paper, we build upon these two approaches. First, we increase the mixture model's flexibility by replacing the fixed mixing weights by an adaptive, input-dependent distribution (specifying the probability of each component) represented by NNs, and by considering uncountably many mixture components. The resulting class of models can be seen as the continuous counterpart to mixture density networks and is therefore referred to as compound density networks (CDNs). We employ both maximum likelihood and variational Bayesian inference to train CDNs, and empirically show that they yield better uncertainty estimates on out-of-distribution data and are more robust to adversarial examples than the previous approaches.