Position: Predictive Uncertainty Is Not Enough -- Joint Distribution for Full Uncertainty Representation

When AI is deployed in safety-critical domains, erroneous and overconfident predictions can have severe consequences. Therefore, comprehensive uncertainty quantification (UQ) should be a foundational requirement for responsible decision-making. Current UQ methods based on epistemic and aleatoric decomposition have been found insufficient for fully understanding the problem. We add that this limitation is further compounded by the systematic isolation of these terms without considering uncertainty about the domain. Our position claims that any meaningful analysis must account for three sources of uncertainty -domain, epistemic, and aleatoric-, and that only the joint distribution $p(x,y|\mathcal{D})$ provides a coherent representation of uncertainty. We begin by mirroring prior findings that show the application of information-theoretic UQ methods to ID and OOD settings is suboptimal, primarily due to the inherent difficulty of disentangling epistemic and aleatoric components. Based on this, we support that modeling the unconditional distribution $p(x|\mathcal{D})$ is required to account for input validity, resulting in a third class of uncertainty: \emph{domain} uncertainty. Finally, by considering both the domain and the conditional distribution $p(y|x,\mathcal{D})$, we argue that their product $p(x,y|\mathcal{D})$ fully encapsulates all sources of uncertainty.