From distribution to sample: Quantifying Uncertainty for Heterogeneous Distributions

Reliable uncertainty quantification (UQ) becomes particularly challenging under heterogeneous distributions. In practice, it can be often observed that many UQ methods exhibit calibration degradation as the proportion of heterogeneous samples increases. We argue that this limitation arises mainly because covariate shift and out-of-distribution (OOD) are only defined with regard to distributions, failing to provide a direct criterion for assessing individual inputs. To address this, we propose a sample-based lightweight plug-in named Dual-signal Coverage-aware UQ (D-CUQ). D-CUQ leverages two key metrics: the Manifold-deviation Signal, capturing where an input lies relative to the ID manifold, and the Coverage-gap Signal, capturing whether the local tangent geometry around it is spanned by training data. We further establish a formal connection between the two levels, showing that distribution-level quantities arise as aggregates of their sample-level counterparts. Extensive experiments on classification and regression benchmarks demonstrate that D-CUQ significantly improves uncertainty estimation for heterogeneous test distributions.