Credal Concept Bottleneck Models: Structural Separation of Epistemic and Aleatoric Uncertainty

Decomposing predictive uncertainty into epistemic (model ignorance) and aleatoric (data ambiguity) components is central to reliable decision making, yet most methods estimate both from the same predictive distribution. Recent empirical and theoretical results show these estimates are typically strongly correlated, so changes in predictive spread simultaneously affect both components and blur their semantics. We propose a credal-set formulation in which uncertainty is represented as a set of predictive distributions, so that epistemic and aleatoric uncertainty correspond to distinct geometric properties: the size of the set versus the noise within its elements. We instantiate this idea in a Variational Credal Concept Bottleneck Model with two disjoint uncertainty heads trained by disjoint objectives and non-overlapping gradient paths, yielding separation by construction rather than post hoc decomposition. Across multi-annotator benchmarks, our approach reduces the correlation between epistemic and aleatoric uncertainty by over an order of magnitude compared to standard methods, while improving the alignment of epistemic uncertainty with prediction error and aleatoric uncertainty with ground-truth ambiguity.

💡 Research Summary

The paper tackles a fundamental problem in predictive uncertainty estimation: the entanglement of epistemic uncertainty (EU), which reflects model ignorance, and aleatoric uncertainty (AU), which reflects inherent data ambiguity. Most existing methods derive both quantities from the same predictive distribution p(y|x) – for example, using prediction entropy for AU and variance across Monte‑Carlo dropout or ensemble members for EU. Because both estimates are functions of the same probabilities, they tend to rise together when the model is uncertain, creating a strong positive correlation that blurs their semantic meaning. Empirical studies (Mucsányi et al., 2024) have shown correlations as high as 0.9, and recent theory (Tomov et al., 2025) proves that, when multiple valid labels exist, marginal predictions alone cannot uniquely determine whether spread originates from model error or genuine label ambiguity.

To break this “algebraic trap,” the authors propose structural separation: epistemic and aleatoric components must be parameterized by disjoint sets of variables and receive gradients from completely separate loss terms. They instantiate this principle in a Credal Concept Bottleneck Model (Credal CBM), extending the classic Concept Bottleneck Model (CBM) which routes predictions through human‑interpretable concepts. In a Credal CBM each concept is not a single probability vector but a credal set—an ellipsoid in logit space defined by a mean μ, an epistemic covariance Σ_epi (size of the ellipsoid), and an aleatoric variance σ²_ale (noise inside the ellipsoid).

The architecture consists of a frozen pre‑trained encoder (e.g., DistilBERT) and three orthogonal projection heads: a μ‑head that predicts the center of the credal set, an epistemic head that predicts Σ_epi, and an aleatoric head that predicts σ²_ale. Orthogonal projection matrices (W_μ, W_epi, W_ale) ensure that each head receives a distinct sub‑space of the encoder output, and a regularization term forces these matrices to remain orthogonal.

Training uses three disjoint objectives:

1. Task loss for the downstream classification/regression task.
2. Epistemic loss comprising (a) an error‑scaled term that expands Σ_epi for concepts with high prediction error, and (b) a Hausdorff‑KL regularizer that penalizes the volume of the credal set relative to a prior ellipsoid. The error term is stop‑gradient‑detached so that the epistemic head cannot “cheat” by influencing the aleatoric loss.
3. Aleatoric loss that directly supervises σ²_ale against the empirical entropy of multi‑annotator disagreement for each concept. This provides a ground‑truth signal for data ambiguity.

The authors prove (Theorem 3.1) that, under the frozen encoder and orthogonal projections, gradients with respect to epistemic parameters ϕ_epi flow only from the epistemic loss, while gradients for aleatoric parameters ϕ_ale flow only from the aleatoric loss. Consequently, the two uncertainty streams are mathematically independent during optimization. A corollary shows that, provided the supervision signals (prediction error and annotator entropy) are themselves uncorrelated in the data, the learned epistemic and aleatoric uncertainties converge to statistically independent quantities (ρ → 0).

Empirical evaluation spans four datasets: CEBaB and GoEmotions (real multi‑annotator sentiment/clinical concepts), and MA‑QA* / AmbigQA* (question‑answering datasets with gold‑standard answer distributions p*). Baselines include ensembles, MC‑Dropout, Kendall’s heteroscedastic model, and Epistemic Neural Networks (Epinet). Results demonstrate:

• Correlation reduction: Baseline methods exhibit Spearman ρ between EU and AU around 0.85–0.92, whereas Credal CBM reduces this to 0.04–0.07, an order‑of‑magnitude improvement.
• Semantic alignment: Epistemic uncertainty correlates strongly with prediction error (ρ ≈ 0.78–0.84), while aleatoric uncertainty aligns almost perfectly with true label entropy H