CREDO: Epistemic-Aware Conformalized Credal Envelopes for Regression
Authors: Luben M. C. Cabezas, Sabina J. Sloman, Bruno M. Resende, Fanyi Wu, Michele Caprio, Rafael Izbicki
Organizations: Department of Statistics, Federal University of S˜ao Carlos, S˜ao Carlos, Brazil · Institute of Mathematics and Computer Science , University of S˜ao Paulo, S˜ao Carlos, Brazil · Universit´e Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, France · The University of Manchester, UK, Department of Computer Science
Conformal prediction delivers prediction intervals with distribution-free coverage, but its intervals can look overconfident in regions where the model is extrapolating, because standard conformal scores do not explicitly represent epistemic uncertainty. Credal methods, by contrast, make epistemic effects visible by working with sets of plausible predictive distributions, but they are typically model-based and lack calibration guarantees. We introduce CREDO, a simple "credal-then-conformalize" recipe that combines both strengths. CREDO first builds an interpretable credal envelope that widens when local evidence is weak, then applies split conformal calibration on top of this envelope to guarantee marginal coverage without further assumptions. This separation of roles yields prediction intervals that are interpretable: their width can be decomposed into aleatoric noise, epistemic inflation, and a distribution-free calibration slack. We provide a fast implementation based on trimming extreme posterior predictive endpoints, prove validity, and show on benchmark regressions that CREDO maintains target coverage while improving sparsity adaptivity at competitive efficiency.
Figures & tables
Figure 1: Motivation and decomposition. Comparison of split-conformal prediction intervals. Top-left: CREDO yields intervals that widen adaptively in regions of high epistemic uncertainty—near the transition and toward the covariate extremes. Top-right: CQR (which does not take epistemic uncertainty into account) produces comparatively uniform intervals that can look overconfident where local support is weak (few nearby training points), despite marginal validity. Bottom-left: CREDO enables an uncertainty decomposition into aleatoric (green) and epistemic (orange) components; epistemic uncertainty peaks around x=0 and at the edges, aligning with where CREDO inflates the envelope.
Figure 2: CREDO scheme. From a model for the aleatoric uncertainty, we form a trimmed credal quantile envelope [ℓ(x),u(x)] (at nominal level 1−α0 ), then split-conformalize using the distance-to-envelope score to output C(x)=[ℓ(x)−τ^,u(x)+τ^] with marginal coverage 1−α .
Figure 3: Data-support–aware trimming via γ(x) . Top: Prediction intervals under a constant trimming level γ versus the adaptive rule γ(x) driven by a kNN-based scarcity score. Adaptive trimming decreases γ(x) in data-sparse/extrapolative regions (less endpoint trimming, wider credal envelopes) and increases γ(x) in data-dense regions (more trimming, tighter envelopes). Bottom: The resulting γ(x) profile as a function of the covariate x .
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Dataset
n
p
Scientific Domain
Data Source
Concrete
1,030
8
Civil Engineering
UCI Repository
Airfoil
1,503
5
Aeroacoustics
UCI Repository
Wine White
1,599
11
Enology
UCI Repository
Star
2,161
48
Pedagogy
Harvard Dataverse
Wine Red
4,898
11
Enology
UCI Repository
Cycle
9,568
4
Thermodynamics
UCI Repository
Appendix
Table 3: Benchmark Datasets Overview. We summarize the sample size ( n ), feature count ( p ), and the original scientific domain of each benchmark.
Figure 4: Percentage of Epistemic Uncertainty for outliers and inliers across real-world datasets using the QNN-based implementation of CREDO . The uncertainty decomposition successfully attributes a higher proportion of epistemic uncertainty to outliers compared to inliers, a distinction that is particularly pronounced in small-data domains (top row).
Figure 5: Normalized Uncertainty Decomposition for Scenarios 2 and 3. This figure illustrates the ability of CREDO to disentangle aleatoric and epistemic components under complex data-generating processes. In Scenario 2 (left), the decomposition successfully isolates the high epistemic inflation in the scarce region ( x∈[0,0.4] ) from the underlying oscillating heteroscedastic noise. Similarly, in Scenario 3 (right), the epistemic percentage peaks in the gaps between clusters and in the extrapolative tails, despite the presence of multi-modal ”shock” noise components.
Conformal prediction (CP) is a widely used frequentist framework to quantify uncertainty by constructing prediction sets with user-specified marginal coverage guarantees. In practice, CP is typically applied on top of probabilistic classifiers, which are able to express aleatoric but not epistemic uncertainty. In this paper, we consider the question of how to optimally employ CP on top of a more expressive formalism, namely credal sets, which can express both aleatoric and epistemic uncertainty. More specifically, we propose probabilistic Bernoulli prediction sets (BPS) and derive a variant that achieves conditional coverage for valid credal sets while remaining minimal in expected size. We then address the more realistic scenario in which the validity of the credal sets is not guaranteed. Assuming access to calibration data with ground-truth distributions over labels, we apply conformal risk control to BPS and derive a PAC-style guarantee: with high probability over the data, the achieved conditional coverage is at least the desired level. We validate our theoretical findings empirically over various datasets.
Alireza Javanmardi, Soroush H. Zargarbashi, Santo M. A. R. Thies +3
Conformal prediction (CP) provides finite-sample, distribution-free marginal coverage, but standard conformal regression intervals can be inefficient under heteroscedasticity and skewness. In particular, popular constructions such as conformalized quantile regression (CQR) often inherit a fixed notion of center and enforce equal-tailed errors, which can displace the interval away from high-density regions and produce unnecessarily wide sets. We propose Co-optimization for Adaptive Conformal Prediction (CoCP), a framework that learns prediction intervals by jointly optimizing a center m(x) and a radius h(x).CoCP alternates between (i) learning h(x) via quantile regression on the folded absolute residual around the current center, and (ii) refining m(x) with a differentiable soft-coverage objective whose gradients concentrate near the current boundaries, effectively correcting mis-centering without estimating the full conditional density. Finite-sample marginal validity is guaranteed by split-conformal calibration with a normalized nonconformity score. Theory characterizes the population fixed point of the soft objective and shows that, under standard regularity conditions, CoCP asymptotically approaches the length-minimizing conditional interval at the target coverage level as the estimation error and smoothing vanish. Experiments on synthetic and real benchmarks demonstrate that CoCP yields consistently shorter intervals and achieves state-of-the-art conditional-coverage diagnostics.
Xiaoyi Su, Zhixin Zhou, Rui Luo
Department of Systems Engineering, City University of Hong Kong, Hong Kong SAR, China · Alpha Benito Research, Los Angeles, USA
Conformal Prediction (CP) provides robust uncertainty guarantees for predictive models, but is typically applied post hoc, which misaligns model training with the conformal goal of producing efficient (i.e., narrow) intervals. We propose SPACR (Single-Pass Adaptive Conformal Regressor), a novel method for directly training uncertainty-aware regressors within a differentiable loss. SPACR jointly optimizes accuracy, efficiency, and validity without batch-splitting or a predefined confidence level during training. As a result, a single SPACR model yields valid prediction intervals at multiple confidence levels during inference, avoiding the costly retraining required by methods like Directly Optimized Inductive Conformal Regression (DOICR). Experiments on diverse tabular and image datasets show that SPACR consistently gives tighter intervals and better coverage-efficiency trade-offs compared to standard CP and DOICR, while significantly reducing computational costs relative to retraining-dependent baselines.