CLEAR: Calibrated Learning for Epistemic and Aleatoric Risk
Authors: Ilia Azizi
Juraj Bodik
Jakob Heiss
Bin Yu
Venue: ICLR 2026 Poster (2026)
Keywords: Uncertainty Quantification
Prediction intervals
Epistemic Uncertainty
Aleatoric Uncertainty
Conditional Coverage
Calibration
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: Current prediction models struggle to optimally combine uncertainty from data noise (aleatoric) and model uncertainty (epistemic). Our method, CLEAR, learns to balance both, yielding more reliable and accurate prediction intervals.

Abstract: Accurate uncertainty quantification is critical for reliable predictive modeling. Existing methods typically address either aleatoric uncertainty due to measurement noise or epistemic uncertainty resulting from limited data, but not both in a balanced manner. We propose CLEAR, a calibration method with two distinct parameters, $\gamma_1$ and $\gamma_2$, to combine the two uncertainty components and improve the conditional coverage of predictive intervals for regression tasks. CLEAR is compatible with any pair of aleatoric and epistemic estimators; we show how it can be used with (i) quantile regression for aleatoric uncertainty and (ii) ensembles drawn from the Predictability–Computability–Stability (PCS) framework for epistemic uncertainty. Across 17 diverse real-world datasets, CLEAR achieves an average improvement of 28.3\% and 17.5\% in the interval width compared to the two individually calibrated baselines while maintaining nominal coverage. Similar improvements are observed when applying CLEAR to Deep Ensembles (epistemic) and Simultaneous Quantile Regression (aleatoric). The benefits are especially evident in scenarios dominated by high aleatoric or epistemic uncertainty. Project page: https://unco3892.github.io/clear/

Published as a conference paper at ICLR 2026
CLEAR: CALIBRATED LEARNING FOR EPISTEMIC AND ALEATORIC RISK
Ilia Azizi
Juraj Bodik 1.2+ Jakob Heiss
Bin Yu
Department of Operations, HEC, University of Lausanne. Switzerland
2 Department of Statistics, University of California, Berkeley, USA
3 Department of Electrical Engineering and Computer Science, University of California, Berkeley
Equal contribution first. was..aasseunn...i jakob. jakob. www.shhsssseeeeee...coooo binyu@berkeley.com edu
ABSTRACT
Accurate uncertainty quantification is critical for reliable predictive modeling. Existing methods typically address either aleatoric uncertainty due to measurement noise or epistemic uncertainty resulting from limited data, but both in balanced manner. We propose CLEAR, a calibration method with two distinct parameters, ?i and "22 to combine the uncertainty components and improve the conditional coverage of predictive intervals for regression tasks. CLEAR is compatible with any pair of aleatoric and epistemic estimators; we show how it can be used with (i) quantile regression for alcatoric uncertainty and (ii) ensembles drawn from the Ceneraaility-Cmmpttabbllll Sttabbibll (PCS) framework for epistemic uncertainty. Across 17 diverse real-world datasets, CLEAR achieves average improvement of 28.3% and 17.5% the interval width compared to the two individually calibrated baselines while maintaining nominal coverage. Similar improvements when applying CLEAR to Deep Ensembles (epistemic) and Simultaneous Quantile Regression (aleatoric). The benefits are especially evident in scenarios dominated by high alcatoric or epistemic uncertainty. Project page: https: //uuco3882 github. iotcc/ar///e////oooooooo
observed
INTRODUCTION
Uncertainty quantification (UQ) is essential for building reliable machine learning systems (Abdar al. 2021 Gawlikowski et al. 2023). Despite their impressive capabilities, modern machine learning methods can give a false sense of reliability; therefore, producing sharp valid prediction intervals remains open problem. Calibration (Kuleshov et al. 2018) and conformal methods (Vovk et al.) 2005|/Vovk/ 2012) Angelopoulos et al.) 2024) adjust prediction intervals to obtain marginal coverage (that is, covering certain percentage of the data average). However, they may suffer from poor conditional coverage, meaning well-calibrated coverage at the individual or subgroup level (Gibbs) al. 2024). In particular, under distribution shift or model misspecification. conditional coverage can degrade substantially, especially in extrapolation regions. Most conformal methods, such as conformalized quantile regression (CQR) (Romano et al. 2019), only capture aleatoric uncertainty while ignoring epistemic uncertainty.
important distinguish between the two main sources of uncertainty, namely epistemic and alcatoric. Epistemicc uncertainty (Hüllermeier & Waegeman 2021) arises from our limited under- standing of the data generation process and the model, encompassing issues related data collection, preprocessing, transformation, and model specification. Notably, this uncertainty is typically large in extrapolation regions where training data is sparse. contrast, aleatoric uncertainty (Kirchhof et al., 2025) reflects the inherent variability within the data (stemming from measurement errors, missing covariates, randomness, or intrinsic noise) that cannot be reduced simply by gathering more observa- tions refining the model unless the data acquisition process itself is improved where more features are measured, Separating the two sources can be beneficial for various applications (Tagasosska & Lopez-Paz 2019: Laves al. 2021). For instance, in active learning, epistemic uncertainty helps in

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20
Aleatoric uncertainty
30
CLEAR
Eplatemic uncertainty
CQR
Ensembles (PCS)
10
-10
3
-5.0
-25
2.5
5.0
X
X
Figure I: Left: Blue represents alcatoric uncertainty, which reflects randomness inherent in the data such measurement noise. Red represents epistemic uncertainty, which arises from limited sample size. Right: Estimated prediction sets using the CLEAR method, which combines both sources of uncertainty in data-driven manner.
selecting which samples to label, while aleatoric uncertainty less relevant (Settles) for prediction tasks, appropriately combining both epistemic and aleatoric parts is necessary to account for the overall uncertainty in the model's predictions (Marques & Berenson 2024).
2012 However,
In this work, we contribute to the existing literature by combining aleatoric and epistemic uncertainties data-driven manner. We consider prediction intervals C(r) of the form
C(x) = f(z) + (7i (x) + x (p))temic_((:)) with 72 = XYI
(1)
where f is point estimate, and 71772 [0, oo) are coefficients selected to 1) calibrate marginal coverage and 2) optimally balance the two types of uncertainty (optimality is defined in terms of quantile loss metric introduced later). The parameter À controls the trade-off between the two uncertainty types: when X = 0, the interval reflects only aleatoric uncertainty, while as + 0, A 22 oo, it reflects only epistemic uncertainty. By allowing an adaptively chosen we ensure more flexible and data-driven trade-off the two components, leading to prediction intervals that both well-calibrated and more informative (Figure 1}. Moreover, estimating À can help practitioners better understand which source of uncertainty the dominant contributor to the overall uncertainty.
71
The combination of epistemic and aleatoric uncertainty is not new: Bayesian methods have long incorporated both components (Kendall Gal 2017 Depeweg al. 2018), and several recent approaches have also explored this decomposition in the context of conformal prediction (Rossellini cet al. 2024: Hofman et al., 2024 Cabezas et al., 2025). However, to the best of our knowledge, existing methods either do not explicitly distinguish between the two types of uncertainty implicitly fix the combination ratio, for instance, setting À = 1 "Lakshminarayanan et al., 2017 Kendall & Gall 2017; Depeweg et al. 2018)) fixing "1 (Rossellini al., (2024). This fixed choice may be suboptimal, as the relative importance of each uncertainty type varies with the data distribution and prediction task (see Appendix A for more details related work).
1.1 CONTRIBUTIONS
1. We are the first introduce two calibration parameters 71 and 72, balance the scales of aleatoric and epistemic uncertainty on the validation dataset.
2. We demonstrate that fitting quantiles on the residuals provides much more sensible estimators aleatoric uncertainty than fitting the quantiles directly on the targets.
3. We are the first to combine the ensemble perturbation intervals of the PCS framework YYu & Kumbier, 2020) with the CQR aleatoric uncertainty estimator, and empirically show the strengths of this combination.
4. We conduct large-scale UQ benchmarking for several models 17 regression datasets.
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2 METHOD
2.1 PROBLEM SCENARIO
Consider classical setting, where an i.i.d. sample (X, Y,), 1,.... . n is drawn from distribution Px x Prjx. The goal of conformal inference is to construct prediction set C(X,+1) S supp(Y) for new data-point (X,+1, satisfying marginal coverage
P P((Y+1 E C(X++1)) 21-a.
(2)
where (0, 1) is for instance 0.05. In order to construct C, data D [(X; Y), 1,... n) can be split into train and calibration subsets Drain, Deal. On the training data, a first estimate of C can be constructed, and then we can use data from Deal to calibrate such that ( satisfied.
In case of CQR, we first estimate conditional quantiles (a/2(4), 91-a/((2) using Draiin and then construct C(x) = (2x) - Y, (2)))(()) + 7]. where the calibration parameter ry is chosen so that the prediction interval (X))) contains Y, for exactly [(1 - ( + 1)] points in the calibration set ceal-
While this procedure guarantees finite-sample distribution-free marginal coverage (Angelopoulos al. 2024), conditional coverage
P(Y++1 E C(X,+1) X,+1 x²) 2
does not need to hold. As pointed out in Lei & Wasserman (2014): Barber et al. (2020), any algorithm with sinite-sample distribution-free free conditional coverage guarantees for all must be trivial C(x) = (-00, oc). However, we aim to design estimators such that conditional coverage holds approximately under reasonable real-world scenarios, even if exact finite-sample guarantees are impossible in general.
2.2 EPISTEMIC UNCERTAINTY
The traditional machine learning approach trains a predictive algorithm on a single version of the cleaned/preprocessed dataset and uses the best-performing algorithm (compared using the validation set) for future predictions. While theoretically sound in the anfinite-sappl limit, this approach ignores the uncertainty stemming from finite sample size and model choice (epistemic uncertainty). Various methods have been proposed to estimate this uncertainty, including Deep Ensembles (Laksh- minarayanan et al., 2017), MC dropout NN (Gal & Ghahramani 2016), Orthonormal Certificates (Tagasovska & Lopez- Paz 2019), NOMU (Heiss et al. (202b)). BNNs (MacKay 1992) and Laplace Approximation (Ritter et al. 2018), among others.
Estimating epistemic uncertainty via PCS: In practice, additional sources of uncertainty arise from subjective choices in data cleaning. imputation, and dataset construction, which we also consider as extended epistemic uncertainty. The Predictability, Computability, and Stability (PCS) framework (Yu & Kumbier (220)) offers holistic point of view on the data-science-life-eyddl without explicitly modeling alcatoric uncertainty. One can obtain an ensemble of m estimators fi...., fm as follows:
1. Split the data into D Dval-
2. Create N₁ differently preprocessed versions of the data and define N₂ different models (e.g.. linear regression, random forest, neural networks). Then, train models for N1 No combinations on Durin and pick the top-k based on their performance on D
... D of D to obtain an 3. Refit each of the top-k models on b bootstrap samples D1 Diain....
ensemble of m b estimators fi...., fm.
Taking the point-wise median of f....,,mm yields the final PCS estimate f. and the point-wise a/2 and a/2 quantiles (denoted fa/2 and f /1-a/2. respectively) define the uncalibrated uncertainty band. The widths of this interval, denoted as ai /T = fi-a/2(z) - f(x) and aepi (T) :=
f(x) - fz), quantify the uncalibrated epistemic uncertainty. Agarwal et al. (2025) extends this by using the combined data set Dirain Drai for training and calibrating uncertainty based
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out-of-bag data. Our main experiments in Sections of the PCS framework. However, in [Section 4.3) we show through a case study that CLEAR can also be applied successfully in the Ni > setting, incorporating uncertainty from data cleaning and pre-processing.
4.1 and 4.2 focus only on the modeling step
2.3 ALEATORIC UNCERTAINTY
Aleatoric uncertainty be estimated by modeling the conditional distribution of the outcome given the inputs. Common approaches include direct conditional quantile regression (Koenker & Bassen 1978), either parametric nonparametric, such smooth quantile regression (Fasiolo et al. 2020) and quantile random forests (QRF) (Mcinshausen 2006); heteroskedastic models that estimate input-dependent noise levels (xx) under Gaussian assumptions (Nix & Weigend 1994 and distributional regression techniques such as simultaneous quantile regression (Tagasovska & Lopez,
Paz 2019). More flexible alternatives include conditional density estimation and deep generative models such as conditional generative adversarial networks (Oberdiek et al., [2022), conditional variational autoencoders (Han et al. 2020). and diffusion models (Chang et al.) 2023).
In this work, we estimate aleatoric uncertainty using quantile regression models fa/2; Po.5, f 11-a/2 (selected in Line from PCS) trained on the residuals Y; - (X,). This approach offers improved stability: underfitting quantile regression directly the y-values can severely distort aleatoric uncertainty estimates. In contrast, extreme underfitting on residuals, at worst, corresponds to assuming homoskedastic noise, which can be an acceptable bias. This improves the stability with respect to hyperparameters. To further improve the stability, we apply a PCS-inspired bagging strategy by taking the empirical median
21-a/2 (x) == Median (f1-a/2(z) - fo.s(z)) + and (a)2(2) == Median (fo.s(2)) - fx)2(x)))
over the ensemble members. Overall, is computationally efficient and straightforward to implement.
2.4 CLEAR: COMBINING ALEATORIC EPISTEMIC UNCERTAINTY
To combine both aleatoric and epistemic uncertainties, we a weighted scheme in Equation ( Specifically, using PCS-type estimator qat and quantile regression estimator golo trained on the residuals Y, f(,). ). we define the prediction interval:
C f Tall//2
cepi f + T+n0t-a/2 724 12//-0/2
72
Given a fixed ratio X 222 we compute 7I on held-out calibration set using the standard split conformal prediction procedure. While the natural choice 71 "/2 may suboptimal. The relative contribution of alcatoric and epistemic uncertainty can vary across datasets, and the corresponding estimators may differ substantially in scale and precision when n 72- Additionally, we adopt this global linear form for simplicity (minimizing overfitting via only two parameters), interpretability (A captures the epistemic/aleatoric ratio), and stability, consistent with standard conformal scaling (Angelopoulos et al.) 2024).
appealing. it is often
To choose À from data, evaluate grid of positive values A. For each candidate À E A, construct the (calibrated) interval Cx- To ensure the best trade-off between uncertainty sources, we select A such that Cy+ performs best under the chosen metric on D We have chosen quantile loss (Koenker& & Bassett 1978) (defined in Algorithm [ which is equivalent to the pinball loss or interval score loss, see Appendixx B.3.4) as simple metric to balance both coverage and width. However, any other metric can also be used. As proper scoring rule, quantile loss incentivizes truthfulness from theoretical perspective (see Appendix B.3). This procedure summarized in Algorithm [
Parameter X* balances aleatoric and epistemic uncertainties: if one estimator fails, X compensates by re-weighting the other. When both estimators qepi and gale are reliable (up to scaling), X interpretable. A large ratio
(x) 41 (11,/(())00")
>
indicates that epistemic uncertainty dominates (reducible with more training observations or stronger assumptions), while a small ratio < 1 indicates alcatoric uncertainty dominates (not reducible by adding more training observations, though sometimes reducible by adding covariates).
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Algorithm 1 CLEAR: Calibrated Learning for Epistemic and Aleatoric Risk
1: Input: Data (X,, Y,) for 1,.... split into training Dusins calibration Deal, and validation val (we consider Denl = vai); grid of values A; significance level a.
2: Step 1: Estimate epistemic uncertainty on Durin-
cpi gopi Example: Estimate stable point predictor f and epistemic quantiles 2a/2 f1-a/2 using PCS ensembles actoss data perturbations.
3: Step 2: Estimate aleatoric uncertainty on Dirain-
Example: train quantile regression model on the residuals Y, (Xx, ) to estimate conditional quantiles /2/2 gale Ti-a/2
4: Step 3: Define prediction intervals for each A.
Define C2 by selecting the smallest value "/ such that the prediction set
pepi f 7/0 + 1 [ f m2a/2 X/19//2
Cx =
contains at least [(1 (caal + 1)] of points in Deat. See Appendix B.5 for implementation. 5: Step 4: Select X by minimizing the quantile loss on Vval- Evaluate the quantile loss of Cx Dial and set
X = arg min QuantileL.oss(Daall Cx); AEA
where QuantileLoss(P) Vaat Ox) == Dail ΣiED,,. (QLa/2(X), + QLL1-a/2(Y,, u(X.))) /2.
with I(x), u(x) denoting the bounds of Cx(r), and QL, = (y - -g)(( -1 -10-oo.)
6: Output: X and calibrated prediction interval Cx-(x)
Lemma 2.1. Let A be compact. Suppose that at least k of the base models used in the PCS ensemble are consistent for the true function f(x), and the quantile regression estimators aple are consistent for both E {x/2,1 10l--//3 a/2}. Then obtain asymptotic conditional validity: for any fixed E X,
holds that
[Daul (Pol Pad P(Yn41 c(Xn+11) Xn41 =x)
lim inf
The proof is given in Appendix B.2 building on (Angelopoulos et al.
2024 Section 5).
Note that these assumptions satisfied by the finite grid A used in our implementation and by many base models, including tree-based methods and neural networks. While the epistemic component vanishes asymptotically, plays a crucial role in finite samples by preventing under-coverage in data-sparse regions (see Section 4. and Appendix [..4.. With an infinitely large calibration set, joint calibration no worse than pangleeparamete baselines (see Lemma B.9.. Appendix B provides further theoretical discussion, including properties of the quantile loss, intuitive motivation for CLEAR, and conformal marginal-covere guarantees for modified variant of CLEAR (Lemma
EXPERIMENTAL SETUP
3.1 DATA
We conduct experiments using both simulations and real-world data. The synthetic experiments demonstrate the main intuition behind our approach. We sample X .N(0,, Ia) and compute the response Y = (xX) o(X) e, where & (0, 1). The mean function ((X) introduces non-linearity through transformations of input features (involving absolute values and fractional powers with random coefficients); its explicit form, alongside a oxX), ), is detailed in Appendixx C The sample size is fixed at n = 5000 and divided into 70-30% training and validation splits. In the univariate case, we generate 100 datasets for d 1, and in multivariate, 100 datasets randomly sampled for d E (2.3, 12.3.20]. We then assess the conditional performance as a function of distance from E[X], where test points randomly generated the surfaces of spheres with varying radii.
For the real-world scenarios, we use 17 regression datasets curated by Agarwal et al. (005). forming of the largest benchmarks for UQ (see Appendixx D.I for details). Categorical features are one-hot
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encoded, with no further preprocessing. To ensure robustness, each dataset is evaluated over 10 random srain-vlldattion standard (using aaa also as Dai) and conformalized (splitting the 20% validation into 10% val + 10% Deat for stronger theoretical guarantees). Note that while the body of the paper only focuses on the standard experiments, the conformalized results provided in Appendix G
splits (60%--0%--20%) We conduct experiments in two configurations:
3.2 BASELINES
We compare CLEAR against its core components, CQR and PCS ensemble (for details, including PCS implementation and further baselines that are relevant only to the appendices, see Appendix Dj. Notably, the underlying uncertainty estimation models from CQR and PCS are reused within the CLEAR framework. Our CQR follows the classical implementation (Romano et al. (219)) with the difference that in our experiments, we primarily utilize an enhanced variant, termed A ALEATORIC, which uses bootstrapping (b 100) on Y; and the model selection from PCS. We further improve ALEATORIC in new baseline called ALEATORIC-R, which models the residuals Y. f(X:) instead, using f from the corresponding PCS ensemble. The CLEAR method combines the uncali- brated aleatoric uncertainty from ALEATORIC-R and the uncalibrated epistemic uncertainty from PCS.
We perform ablation studies by exploring three different variants of base models for PCS (epistemic) and CQR (aleatoric). Our main approach, which we refer to as variant (a), uses quantile PCS for estimating epistemic uncertainty. We employ a diverse set of models designed to estimate conditional quantiles, namely quantile random forests (QRF) (Meinshausen, (2066), quantile XGBoost (QXGB) (Chen & Guestrin 2016), and Expectile GAM (Servén & Brummitt 2018). Then, the po--efformmng model (k: 1), in terms of the RMSE of f on D is selected and bootstrapped (b 100) to generate the epistemic uncertainty estimate qeri and the median f. CLEAR then combines this bootstrapped q. with the bootstrapped aleatoric estimate from ALEATORIC-R, ensuring ALEATORIC R also uses the same selected quantile model type. The other two variants are explained (Appendix D.4 both using the same b and & as above and only modifying the models. Variant (b) restricts quantile PCS as well as CQR baselines to only QXGB to remove any impact on CLEAR's s evaluation due to the choice of the base models. In contrast, variant (c) uses the standard PCS models to estimate the conditional mean. In all cases, PCS intervals are calibrated using the multiplicative method.
To further validate the generalizability of CLEAR, we also evaluate its application to other un- certainty estimators. This is a separate setup, where for epistemic uncertainty, we employ Deep Ensembles (DE) (Lakshminarayanan et al. 2017
ensemble of neural networks trained with diverse initializations and for aleatoric uncertainty. we use Simultaneous Quantile Regression (SQR) (Tagasovska & Lopez-Paz (009)), which directly models multiple conditional quantiles (more detail in Appendix D.3). These represent state-of-the-an deep learning approaches for uncertainty quantification, complementing our primary PCS and CQR baselines.
In all cases, CLEAR parameters (1, are optimized via quantile loss on the validation set. is chosen from dense grid A combining linearly spaced values from 0 to 0.09 and logarithmically spaced values from 0.1 to 100 (totaling over 4000 points), and ?1 is determined via conformal calibration for the chosen X. All intervals are evaluated at 95% nominal coverage. In our benchmarks, we skip the data pre- pre-processing steps of PCS (i.c.. set N₁ = 1), as Agaaawa et al. (2025), addressing only the model perturbation step of PCS. However, we show an example in Section of3 how CLEAR
can be applied in the N1 > 1 setting from Yu & Barter (204 Chapter 13).
3.3 METRICS
To evaluate the quality of our prediction intervals, employ interval coverage (PICP), normalized interval width (NIW), average interval score loss (AISL) and the quantile loss that are common in the interval prediction literature (Pearce al. 2018 V yugin & Trunov 2019 Azizi et al. 2025). We
also evaluate our work on Normalized Calibrated Interval Width (NCIW), defined as:
NCIW(f,1, NIW (i Cust.cal, , Castenal))

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with calibration constant
arg min PICP (f-d. cu 2 1 a1
Cest-cal
In the standard configuration, all methods are calibrated on the validation set; in the conformalized configuration, calibration uses a separate 10% calibration split. In both cases, the methods are already calibrated before evaluation on the test set, resulting Gest-cal R2 We primarily discuss NCIW and quantile loss, but provide the results for all the available metrics.
4 RESULTS
4.1 SIMULATIONS
Figure 2 shows conditional coverage and interval width for the univariate homoskedastic case. ALEATORIC-R achieves good coverage in high-density regions but undercovers in low-density or extrapolation regions. Conversely, calibrated PCS intervals slightly undercover data-rich regions but widen too much in extrapolation regions. Only CLEAR maintains approximate conditional coverage throughout, adapting interval width as needed. Additional results for heteroskedastic and multivariate settings in Appendix C.2.2 show similar findings.
1.0
Conditional Coverage
0.9
Average wädth
0.6
0.5
5
-6
-2
2
X
CLEAR
PCS-EPISTEMIC
ALEATORIC-R
Figure 2: Results for univariate homoskedastic case averaged over 100 simulations: On the left, conditional coverage, and on the right, mean width, for X æ. It compares CLEAR, PCS, and ALEATORIC-R (bootstrapped CQR trained residuals Y; f(X,)). The dashed horizontal line is the target coverage level of 0.9. CLEAR adapts to maintain target coverage across the input space.
4.2 REAL-WORLD DATA
Figure 3 shows the Normalized Calibrated Interval Width (NCIW) and Quantile Loss for 95% prediction intervals across all datasets for our main approach, CLEAR (variant a), compared to several baselines. CLEAR consistently demonstrates superior performance, achieving better or comparable interval width and Joss metries while consistently maintaining nominal coverage. The inset boxplots show that CLEAR (a) compared to PCS, ALEATORIC, and ALEATORIC-R has an improved quantile loss of 15.8%. 24.4%, and 9.4%, respectively. Similar relative increases are observed for NCIW, with PCS, ALEATORIC, and ALEATORIIC- exhibiting increases of 17.5%, 28.3%, and 3%. Moreover, CLEAR (a) was, in fact, the top-performing method on 15 of the 17 datasets, while remaining the most stable compared to the baselines. These trends hold across other model variants as well (Appendix for standard and Appendix for conformalized results): both CLEAR (b) and CLEAR (c) exhibit very similar relative improvements (Figures 7 and 8 in Appendices F2 and F3) with variant (a) remaining the strongest and most robust configuration.
While observed that setting
could marginally improve results if one had prior knowledge of uncertainty components an unlikely scenario in practice- fulll optimizing both parameters offers greater robustness. A limited size of the validation dataset lead to
?1
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3.50
PCS-EPISTEMIC
ALEATORIC
ALEATORIC-R
1075
Quantile Loss (Noraaiizd medalizze for CLEAR =)
158%
34.4%
9.4%
2975
3.00
23015
Relative incessse
2.50
1376
5056
2,00
.016
average increase CLEAR
1.50
1.00
0.65
CLEAR
PCS-EPISTEMIC
ALEATORIC
ALEATORIC-R
3.50
PCS- DIS-SPISTEMCC
ALEATORIC
ALEATORIC-R
3.0%
125%
NCIW Normalized for CLEAR = -1)
3.00
1076
2.50
Relotive
3056
2.00
.25%
Valurs show average CLEAR
1.50
0.65 Powerplant
Navill Propulsinn
Elerator
Allerona
Quar
KiniNin
Parkinsum
Allstto
Sulfus
Inaurance
Soparcandustor
Airfoil
Concrate
Miami Rausing
Co Housing
Energy Efficiency
Computer
Figure 3: Results for real-world data: Quantile loss and NCIW performance of different methods over 10 seeds normalized relative CLEAR (baseline 1.0) with error bars are =1 a. Lower values are better. The inset boxplot shows the average (%) relative increase of the metric over CLEAR. EPISTEMIC is PCS-UQ, ALEATORIC bootstrapped CQR, and ALEATORIC-R uses residuals,
overfitting of the two parameters, and incorporating some prior on them could further improve the results. Importantly, the absolute value of À is relative to the pre-calibrated scales of the uncertainty estimators and dataset noise; its interpretation is best contextualized by observing its behavior when systematically varying dataset characteristics, such the number of features observations. While fixed parameters (A = or ?1 = 1) could marginally improve results with prior knowledge, fully optimizing both parameters offers greater robustness. CLEAR's -arammee calibration enhances stability by adaptively re-weighting potentially unreliable uncertainty components, as evidenced in datasets like energy ef efficiency, where baselines show markedly larger NCIW. The dataset-
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dependent variability in optimal À underscores the need for adaptive selection over fixed heuristics. The calibration runtime (grid-search) is also extremely negligible in practice (Appendix F.5).
Empirical comparison with UACQR (Rossellini al., 2024). The comparison summarized in Table 1, showing the percentage improvement of (standard) CLEAR over both UACQR variants across all 17 datasets and three metrics (detailed metric-specific tables for the conformalized version provided in Tables 43 to 47p. The performance of CLEAR is much more reliable across the considered datasets and metrics. For example, on the airfoil, energy efficiency, and naval propuls s datasets, CLEAR significantly better (40-70%) in metries such NCIW, AISL,, and average width, while still exceeding 94.5% coverage. In contrast, UACQR outperforms CLEAR across these metries on only one dataset (insurance). In some instances, UACQR-P can output infinitely wide predictive intervals (Rossellini et al. 2024 5), which we observed for energy effffiiiense in our experiments. CLEAR conclusively outperforms both versions of UACQR on 14 out of 17 datasets. These large differences in performance can partially be explained by our approach for fitting aleatoric uncertainty to the residuals. We hypothesize that CLEAR is more stable and robust because it can more easily compensate for the shortcomings of the base models. If aleatoric uncertainty over- or underestimated, CLEAR can correct scale by adjusting 71-
Table 1: Improvement (%) standard CLEAR variant (c) over UACQR-S and UACQR-P at 95%
coverage across 17 datasets. Bold values with indicate CLEAR outperforms the baseline. +oc. 1/10 denotes that UACQR-P produced infinitely wide intervals for out of the 10 seeds, and the mean improvement the remaining 9 seeds
Datasel
UACQR-S
UACQR-P
NCIW
AISL
Width
NCIW
AISL
Width
ailerons
-1.5%
+3.2%
-0.4%
-9.3%
+2.7%
-18.5%
airfoil
+40.3%
+44.5%
+49.4%
+40.7%
+43.3%
+45.7%
allstate
+6.8%
+1.2%
+8.8%
+0.4%
-3.2%
+1.9%
ca housigg
+23.4%
+8.8%
+22.4%
+15.0%
+5.7%
+13.9%
computer
+20.4%
+9.5%
+21.1%
+6.7 %
+6.6%
+3.4% %
conerete
+21.3%
+21.0%
+31.4%
+22.0%
+18.1%
+26.4%
elevator
+36.5%
+29.1%
+36.1%
+24.5%
+27.6%
+19.2%
energy efficiency
+69.6%
+62.4%
+72.4%
+20
+oc/10
insurance
-15.5%
-1955%
-29.1%
-18.6%
-23.5%
-32.5%
kinSnm
+27.1%
+26.7%
+28.2%
+25.8%
+24.1%
+26.8%
miami
+19.6%
+20.0%
+23.0%
+13.4%
+15.2%
+15.5%
naval propulsion
+55.8%
+52.0%
+55.4%
+7.4%
+59.2%
+6.4%
parkinsons
+20.5%
+5.1%
+20.6%
+8.8%
-4.6%
+7.7%
powerplant
+19.9%
+13.5%
+22.7%
+15.6%
+10.1%
+18.8%
qsar
+21.5%
+10.3%
+23.1%
+15.0%
+5.5%
+16.1%
sulfur
+13.5%
+11.3%
+11.8%
+5.0%
+7.0%
+3.2%
superconductor
+17.7%
+6.4%
+16.6%
+12.6%
+3.8%
+11.1%
Empirical comparison with DE and SQR. Beyond PCS and CQR, CLEAR demonstrates substan- tial improvements when applied to DE and SQR TTable 2). When using DE for epistemic uncertainty and SQR for aleatoric uncertainty 95% nominal coverage, CLEAR achieves average width re- ductions (NCIW) of 28.6% and 13.4%, respectively, with similar improvements in quantile loss (24.0% and 13.7%) The gains persist even after conformal calibration of the baselines, particularly relevant to the aleatoric component (SQR). The results underscore CLEAR's ability to balance the two uncertainty sources well, regardless the underlying estimators. This consistency across both PCS and neural approaches validates the generality of CLEAR (full results in Appendix D).
4.3 CASE STUDY: ACCOUNTING FOR DATA UNCERTAINTY
We consider case study the Ames Housing dataset, detailed Appenddi H where demonstrate the full PCS pipeline and vary both the number of predictor variables and training sample size to explore how aleatoric and epistemic uncertainty respond to different data characteristics. We induce changes alcatoric uncertainty by restricting features (80 vs. 2) and epistemic uncertainty by subsampling training data (100%, 50%, 20%). Tables and show the results. CLEAR correctly

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Published as a conference paper at ICLR 2026
Table 2: Mean (%) improvement of CLEAR over DE & SQR across all datasets 95% coverage (higher is better). Bold values indicate CLEAR outperforms the baseline.
Metric
DE
SQR
DE-conformal
SQR-conformal
PICP
+0.05
-0.66%
-0.09%
-0.15%
NIW
+28.81 %
+17.38 %
+29.55%
+14.07 %
NCIW
+28.57 %
+13,36 %
+27.90%
+13.23 %
QuantileLoss
+23.98%
+13.66 %
+24.08%
+10.12%
identifies the dominant uncertainty source: in the low-information regime (2 features), prioritizes
aleatoric uncertainty (A 0.64), whereas in the high-feature but low-data regime (20% samples), it drastically shifts weight to epistemic uncertainty (A 100, calibrated E/A ratio x 250). This
adaptive weighting allows CLEAR outperform baselines or perform comparably them.
Table 3: Ames Housing results with varying the number of predictors (90% coverage target).
Experiment
Method
NCIW
Quantile Loss
Average Width (S)
Coverage
PCS
0,214
3,818
107,880
0.87
2 features
CQR
0.186
3,448
104,741
0.90
CLEAR
0.171
3,131
95,177
0.89
PCS
0.105
1,922
57,594
0.89
All features
CQR
0.117
2,194
62,398
0.88
CLEAR
0.103
1,923
55,910
0.88
Table 4: CLEAR calibration parameters across experimental scenarios (90% coverage target).
Experiment
711
Epistemic/Aleatorie Ratio
2 features
0.64
0.99
0.03
All features
14.45
0.13
7.72
All features, 50% data
17.97
0.09
14.09
All features, 20% data
100
0.01
250.5
5 CONCLUSION, LIMITATIONS & FUTURE WORK
This paper introduces CLEAR, a novel framework for constructing prediction intervals by adaptively balancing epistemic and aleatoric uncertainty. Through calibration process involving two distinct parameters, "/1 and "22 CLEAR offers an improvement over classical methods that often address these uncertainty types in isolation rely their fixed, non-adaptive combination. Our evaluations using CQR (and SQR) for aleatoric uncertainty and PCS (and DE) for epistemic uncertainty show that interval width and quantile loss improve both simulated and the 17 real-world datasets. CLEAR consistently achieves improved conditional coverage, notably by adapting interval widths appropriately extrapolation regions, while yielding narrower interval widths.
Limitations remain despite CLEAR's significant advantages, and these warrant further discussion. When scaling CLEAR significantly larger datasets and models, specific hyperparameters (e.g.. number of bootstraps) can be adjusted to save computational costs. The accuracy of CLEAR is intrinsically linked to the quality of the base estimators for epistemic and aleatoric uncertainty. All our calibration datasets contained at least 150 data points, but for smaller calibration datasets, overfitting 'n and À could be a problem. Epistemic uncertainty, as demonstrated the case study, must be addressed through careful judgment calls, in line with the principles of the PCS framework. The additive combination of scaled uncertaintics, while powerful, is also specific structural choice.
Future work could explore extension to classification tasks (Heiss et al. 2026), alternative À selection techniques, integration with active learning, and more scalable epistemic UQ approaches (Tagasovskal et al. 2023). Extending CLEAR to time series settings could also be valuable, particularly for capturing the dynamics of temporal uncertainty. Finally, a deeper study of the interpretability of learned parameters could provide insights into the dominance of the two uncertainty sources.
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Published as a conference paper at ICLR 2026
ACKNOWLEDGMENTS
We are grateful to Abhineet Agarwal and Michael Xiao for their valuable feedback, helpful discus- sions, and for sharing early previews and detailed explanations of their code. We also thank Natasa Tagasovska, João Vitor Romano, Erez Buchweitz, Omer Ronen, Jakob Weissteiner, Hanna Wutte, Florian Krach, Markus Chardonnet, Josef Teichmann, Bertrand Charpentier, Janis Postels, Alexander Immer, James-A. Goulet, and Hans Bühler for insightful and stimulating discussions. We are also grateful Valérie Chavez-Demoulin and Marc-Olivier Boldi for their continuous support.
Ilia Azizi acknowledges support from the HEC Research Fund for research that enabled and facilitated computations on DCSR clusters at the University of Lausanne.
Juraj Bodik acknowledges support from the Hasler Foundation [grant number 24009]. Part of this work was conducted during his research visit to the UC Berkeley, supported by the Hasler Foundation. He thanks the Department of Statistics at UC Berkeley for their kind hospitality.
Jakob Heiss gratefully acknowledges support from the Swiss National Science Foundation (SNSF) Postdoc. Mobility fellowship [grant number P500PT 2253566 and the Department of Statistics at UC Berkeley. He also wishes to thank the Department of Mathematics at ETH Zürich where the project's early ideas originated.
Bin Yu gratefully acknowledges partial support from NSF grant DMS-2413265. NSF grant DMS 2209975, NSF grant 2023505 Collaborative Research: Foundations of Data Science Institute (FODSI), the NSF and the Simons Foundation for the Collaboration on the Theoretical Foundations of Deep Learning through awards DMS-2031883 and 8 814639, NSF grant MC2378 to the Institute for Artificial Cyber Threet Intelligence and OperatioN (ACTION), and NIH (DMS/NIGMS) grant R01GM152718.
REPRODUCIBILITY STATEMENT AND USAGE OF LARGE LANGUAGE MODELS
All code and datasets used in this work are provided at htthtps:// /etp:://tttti thub. com/ Unco 38 92 c lear r to ensure full reproducibility of our results. We declare that we used large language model for grammar and language polishing. well as for limited coding assistance (c.g., boilerplate code and debugging). All conceptual and thcoretical contributions, experimental designs, and conclusions
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Table 5: List of notable abbreviations used in the main paper.
Abbreviation
Full Term
Description Citation
PCS
Predictability, Computability, and Stability
Framework veridical data science
Yu &
Kumbier 2020 Yu & Barter 2024).
CQR
Conformalized Quantile Re- gression
Distnbution- free prediction intervals (Romano et al. 2019)
ALEATORIC
Bootstrapped CQR
CQR variant, where we compute CQR on boot-
strapped data and taking the median final estimate. Follows from PCS framework (Yu & Barter 2024
ALEATORIC-R
Residual-based Bootstrapped CQR Uncertainty aware CQR
ALEATORIC applied on residuals Y; f(X...
where f obtained from PCS.
UACQR
Method from (Rossellini et al. (2024) with two vari-
ants UACQR-S and UACQR-P
QRF
Quantile Random Forests
Tree-based model for quantile estimation MMein-
shausen 2006).
QXGB
Quantile XGBoost Generalized Additive Model
Gradient boosting quantile estimation.
GAM
Used for expectile or quantile regression (Servén &
Brummiti 2018).
DE
Deep Ensembles
Esnsemble of neural networks for epistemic uncer-
tainty (1 aakshminarayanan
(017). 7).
SQR
Simultaneous Quantile Regres- sion
Neural network approach for multiple quantile esti- mation (Taaasvsska & Lopez-Paz 2019).
AISL
Average Interval Score Loss
Computed from (Rossellimi et al. 2024 and de- fined in Appendix B.3.4]
PICP
Prediction Interval Coverage Probability
Fraction targets withm predicted intervals.
NIW
Normalized Interval Width
Average interval width normalized by target range. NIW after test-time calibration.
NCTW
Normalized Calibrated Inter- val Width
RMSE
Root Mean Squared Error
Square root the average squared prediction errors. Average of absolute prediction error..
MAE
Mean Absolute Error
(O, 1)
Standard Normal Distribution
Gaussian distribution with mean 0 and variance 1. Support of distribution function.
supp
support
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LIST OF APPENDICES
A
Related literature
22
Literature that implicitly assumes %/1
22
A.2
Literature that implicitly assumes X
22
A.3
Bayesian models
24
A.4
Contormal literature that does not account for epistemic and alcatoric uncertainty
25
Other calibration methods
25
A.6
Methods that mainly focus distributional alcatoric uncertainty
25
A.7
Further related literature
26
Theory: coverage guarantees and theoretical justifications
28
B.1
Finite-sample marginal coverage for CLEAR
28
B.2
Asymptotic conditional coverage for CLEAR: Lemma 2. Proof Assumptions
30
B.3
Properties of the quantile loss
30
4
Intuitive theoretical motivation of CLEAR
35
B.5
CLEAR algorithm: details
36
B.6
Asymptotic optimality of joint calibration relative to single-paranen parameter baselines
36
Simulations: details
39
Heteroskedastic case
39
Multivariate case
39
Experimental setup: details
D.I Real-world datal
41
D.2
Baselines for variants (a), (b), (c)
D.3
Deep ensembles and simultaneous quantile regression implementation
41
D.4 PCS implementation: details
42
D.5
Metries
44
E
CLEAR with DE and SQR: results real-world data
45
CLEAR with PCS and CQR: results real-world data
48
Variant (a)
50
F2
Variant (b)
52
................................
Variant (c)
55
F4
Comparing variants of CLEAR
58
FS5
Runtime
58
Conformalized CLEAR with PCS and CQR: results on real-world data
61
G.1
Variant (a) conformalized
61
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G.2 Variant (b) conformalized
64
G.3 Variant (c) conformalized
67
H
Case study: house price prediction with varying number of predictors
71
On the role of relative and absolute uncertainty in coverage guarantees
72
Metries: relevance
72
I.2
Applications
72
Methods
73
1.4
Calibration techniques
73
1.5
Achieving conditional coverage
73
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A RELATED LITERATURE
In the main paper, Section provides a high-level overview of uncertainty quantification in ML. Section 2.2 contains an overview of epistemic uncertainty, and Section 2.3 discusses the aleatoric un- certainty. In this appendix, we discuss these concepts in more depth. For an introduction to epistemic and aleatoric uncertainty, see |Hällermeier Waegeman (2021), which provides a comprehensive overview.
A.1 LITERATURE THAT IMPLICITLY ASSUMES 91
UACQR, introduced by Rossellini et al. (024), is conceptually closest to CLEAR. Conceptually on high level, this method corresponds variant of CLEAR where 71 1 is fixed, and only A is tuned. This is justified when aleatoric uncertainty is well-calibrated, which may often hold asymptotically. When the validation dataset very small, setting "1 can even bring small advantages compared to tuning ?1 In practice, however, aleatoric uncertainty is often miscalibrated due to over- or under-regularizatioo. Tuning ?1 helps correct its scale. Furthermore, it can happen in practice that the estimator of the aleatoric uncertainty has much lower much higher quality than the epistemic uncertainty. In this case, optimizing both parameters YI and À allows us to compensate for the failure of one of the two uncertainties to extent by putting more weight the other type of uncertainty without changing the empirical marginal coverage. This results in higher robustness and stability of CLEAR.
A.2 LITERATURE THAT IMPLICITLY ASSUMES À
Most of the literature that combines epistemic and aleatoric uncertainty implicitly assumes that À = 1, when they simply combine epistemic and aleatoric uncertainty in the ratio 1:1. On top of the resulting uncertainty, one
(conformal) calibration, which corresponds to CLEAR's 's calibration of 71- However, in contrast to CLEAR, they do not rebalance the ratio of epistemic and alcatoric uncertainty with X.
In practice, it is possible to underestimate the aleatoric uncertainty and overestimate epistemic uncertainty. This results in one-dimensional calibration via 7/, and consequently, narrow intervals in regions of dominating aleatoric uncertainty or in wide intervals in regions of dominating epistemic uncertainty. 7i alone cannot solve this. CLEAR can deal well with such situations by compensating for such an imbalance via X.
A.2.1 DEEP ENSEMBLES
While it quite common to refer to deep ensembles (DE), whenever one uses an ensemble of neural networks (NNs) for uncertainty estimation, is important to note that thaKhhahhminaaaanaaa introduced DE as a method that both estimates epistemic and aleatoric uncertainty. For regression, they train each NN with 2 outputs estimating μ and a via a Gaussian Maximum- Liw.uuu---iiiiiiooo-leee (as in Nix & Weigend (1994)). where is responsible for the aleatoric uncertainty, which they refer to as "ambiguity in targets for given x". Moreover, they use the ensemble diversity to estimate epistemic uncertainty. which they refer to as "model uncertainty". Although DE is mainly known for its ensembling approach, [Lakshminarayanan et al.] 2017 Table 2 in Appendix A. clearly shows that both the aleatoric and epistemic parts crucial terms of empirical performance. In their paper, the authors do not apply any calibration on top, i.c., as it is implicitly assumed that 22 = 71 = 1 = X. However, it is common to apply (conformal) calibration on top. The commonly used calibration techniques only calibrate Y1, while keeping À = 1 fixed, in contrast to CLEAR. In particular, for DE, both the diversity of the ensemble and the bias on aleatoric are very sensitive to various hyperparameters. One type of uncertainty may be strongly underestimated. In contrast, the other type is strongly overestimated, which motivates the need explicitly calibrate in a data-driven way.
al. (2117)
Technical details on adversarial attacks for deep ensembles. The original paper Lakshmi marayanan al. (2017) is written if applying adversarial attacks is an integral part of DEs and one of the paper's 's main contributions. However, to the best of our knowledge, many practitioners refer to "DEs" without implying adversarial attacks during training, and adversarial attacks during training
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seen optional add-on to DEs. but not important part of DEs. Furthermore, the regression results in (Lakshminarayanan) et al. 2017] Table 2 in Appendix A.1) do not suggest that adversarial attacks during training are particularly beneficial to the performance.
Technical details on measuring deep ensemble's diversity. Intuitively, one should estimate high epistemic uncertainty if there is significant disagreement among the ensemble's predictions and small epistemic uncertainty if they agrec. While (Yu & Barter 2024, Chapter 13) and Agarwal et al.
(2025) suggest using quantiles, Lakshminarayanan et al. (2017 suggest using the empirical standard deviation to estimate DE's disagreement among ensemble members' predictions. It seems plausible that in Lakshminarayanan et al. (2017)'s 's case of 5 ensemble members, using the standard deviation and some Gaussian assumptions can be more appropriate while in (Yu & Barter 2024] Chapter 13)'s and Agarwal et al. 2025) 's case of 100 or more ensemble members, quantiles can be more accurate.
Variations of deep ensembles. There exist several modifications of DE that, for example, promote the ensemble's ss diversity on the function space via additional loss term during training (Wang ct al. 2019), ensemble over multiple different hyperparameters (Wenzel et al. (2020b). or reduce
the computational training cost (Kendall & Gal. 2017 Gal & Ghahramani 2016 Wen al. 2020
Havasi et al. 2021 Rossellini et al.
et al. 2025 Agarwal et al. 2025
A.2.2 MONTE CARLO DROPOUT
Gal & Ghahramani 2016) originally studied Monte Carlo Dropout (MC Dropout) without explicitly modeling alcatoric uncertainty, which was then extended by Kendall & Gal (207)) to also explicitly model alcatoric uncertainty. While Gal & Ghahramani (016) see MC Dropout as an approximation of Bayesian neural network, one also it another ensemble method. The main difference to DE is that MC Dropout only trains one NN with dropout and obtains an ensemble after training by randomly setting weights of the model zero inference time. Analogously DE, MC Dropout also adds epistemic and aleatoric uncertainty in the ratio 1:1 with the same disadvantages as described before in Appendix A.2.1
A.2.3 "A DEEPER LOOK INTO ALEATORIC AND EPISTEMIC UNCERTAINTY DISENTANGLEMENT
Valdenegro- Toro & Saromo 2022) empirically concludes that ensembles have the best uncertainty and disentangling behavior of epistemic and aleatoric uncertainty. In their paper, the authors do not use any form of calibration. This would correspond to 7/2 = 71 = = A in our notation. Their paper suggests different loss function, which they call 3-NLL., for training mitigate the underestimation of alcatoric uncertainty to some extent in their experimental setting without comparing this approach calibration. In (Valdenegro- Toro & Saromo 2022 Figure 6), one can observe that even without the 3-NLL,, both epistemic and aleatoric uncertainty already have a good shape (that is, good relative uncertainty, see AAppendi 1p for deep ensembles (DE). The main problem of DE in (Valdenegro- Toro & Saromo, 2022 Figure 6) is that the epistemic uncertainty is too small by very large factor (that is, poor absolute scale of uncertainty, see apeeAdip T). while aleatoric uncertainty has already almost the correct scaling. From our perspective, applying CLEAR on their DE would probably largely fix their problem of DE if CLEAR chooses ?1 x 1 and X > 1. However, they show that for this specific experiment, 3-NLL also fixes the problem. In general, where epistemic uncertainty might be too small or too large across the domain, we strongly recommend simply applying CLEAR on top of the already trained ensemble instead of retraining all the models with a new training pipeline. The concept of CLEAR can be implemented in a few minutes and calibrates an already trained ensemble in a few seconds. We considered an interesting open problem to study if B-NLL can improve the relative epistemic and aleatoric uncertainty Asee/Appendi I).
suspects that the alcatoric or the
A.2.4 "RECALIBRATION OF ALEATORIC AND EPISTEMIC REGRESSION UNCERTAINTY MEDICAL IMAGING"
Laves et al. 12021) also combines epistemic and aleatoric uncertainty in the ratio 1:1 and applies single constant (corresponding to 7 in our notation) to scale the total predictive uncertainty, which corresponds to fixing 1, resulting in the same potential for problems as mentioned before.
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A.3 BAYESIAN MODELS
If one had access to perfect prior, perfectly computed Bayesian inference, it would provide well- calibrated epistemic and alcatoric uncertainty, at least theory. However, in practice, the ratio of estimated epistemic and aleatoric uncertainty can be very wrong, and the uncertainty can be miscalibrated.
A.3.1 GAUSSIAN PROCESS REGRESSION (GPR)
Gaussian Process regression (Edward 2003 provides a closed form for exact posterior epistemic uncertainty for given Gaussian process as prior and for given known Gaussian noise distribution. If the (scale of the) prior the scale (of the noise) is misspecified, the ratio of estimated epistemic and aleatoric uncertainty can be arbitrarily bad, and the uncertainty can be miscalibrated. Therefore, it is common to optimize hyperparameters like the noise scale and the prior scale (typically in non-Bayesian way). The main differences to CLEAR are that 1) for GPR, one has to refit the model for every considered possibility of hyperparameters, while CLEAR optimizes I and À after fitting the model, resulting in much lower computational costs; 2) CLEAR can be applied to other base models as well such as tree-based models which are more popular many applications: 3) standard- implementations of GPR fit the hyperparameter on the training data rather than on the validation data.
A.3.2 BAYESIAN NEURAL NETWORKS (BNNS)
Bayesian neural networks MacKay 1992); Neal 1996) offer principled Bayesian framework for quantifying both epistemic and aleatoric uncertainty through the placement of prior distribution on network weights. As for GPR, the ratio of estimated epistemic and aleatoric uncertainty in BNNs is highly sensitive to the choice of prior. Consequently, already trained BNN, calibrating both uncertainty types via scaling factors and 7/2 with negligible additional computational overhead. While exact Bayesian inference large BNNs is computationally intractable, numerous approximation techniques have been proposed (Graves 2011 Blundell et al.
advocate applying CLEAR to
2015 Heenádeezz-oobattt & Adams 2015 Gal Ghahramani 2016: Lakshminarayaaan et al. | 7
Ritter al. 201
|0. 8: Daxberger et al. 2021 Heiss al., 2022b Wenzel et al. [2020a] Nguyen & Goulet
2022b a Cong et al., 2024) Shen et al. 2024). Interestingly, theoretical (Heiss et al. 2022a] Heiss
2024) and empirical (Wenzel al. 2020a) studies suggest that some of these approximations
actually provide superior estimates compared to their exact counterparts, due to poor choices of priors, such i..dd Gaussian priors, in certain settings.
A.3.3 EPISCORE
The recent work by Cabezas et al. (2255) introduces EPISCORE. Similar to our work, this method addresses the limitations of standard conformal prediction in capturing epistemic uncertainty. EPIS- CORE focuses on enhancing existing conformal scores with Bayesian epistemic uncertainty. They also compute the average interval score for 95% predictive intervals on the datasets airfoil (where their best method out variants performs more than times worse than the worst of the 3 variants of CLEAR), concrete (where their best method performs more than 1.5 times worse than the worst variant of CLEAR) and Superconduce ivity (where their best method performs approximately 1.3 times worse than the worst variant of CLEAR).
A.3.4 TABPFN
TabPFN (Müller al. 2021 Hollmann et al. 2025) a transformer trained to emulate Bayesian inference over diverse prior of realistic tabular problems, achieving strong predictive uncertainty estimates with a single forward pass. Its prior spans diverse noise structures and function classes, yielding uncertainty estimates that inherently mix aleatoric and epistemic components. Although recent variants such as TabPFN-TS (Hoo et al. 2025) extend its applicability to time series, the method is limited tabular datasets of limited size and does not disentangle uncertainty types. In contrast, our approach explicitly separates and calibrates epistemic and aleatoric uncertainty and scales to arbitrary modalities and data sizes.
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A.4 CONFORMAL LITERATURE THAT DOES NOT ACCOUNT FOR EPISTEMIC AND ALEATORIC UNCERTAINTY
The classical conformal prediction literature (Vovk al., 2005 2009 Vovk 2012 2013 Lei &
Wasserman 2014 Romano et al., 2019 Kivaranovic et al. 2020; Angelopoulos et al. 2024) primarily focuses on achieving marginal coverage, often with attention to asymptotic conditional coverage. However, it often overlooks epistemic uncertainty, which is especially critical in finite-sample settings. While|Bareer et al. (2020) established the impossibility of achieving exact distribution-Fii conditional coverage finite samples, several recent works attempt to improve coverage guarantees in more restricted settings under certain assumptions. For instance, Gibbs et al. (2024) propose coverage guarantees
subclass of distribution shifts, effectively interpolating between marginal and conditional coverage. Others, such as Guan (2022) and Dwivedi et al. (2020), provide guarantees over finite set of prespecified subgroups. We argue that to obtain reliable conditional coverage, one must model the uncertainty arising from each stage of the data science life cycle (Yu & Barter (2024). and appropriately integrate these uncertainties to achieve meaningful coverage guarantees.
A.5 OTHER CALIBRATION METHODS
Predictive uncertainty can be expressed either as a predictive set for a given level a, as a predictive distribution, or numerical value quantifying the level of uncertainty (e.g., the entropy of the pre- dictive distribution). Our implementation of CLEAR yields predictive intervals, making it compatible with both base models that output intervals (such as QR) and with base models that output predictive distributions from which one can easily obtain predictive intervals. Conceptually, CLEAR could also be extended to predictive distributions or numerical values.
In the applied literature on predictive intervals, it is quite common to calibrate a constant factor to rescale the width of predictive intervals (Laves et al. 2021 Heiss et al.] [0226) Yu & Barter
2024] Agarwal et al., 2025). Independently, the conformal community came up with almost identical methods based on theoretical considerations, as already discussed in Appendix A.4
In the literature on predictive distributions, early works suggested more sophisticated non- non-linear transformations for the predictive distributions (Kuleshov al., [018 Song et al. 2019: Kuleshov & Deshpande 2022). However, Levi et al. (2022) demonstrated that simply calibrating a constant scaling factor performs on par with these more sophisticated calibration methods.
To summarize, there are multiple (slightly) different methods to calibrate uncertainty with similar em- pirical performance. These methods only monotonically transform the uncertainty without changing the ranking of the uncertainties (i.e., if one was more uncertain on than on ã before the calibration step, one will also be more uncertain on x than on 7 afterwards; see/Appendis Ip. contrast, CLEAR and UACQR can also change the ranking of the uncertainties (e.g., in the situation of Figure small value of À would assign more uncertainty to the center around x = 0 compared to the region around x -6, whereas large value of X would assign more uncertainty to -6 than to 0).
A.6 METHODS THAT MAINLY FOCUS ON DISTRIBUTIONAL ALEATORIC UNCERTAINTY
In this section, we provide more details on the literature overview on aleatoric uncertainty from Section 2.3
One can directly estimate conditional quantiles for a given level a via quantile regression (QR)
(Koenker & Basseti 1978), either parametric or nonparametric, such as smooth quantile regression (Fasiolo al. (2200) and quantile random forests (QRF) (Meinshausen) 2006). This is computationally cheap if you a priori know which level a is of interest for your predictive intervals.
Alternatively, one can also try to estimate the conditional distribution. This can initially be computationally more expensive, but once the conditional distributions are estimated, one can easily obtain conditional quantiles for multiple different levels a. One the computationally cheapest ways to estimate conditional distributions is to estimate parameters of specific distribution (e.g., μ((x) and a (z) of a Gaussian distribution) NNx & Weigend 1994). There are semi eemi-parametric extensions of this with universal approximation properties (Kratsios] 2023). There
many similar approaches to estimate conditional densities (Rothfuss et al. 2019). Another way to obtain a conditional distribution is to estimate the conditional quantiles for all levels [0, 1] simultaneously, in simultaneous
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quantile regression (SQR) (Tagasovska & Lopez- Paz
2019) and its extension MAQR (Chung
let al. 2021). Especially for upplications where the model output is digh-dimmnsional (e.g., models that output images), more modern deep generative models have become popular, e.g.. conditional generative adversarial networks (Oherdiek et al. 2022), conditional variational autoencoders (Han
al. ) and diffusion models (Chang al.
2023 Chan et al. 2025).
Limitations and empirical comparison. The majority of these works fail to properly account for epistemic uncertainty and don't include post-hoc calibration step. Each of these methods could be incorporated into CLEAR as an alternative to our recommended ALEATORIC-R module. For example, in some preliminary experiments combining SQR (Tagasovska & Lopez-Paz, 2019) with ensemble-diversin (Lakshminarayanau et al. 2017) via CLEAR's calibration step results in significant improvements over the (calibrated versions) the individual models. This shows the applicability of CLEAR's calibration step pure deep-learning pipelines. Furthermore, when compare the results of our entire CLEAR pipeline reported our paper with the results of the entire SQR- SQR-pipeline and MAQR Anii-liie reported in (Tagasovska & Lopez- Paz 2219 Chung et al. 2021), we can see that CLEAR massively outperforms their pipelines. This shows that our PCS- based approach and novel variant of QR already highly recommendable choices, resulting in state-of-the-art performance of the CLEAR-pipellig directly out of the box.
A.7 FURTHER RELATED LITERATURE
A.7.1 "BEYOND PINBALL LOSS: QUANTILE METHODS FOR CALIBRATED UNCERTAINTY
QUANTIFICATION
Chung al. 2021 explores the limitations of the pinball loss (which we call Quantile Loss), criticizing that its direct minimization does not guarantee correct marginal coverage. In CLEAR, we address this by framing the optimization of 7/1 and
constrained optimization problem (see Equation (4) in Appendix B.3): we minimize the pinball loss on a calibration dataset Deal (see Line in Algorithm [ while constraining the marginal coverage on Deal (see Line 4 in Algorithm T This constrained approach directly mitigates the criticism of Chung et al. (2021).
A.7.2 UNCERTAINTY QUANTIFICATION FOR CONDITIONAL IMAGE GENERATION
Chan et al. (2025) suggests combining a diffusion model for aleatoric uncertainty with hyper- network-generated ensemble for epistemic uncertainty to quantify the uncertainty in conditional image generation. Diffusion models are particularly well-suited for conditional image generation, and using a shyeernetword can computationally more efficient way to obtain an ensemble. However, they combine these two sources of uncertainty simply in the ratio 1:1 without any form of calibration (corresponding to hard-coding = 71 = 722 in our notation). would be interesting future work to apply calibration in the spirit of CLEAR on top of Chan et al. (2255) to extend the idea of CLEAR to conditional image generation.
A.7.3 UNCERTAINTY QUANTIFICATION FOR PRETRAINED MODELS
Wang & Ji 2024) estimates the epistemic uncertainty for pre-trained classification models, while CLEAR focuses on both epistemic and aleatoric components for regression. It would be interesting future work to extend CLEAR to also applicable to already pre-trained models using techniques
presented in Wang & Ji (024).
A.7.4 UNCERTAINTY QUANTIFICATION AS A BINARY CLASSIFICATION PROBLEM
Altieri et al. (2044) do not provide predictive intervals or distributions. Instead, they partition test data into "good" (more certain) and "bad" (less certain) points. As future work, one could derive binary classifier from CLEAR by thresholding the predictive interval width and then evaluating it under their proposed metrics.
Chan et al. 2025) include epistemic uncertainty in the ratio 1:1: Chung et al. 2021) conduct limited experiments on including epistemic uncertainty in their appendix: Tagasovska & Lopez- Paz (219) proposes Orthonormal Certificates for estimating epistemic uncertainty, hut explicitly leave the combination for future
work.
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A.7.5 CREDAL SETS
Hofman al. (2024) describe epistemic uncertainty via credal sets on the space of distributions. This approach is quite natural for classification, where the space of distributions over K classes is (K - 1)-dimmensi.n However, they do not provide any concrete algorithms for regression, where the space of all distributions over continuous set is suggest multiple ideas on how one can translate credal sets into two real-valued numbers describing the magnitude of the epistemic and the aleatoric uncertainty. JJvanmardi et al. (2025) proposes method that can provide conformal predictive sets for classification with guaranteed input-conditional coverage under the assumption that one already has access credal sets that guaranteeably cover the true input/conditional distribution. They also briefly discuss that for real-world applications, this assumption is not satisfied.
[Hofman et al. (2024)
A.7.6 SA-PAE CALIBRATION YIELDS CONFORMAL CALIBRATION GUARANTEES
For the distributional regression, Allen al. 2025 suggest using conformal binning or confor- mal isotonic distributional regression and prove theoretical guarantees under the exchangeability assumption.
A.7.7 ENGRESSION AND EXTRAPOLATION
Shen & Meinshausen 2024) uses generative methods for uncertainty estimation that, in particular, take advantage of pre-additive noise (i.e., noise that directly added to .r). [Bodik] (2024) uses extreme value theory for dimension reduction in the extrapolating region in causal setting.
A.7.8 UNCERTAINTY QUANTIFICATION FOR TIME SERIES
Adapting CLEAR to temporal settings offers promising avenue for capturing complex uncertainty
dynamics. Existing approaches tackle temporal uncertainty from various angles: Xu & Xie (2021)
apply conformal methods, while Bodik & Pasche (2024) study uncertainty in time series stemming
from extreme events from causal perspective.
irregularly observed data, Herrera et (002) )
Krach et al. (2022): Andersson et al. (2023); Krach & Teichmann (2024): Heiss et al. (2024). Krach
(2025): Crowell et al. (2025) utilize neural jump ordinary differential equations (NJODEs) to forecast trajectories and estimate the conditional variance, providing natural estimators of aleatoric uncertainty. However, existing NJODE literature does not combine this aleatoric component with epistemic uncertainty estimation. An extension of CLEAR could address this gap by using NJODE-batt based variance estimates as the aleatoric component and combining them with PCS-inspired ensemble- epistemic uncertainty estimates via CLEAR's s dual-parameter calibration. For example, AAdamvv et al. (025) uses NJODE's uncertainty estimation to detect outliers-a setting where disentangling epistemic and aleatoric sources could further improve detection. Similarly, recent works leverage foundation models for time series forecasting (Hoo et al. 2025 Moroshan et al., 2026). None of these works explicitly decompose the uncertainty into epistemic and aleatoric components, suggesting that CLEAR's 's calibration framework could complement them as well.
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B THEORY: COVERAGE GUARANTEES AND THEORETICAL JUSTIFICATIONS
B.1 FINITE-SAMPLE MARGINAL COVERAGE FOR CLEAR
While conformal methods typically offer finite- sampl marginal coverage guarantees, our implemen- tation of CLEAR does not strictly adhere to these guarantees for two reasons:
We reuse the validation dataset Vaal as the calibration dataset, i.c., Deal = D
2. We optimize two parameters "1 and À on the calibration dataset Deal-
However, empirical results in Tables 19 and 24 show that for reasonably sized datasets, this theoretical discrepancy does not visibly impact our practical marginal coverage. From a theoretical perspective, CLEAR still achieves asymptotic marginal coverage (as consequence of asymptotic conditional coverage, see Lemma 2.1
By slightly modifying the CLEAR procedure, can obtain theoretical finite-sample guarantee for marginal coverage under the standard exchangeability assumption, as in conformal inference, without sacrificing asymptotic conditional coverage. The key idea is optimize the parameter À using only validation dataset val. and then calibrate 71 using a separate, previously unseen calibration dataset Deal-
Definition B.1 (Conformalized CLEAR). Let the available data be split into training set Dirain and an old validation set val- We first split Deld into two disjoint sets: a new validation set Daal and a calibration set Deat. The procedure is as follows:
Train and Optimize À by running Algorithm on Dirain and Dval: The model f and the uncertainty estimators trained Duain (Step and 3 of Algorithm IT. The optimal value X is found by optimizing the QuantileLoss on D without using any data from Deal (Step 4 and 5 of Algorithm 1-
2. Compute Conformity Scores: For each data point (X,, Deat, compute the conformity score S.a::
f(X.) -Y
Y, f(X,)
S.¹ max f(X.) 1x-((x)) ux- (X,) - f(x,)
where Ix- : f -Galy X
epi Ja/2 and ūX- == f oft + x epp 01-0/2"
3. Calibrate the Prediction Interval: Set the calibration parameter fी to be the [(1
(Daal + 1)] smallest value among the conformity scores S.¹ from Deal- If [[1
a) + +1] > [Peal]. set yi xo.
4. Form the Final Prediction Interval: For new test point Fnew. the final - (1-a) prediction interval is given by:
C'Inew) [[(now - 7 ( + X 0 f(xnow) + trt ( + X epi
01-a/2)
This modified version of CLEAR, which we call Conformalized CLEAR, satisfies the standard finite-sampi sample marginal coverage guarantee of conformal prediction.
Lemma B.2. Under the assumption that the data points in the calibration set Deat and the test point (Xnew, Ynow) are exchangeable, the prediction interval C(Xnew) generated by the Conformalized CLEAR procedure satisfies:
P(Ynew E C(X 2
Proof. The proof is direct application of the standard theoretical guarantees for split conformal pre- diction. Since X" is determined using only data from Deat. is fixed with respect to the calibration set
Deat- The conformity scores S.¹ are therefore exchangeable for all (X,, Y2) E Deal U {[Xnew Ynew)) )- The choice of Y as the empirical (1 - a1 + 1/(Deal) -quantile of the calibration scores ensures that the resulting prediction interval achieves at least 1 a marginal coverage, as established by (Angelopoulos al. 2024 Theorem 1.4).
n
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However, in practice, we recommend using our default implementation of CLEAR, where the valida-
tion data is reused for calibration (Deat = val). This improves data efficiency while still achieving strong approximate marginal coverage in our experiments (see Tables 14 19
and 24). Avoiding split
allows more data for both validation and calibration, which improves model selection and stabilizes marginal and conditional coverage. Even when marginal coverage TewYnew E (C(Xeew))
is satisfied, the actual coverage P(Ynew E |Pmin)) Deal, Deal) can be significantly lower than the target coverage
for fixed calibration dataset Deal. due to the inherent variation of Peal. Only with sufficiently large calibration datasets Deat can we expect IPY E C(X Dirrinn Daat, Deat) to be reliably close P(Yneew new E (XXeew)
B.1.1 LIMITATIONS OF CONFORMAT. MARGINAL COVERAGE GUARANTEES
The conformal theory heavily relies on the assumptions of exchangeability. Exchangeability means that the joint distribution of calibration and test observations is invariant to permutations (e.g.- iid observations satisfy this assumption). While exchangeability is theoretically convenient, it is unrealistic in many real-world settings. Models are typically trained on past data and deployed in the
future, where the distribution of Xeew usually shifts, i.c., ["[ o [P[X]. Even if (en [ee
P[Y[X] remains fixed, such marginal shifts in Xnew can cause conformal methods to catastrophically
fail to provide valid marginal coverage. In Section 4. and Appendix [ (Figures 2 and 4 to 6
CLEAR empirically remains robust, while CQR and Naive-Conformal fail under distribution shifts in X new E.g., Figure suggests P(ynew E CNaive nenww) ) ne 2 2] S 70% < 90% = 11 - a, thus marginal distribution shift of Xneww that strongly increases the probability of [Xnew] > 2. would lead to large drop of marginal coverage for (Xnow: Yusw). ). CLEAR likewise lacks formal guarantees under extreme shifts, but consistently performs more reliably across our experiments. In the case study (Section 4.3). a realistic temporal split (see Yu & Barter 2024, Section 8.4.3)) also violates exchangeability, and CLEAR outperforms conformal baselines in marginal coverage. Caution is required when trusting conformal guarantees, as the assumption of exchangeability is often not met in practice, and some conformal methods catastrophically fail for slight deviations from the exchangeability assumption.
Even under the assumption of exchangcability, conformal guarantees have further weaknesses:
1. The conformal marginal coverage guarantee
PeWnew E C(X = E Dea Neew E (e Deall 2 21-4 aa
does not imply that P[Ynew E [[ Deal] 2 1 for a fixed realization of the calibration set Daa,, as already discussed before. If the calibration residuals are small by chance, conformal intervals may be too narrow, especially with tiny calibration datasets. Guaranteed reliable calibration is generally unattainable with tiny calibration datasets: Even if the exchangeability assumption is satisfied, even methods with conformal guarantees often strongly undercover, i.e., PD [P[Ynew new E &Coonfommll (Xnew) DDiain: Deai] << 1 - o. > 0. Therefore, caution is required when communicating conformal guarantees to practitioners. Kivaranovic et al. (2020) also discuss this problem and suggest method to mitigate this via
Probably Approximately Valid prediction intervals (Kivaranovic et al., 2020 Def. 2).
2. Beyond marginal coverage, CLEAR is designed to improve conditional calibration: - a. This is crucial in numan-in-the-ee interventions are prioritized based on an accurate ranking of predictive uncertainty across data points (see Appendix [ Marginal coverage guarantees offer no guarantees for such rankings nor for conditional coverage. A method could have perfect marginal coverage but rank uncertainties arbitrarily. In other words, marginal coverage guarantees only address one specific performance metric (marginal coverage), while ignoring many other metrics that are often more important in practice.
P (YYnew E C(Xnew)) [Xaww) R
settings, where
To summarize, conformal marginal coverage guarantees (such as Lemma B.2 say very little about the overall quality of an uncertainty quantification method. Conformal marginal coverage guarantees only
"Conformal theory provides theoretical guarantees for P(Y.ow E C(Xiew)) 2 1 a and P(Ynow E C(Xnew) ) Darin, D > 1 under the standard exchangeability assumption However, it does not pro- vide finite-sample guarantees for P(Y E C(Xzev) | Durain D,a, Dea) and P(Ynzw E C(Xnww) 'Daa) for fixed calibration dataset, even under the standard exchangeability assumption.
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shed light very specific aspect of uncertainty quantification and only under the quite unrealistic assumption of exchangeability.
B.2 ASYMPTOTIC CONDITIONAL COVERAGE FOR CLEAR: LEMMA 2.1 PROOF & ASSUMPTIONS
Proof of Lemma 2.1 [Drin|,| + oo ensures that model selection on Dval identifies & consistent base models among the candidates, since the validation loss converges to the population loss. The consistency of the predictors implies 9zt
O, and the consistency of the quantile estimators implies (x)(xx) - 9² point-wise. Hence, the estimated pre-calibrated intervals converge to their population analogues. Since suPжcd Agr cpll (z) + O converges to O uniformly over À E A (due compactness of A), therefore C(x) is asymptotically equal to true conditional quantiles (2))2(()),111/2((2) and necessarily limppall-oss +oo 1. as is the case in classical CQR (see e.g.
(Angelopoulos et al. 2024 Section 5)).
The epistemic component vanishes asymptotically in this proof. However, for finite samples, the epistemic uncertainty plays crucial role in preventing under-coverage in out- -ut-of-sample regions as we intuitively explain in Appendix (aaadixxx44aaa and empirically demonstrate in Section 4.1 and Appendix C While these synthetic datasets provide the most direct and intuitive empirical evidence of the im- portance of combining epistemic and aleatoric uncertainty, all our experiments (Sections and and Appendices to support this hypothesis.
The assumptions underlying Lemma 2.1 are generally mild and often satisfied in practice. The assumption that A is compact is particularly mild. In experiments, example, a combination of linearly spaced values from 0 to 0.09 and logarithmically spaced values from 0.1 to 100, resulting in approximately 4010 points. We conjecture that even if A were unbounded, the result of Lemma would would not lead excessively large values of X in the limit.
finite grid; for
would still hold, since QuantileLoss is a strictly proper scoring rule and thus
Lemma 2.1 also assumes consistency the estimators. Many estimators satisfy this condition for both regression and quantile regression tasks:
1. Quantile Random Forests have been shown to consistently estimate quantiles under rea- sonable conditions, such as by regularizing the minimum number of samples per leaf (Meinshausen) 2006). Similarly, classical Random Forests known to be consistent under standard assumptions (Scornet et al. 2015).. Both QRF and Random Forests are available in our implementation of CLEAR.
2. Boosting methods trained on general loss functions can be made consistent under certain conditions, especially when regularized through early stopping (Zhang & Yu 2005). This suggests that XGBoost and its quantile version, QXGB, can be consistent for suitable choices of hyperparameters. Both are included in our implementation of CLEAR.
3. More generally, regularized minimization of QuantileLoss, over sufficiently expressive function class on Derain yields consistent quantile estimators under broad assumptions (Steinwart Christmann 2011).
In our experiments, we set k so Lemma 2.1 requires only one of the base models to be consistent.
Finally, note that Lemma 2.1 limpicitll assumes i.i.d. data, as described in Section 2.1 For real-world datasets, this i.i.d. assumption often the most difficult justify in practice and may pose a greater challenge than the other assumptions in the lemma.
B.3 PROPERTIES OF THE QUANTILE LOSS
We define the QuantileLoss (D,C) [D| E(wDeD (QLa/2(y. QL1-a/2 u(x)) /2
(also denoted "pinball loss". "quantile loss". "asymmetric piecewise linear loss". "linlin loss". "hinge loss", "tick loss", or "newsvendor loss" (Gneiting 201T)), with 1(x),, u(x) denoting the bounds Note that the majority of the literature defines the quantile loss as Q whereas QuantileLooss already aggregates Q over both the upper and
of Czz), and QL, (y. q) = (y (T 1 (-oo.1)) (v)). (1-))
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the lower bound of the intervals C and over the data points in D, similarly to the interval score loss (see Appendix B.3.4p.
We the QuantileLoss for three different purposes in this paper:
1. Some QR-methods in Step 2 of Algorithm D use the Quantilel Loss to train their models.
2. In Step 4 of Algorithm we minimize the QuantileLoss to determine X In other words, Step and 4 of Algorithm ! together (approximately) solve
(7i -
arg min (n (E,,0) XA st. (t.)(( (e
QusstileLoss(Peall Cnid) (4)
epi where C f /
-
Amii-a/9
cpi
3. We use the Quantilelooss as an evaluation metric on the test dataset D Pesst-
In the following, we will discuss multiple favorable properties of the QuantileLoss (which equivalently also hold for the Interval Score Loss, see Appendix B.3.4).
B.3.1 INTUITION BEIIIND THE QUANTILE LOSS
In many real-world applications, the severity of a prediction error often depends on the magnitude of the deviation from the predictive interval. For example, in financial portfolio management, massive drop below the predicted lower bound can be significantly more damaging than a slight deviation. Similarly, for flood protection systems where dam heights are based on an upper predictive bound, a large amount of overflow causes substantially more damage than small overflow. Traditional metrics like PICP treat coverage as a binary outcome, distinguishing only between points that are covered and those that are not. The QuantilelLoss, however, offers more nuanced evaluation by penalizing the magnitude of data point's distance from the predictive interval when it not covered.
Intuitively, the QuantileLoss strongly penalizes the distance to the predictive intervals for data points outside the predictive interval, and gently penalizes the width of the predictive intervals that do not cover the data points. This incentivizes the predictive intervals to widen in regions of large uncertainty and to adaptively narrow down in regions of low uncertainty, incentivizing good relative uncertainty (see Appendix (b. Simultaneously, the QuantileLoss incentivizes good absolute uncertainty (i.e., marginal calibration, see Appendix [): If the proportion of data points below the upper bounds is less than 1 - 5. then increasing the upper bounds by constant improves the Quantileloos until the proportion reaches 1 S. i.e., for all c >>> 0,
Hl v) E Deal : S u(x) + +cJl < (1 /
a [Paal
QuantileLooss(D, [1, x)) QuantileLoos((), [1 c)),
and vice versa if the proportion of data points below the upper bounds is more than 1 3. i.e., for all e > 0,
K(x, v) E Deat y [(=))) (1 -
0r |Daal
QuantileLoss(D) [1, ul) QuantileLoss [1 + c)),
and analogously, the QuantileLoss incentivizes the lower bound to be above a/ Deal data points and below (1 Deal data points. The Quantilel.oss simultaneously evaluates multiple different properties predictive intervals and cannot be easily tricked, in contrast to other metrics: PICP can be easily maximized by infinitely wide intervals, which are completely useless in practice: NIW can be minimized by zero-width intervals, which are useless in practice; NCIW only measures the relative uncertainty while completely ignoring the marginal coverage.
B.3.2 THE QUANTILELOSS MEASURES INPUT-CONDITIONAL CALIBRATION
In the following, we will mathematically argue why the QuantileLoss measures input conditoon calibration. Within this paper, we denote input conditional calibration P[Ynew E C [(Xnew 1 a simply conditional calibration.
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The true input conditional quantiles minimize the expected QuantileLoss, i.e. (Nan Yazu) [QuantileLoss( {(Xnew: Ynew) n. u.)]
(1-/2:91-0/2)
arg min (1,x) xy*
(5)
Any minimizer of the QuantileLoss satisfies input-conditional coverage almost surely (a.s. i.e., any solution (t", u²) of the minimization problem (5) satisfies that (X) an input-conditions a/2-quantile a.s. and u² (X) is an input-conditional
a/2-quantile a.s., thus, P|Y
0..
[[Xnne):: u(XXnew [[[-Xnew] 2 11-a. a. In other words, any deviation from the true quantiles gets penalized by the expected QuantileLoss, as it is strictly proper scoring rule (Koenker 2005). If the true conditional CDF is a.s. continuous, then any solution (I", u²) of (5) satisfies nppt-conditional calibration P P E [[(")new,, ne] neww = 11-a: a. In other words, any deviation from input- conditional coverage gets penalized. Applying the Quantilel on finite dataset DD can be seen as a Monte-Carlo approximation of the expected Quantilelooss
Another common evaluation method is to compare the average interval width NIW (or NCTW) among methods that approximately obtain the targeted marginal coverage. However, this evaluation method can be exploited: Even if you perfectly know the true distribution, reporting intervals that do not satisfy input-condition that over-cover in regions with low uncertainty and under-cover in regions of high uncertainty. as demonstrated in the following examples (Fxamples B.3 and B.4 are easier to derive, but Examples B.5 and B.6 are slightly more insightful).
coverage would be optimal. This evaluation method prefers intervals
Example B.3. Let X E {1. 2}, with with = 1| = 0.5 and P[X = 2] = 0.5. Let Y = ~ 1, and X U( 2, For target coverage of
0.9, the true conditional intervals are:
[-0.9,0.9] when X (width=1.8, coverage=0.9) *
* 1-1.8, when X (width=3.6, coverage=0.9)
The average width of the true intervals is [iitth] = 0.5 2- 2-0 0. 2.7. The marginal coverage exactly 0.9. Now, consider an alternative method that sacrifices conditional calibration to minimize average width. This method could report the following intervals:
[-1, when X = (width=2, coverage=1)
* [-1.6, when X = (width=3.2, coverage=0.8)
The marginal coverage of this method is IP[covered] = 0.5 x 11 0.5 x 0.8 = 0.9. It still achieves the target marginal coverage of 90%, but its average width is [[[idth] 0.5 x + 2 1.6 2.6. Since 2.6 < 2.7. this method would be preferred, demonstrating that NIW can incentivize deviations from input-conditional calibration.
Example B.4, Under the setting of Example B.3
the situation gets even more extreme if we change 0.5, the true conditional intervals are:
to = 0.5: For a target coverage of
0.5, 0.5] when X = (width=1, coverage=.5)
* [_1,1] when X = (width=2, coverage=0.5)
The average width of the true intervals is E[iidth] = 0.5 x 1 exactly 0.5. Now, consider an alternative method that sacrifices conditional calibration minimize average width. This method could report the following intervals:
1.5. The marginal coverage
[-1, when X (width=2, coverage=1)
[127, 127] when X (width=0, coverage=0)
The marginal coverage of this untruthful method is IP[covered] 0.5 x 1 0.5. still
achieves the target marginal coverage, but its average width is [[[idth]] 0.5 x 0.5 x 0 11
'A statement holds almost surely holds with 100% probability. F.g., a standard normally distributed random variable X Ao 1) is not exactly equal 22, i.c., X # V2.
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Since < 1.3, this untruthful method would be preferred, demonstrating that NIW can incentivize strong deviations from input-conditional calibration, Here, the interval that minimizes NIW (and NCIW) under the constraint of maintaining marginal coverage, outputs much wider interval for x= 1 than for = 2, while there is obviously more uncertainty for X = 2 than for X =1 1.
Example B.5. Let P[x = 1] 0.5 = P[X 2/. and Y|X = Ni 0, z). Then for a 5%, the true conditional quantiles would be (0.975(x) = 4 -01(0,)7)) R 1.96x and 20.025 (z) - (0.025)): s - 11966z, thus [go.025, 40.975] exactly satisfy input-conditional calibra- u.s. 95%. However, the average narrowest tion 1 P|Ynew new E 190.025 (Xnew), ) 90.975 (Xnew)] neww]
intervals satisfying marginal coverage would be approximately
1-216337, 2.16357] when X (width = 2 2...2.1.331 2.16357, coverage = 0.969502)
. [-1.81514 2, 1.81144 2) when X = 2 (width 22 1.81514 2, coverage 2 0.930498).
These intervals fail input-conditional coverage while maintaining marginal coverage and result in a narrower average width 5.79385 5.88 than the intervals [go.025, 90.975]
Example B.6. Let IP(X- =1] 9 PP[X = 2). P(X = 11] = Ig and Y|X x Nip
19
0,0 z). Then for a = 10%, the true conditional quantiles would 90.95 (x) = -- -0.95)) R3 1.645x and (x) Φ (0.05))
- 1.645x, thus [4o.us 05 00.9s] exactly satisfy input-conditional
(>X neX neew] a. 90%. However, the on average narrowest calibration IP [Yeew E 190.0 (Xnnew), 90.95
intervals satisfying marginal coverage would be approximately
2.116357, 2.16357] when X = (width 2 16357, coverage 2 0.969502)
*
[11811514 2, 1.81144 2) when X = 2 (width 2 1.81514 2, coverage 0.930498).
[12345, 12345] when X 11 (width 0, coverage 0).
These intervals fail input-conditional coverage while maintaining marginal coverage and result in narrower average width 5.48891 < 66.799455 than the true intervals [10.05: 90.95].
Interpretation of Examples B.3 B.6 While QuantileLoss only minimized for intervals
consisting of the true quantiles 9 91-99-199 y the Examples B.3 to B.6 showed that metrics like PICP. NIW, NCIW, and their combinations can misaligned with input-conditional coverage. This misalignment can be very severe in Examples (aadlesB.4anng [.44and [.6 whereas it is less severe in Examples B.3 and B.5 Intuitively, the misalignment can be particularly extreme for large values of a (as in
Example B.4 or for inputs that are much more uncertain than the average uncertainty (as for in Example B.5). In our experiments on real-world data in Sections 4.2 and 4.3 and Appendix F compared to these extreme examples, the different metrics shows superior performance across all considered metrics. Reasons for this empirically observed alignment could be that 1st none of the compared methods actively tries to exploit the weaknesses of PICP, NIW, NCIW; 2nd in our experiments, we use only small values of a 5% (and a = 10%); 3rd there might not be many regions of extremely high uncertainty in these datasets.
aligned, in the
that CLEAR
Now we have concluded that under the QuantileLoss, it is optimal always to report the true quantiles if the true underlying distributions are known. In the next subsection, we will discuss how this translates to situations where the true distributions are not known.
B.3.3 THE QUANTILELON LOSS INCENTIVIZES TRUTHFUL EVEN UNDER INCOMPLETE INFORMATION (FROM BAYESIAN POINT OF VIEW)
Proper scoring rules are mathematically guaranteed to incentivize reporting the true distribution, but their application to uncertainty quantification requires care in interpreting what kind of uncertainty they incentivize. Buchwweizz et al. (2025) show that some proper scoring rules impose asymmetric penalties for over- versus under-estimation. While this may appear induce bias, argue that such asymmetry faithfully captures epistemic uncertainty rather than distorting the estimate of total predictive uncertainty.
In our case, the relevant scoring rule is the QuantileLoss. For given quantile, e.g., 90.975 the loss penalizes underestimation more heavily than overestimation. This asymmetry not a flaw; rather, it
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incentivizes appropriately wider predictive intervals in the presence of epistemic uncertainty. Specifi- cally, this asymmetry encourages intervals to widen more in regions of high epistemic uncertainty than in regions of low epistemic uncertainty.
This phenomenon can be described more quantitatively from a Bayesian perspective. The posterior predictive distribution,
P[y new
X Dirain; m7| E [P[Y
new 0] D Dirain:
optimally reflects both aleatoric uncertainty P|Ynew X (e.g., noise) and epistemic uncertainty Marginalizing over the posterior naturally yields wider distribution than relying on single point estimate, with greater widening effect in regions of high epistemic uncertainty. Let Pr.P denote the --quantile of a distribution P. The posterior predictive quantiles (9a/2,7(Yeer N , Puii 7). 11 91-a/2,P(Yme N mu minimize the expected quantile
P|o Diann π] (e.g., over parameters) given a prior π
loss:
E QuantileLoss (eewew Ynow)) }, [4, ul) Duain x]
Since the posterior predictive distribution includes epistemic uncertainty, its quantiles also appropri- ately include this component. This stands in contrast to other estimators, such as 9 41-a/2.P(Ye: Xnex.0
DDrainn π] (similar to how we es- timate the pure aleatoric uncertainty in Section 2.3). which ignore epistemic uncertainty.
//1-//2.7 F..F/ (X 0] Durain π. or Median [q] /1,00/2, P2.e///
Example B.7. Let Y|X
(((xz), a 1) with an unknown mean ((x) E R and a known
standard deviation of 1. Further, assume that the posterior distribution of 0(xr) is >N -10)
due to large epistemic uncertainty. Then, the predictive posterior distribution of Y|X = x
N (0, = v10² + I), which includes both aleatoric and epistemic uncertainty.
40.975. (0.,97 Xwew Xeee, Du)n. (0.3755(000-1000411) 1.96 /102 appropriately takes epistemic uncertainty into account.
(o. 0] Co 40.975.X ((x) + 1.96 ignores the large epistemic uncer-
tainty (with interval width of 1.96 40.025, Plaael R 0(z) 1.96).
E (0..77.PPYY... X.... D Diain: π
E V Pinin π
E [A(x)) 1.96 D Dirain Ti] = 1.96 ignores the large epistemic uncertainty.
Median 99..975 Xzc. Dusin π = Median [90.975.. Durann T
Median (0(x) + 1.96 D Paiin m| 1.96 ignores the large epistemic uncertainty.
We fully agree with Buchwwetz et al. (2025), that minimizing the QuantileLoss leads to a biased estimator of aleatoric uncertainty alone (as in Example B.7 the aleatoric uncertainty corresponds to an interval width of 2 1.96). However, minimizing the Quantilel.oss yields a principled estimator for total predictive uncertainty, it is optimal from Bayesian point of view, taking epistemic uncertainty into account. An unbiased estimator for total predictive uncertainty must also incorporate the epistemic uncertainty and is therefore naturally biased for estimating aleatoric uncertainty.
Thus, the asymmetry of the QuantileLoss is precisely what makes appropriate for evaluating models that estimate total predictive uncertainty. This justifies its use both in Step 4 of our Algorithm [ and as an evaluation metric on the unseen test dataset Drest-
B.3.4 THE INTERVAL SCORE LOSS EQUIVALENT TO THE QUANTILE LOSS UP TO MULTIPLICATIVE FACTOR.
Our definition of the quantile loss
QuantileLoss((, (1, u)) =
QQaae (x)) + +LI1-/2221 u(x)) /2
D
(a.y)ED
Mathematicians (with hackground in thesure should read "P(Y| X]" "P(Y (+) x]".
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only differs from the average interval score loss (Dunsmore 2007 Brehmer & Gneiting 2021)
1968
Winkler
1972
Gneiting & Raftery
(1(x) y)
y ((x)
AISL(D, (1,
Σ
(u(x) - ((x))
2 (g-u(z))
if y > u(x)
[D]
(2.9)ED
0
clse
by the constant factor of a/4, i.e., QuantileLoss(D, (1, u)) AISL(D, (I,
Traditionally the terms "quantile loss". "pinball loss", "asymmetric piecewise linear loss". "linlin loss". "hinge loss", "tick loss", or "newsvendor loss" refer to the loss QL, q) (y (T
(x)) that only evaluates single approximate quantile q (Koenker & Basseu 1978 Gneiting
2011). while the interval score always evaluates two approximate quantiles (7, u simul taneously
Dunsmore 1968) Winkler 1972 Gneiting & Raftery 2007 Brehmer & Gneiting 2021). However,
within this work, we also denote the average QuantileLoss(P) (D, (I,u)) as quantile loss. This shouldn't t
create any confusion, as we never evaluate single quantiles within this work.
B.4 INTUITIVE THEORETICAL MOTIVATION OF CLEAR
PCS achieves empirically good results and nicely captures common sense, with limited theoretical guarantees, while CQR satisfies theoretical guarantees, but misses crucial common sense. In our work, we combine practical experience, theoretical insights, common sense, and empirical results, always with the goal in mind to obtain useful, veridical, reliable, stable, high-performing method for practical real-world applications.
Note that the original version of PCS did not have any theoretical guarantees for uncertainty quantifi- cation Yu & Barter 2024), and Agarwal et al. (2025) mentioned a modified version of PCS-UQ that satisfies conformal marginal coverage guarantees. However, no version of PCS offered theoretical guarantee for asymptotic input-conditional coverage.
Via CLEAR, we the first to propose extension of PCS-UQ that provably satisfies asymptotic input-conditional coverage guarantees (see Lemma 2.1 All previous versions of PCS did not only lack proof for input-conditional coverage guarantees, but actually do not satisfy input-conditional coverage guarantees. Pure PCS-UQ has the systematic bias of under-coverage in regions with many observations and large noise. This was a central motivation for CLEAR. While this theoretical bias is quite obvious from theoretical common sense point of view, we conducted our large-scale experimental evaluation to see that this bias can actually have a significant impact on the performance, which can be substantially mitigated by CLEAR.
On the other hand, CQR (or QR) has major problems based on common sense. If one looks closely into the proof of the asymptotic nnput-conditionl coverage of CQR (or QR), one can see that the main idea of the proof is that, asymptotically, you will have infinitely many training data points in close proximity to every point resulting in accurate quantile predictions. However, for finitely many data points, there will always be regions in the input space with too few data points. these regions (C)QR cannot accurately estimate the quantiles. However, (C)QR will sometimes output very narrow intervals in these regions, which are fundamentally unreliable and will undercover substantially in these regions. We can see this phenomenon across all different synthetic data- generating processes that we considered (see Sections 3.T and Appendix C. When we sample X ~~ (0, Ia) standard Gaussian, we can see that even for n 5000 training data points, we still heavily under-cover in low-density regions, i.e., P|Y E CCOR ||X||2 > 4] < 1 a. Based on common sense, it is strongly expected that for every sample size n, there exists radius r(n) beyond which P [Y E CooR ||x||2 > r[n)] <> 1 - since the intervals of (C)QR do not have any reason to become wider sammle while their predictions become more unreliable the farther you move away from the training data.
Hypothesis B.S. We hypothesize that under some fairly general and mild assumptions, for every "non-trivial" data-generating process,
3c > 0 Vn N 3R(n) c0) Vr RY Ccor ||||||2 > > <
holds. We think that this claim strongly agrees with basic intuition, and all our empirical results strongly support this hypothesis. However, we leave the proof future work. The main technical challenge is probably to define "non-trivial" appropriately.
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This does not contradict asymptotic input-conditionll coverage guarantees lima- r(n) co. However, for any finite sample size n, this is a serious problem both from a common-sense perspective and from what you members heavily disagree on the predictions (usually in regions with few training data), (C)QR would be substantially over-confident. CQR tries compensate for this overconfidence in the calibration step, which results in CQR slightly over-covering in regions with many training data points while still considerably under-covering in regions with few training data points (see Figures 2 and to 6
from empirical results. In other words, in regions where the ensemble
This is the reason why PCS-UQ and CQR form a particularly effective symbiosis. In regions with many training data points, the ensemble predictions coming from different bootstraps of the training data will be close to each other. PCS-UQ has a systematic bias to under-cover in regions with many training data points and to over-cover in regions with few training data points, whereas for CQR, it is exactly the other way around. If you apply any monotonic transformation of the uncertainty of PCS (multiplicative (conformal) calibration, additive (conformal) calibration, or the calibration methods suggested by Kuleshov et al. (2018); Song et al. (2019); Kuleshov & Deshpande (2022); Levi et al. (022)), you will always have the systematic bias that any monotonically calibrated version of PCS-UQ will under-cover in regions with many training data points and will over-cover in regions with few training data points, whereas it is exactly the other way around for any monotonically calibrated version of CQR. By combining the of them, we can mitigate this bias (see Figures [ and 4 to 66. This aligns very well with classical statistical results, telling us that for predictive intervals, one needs combine the estimated noise structure with epistemic uncertainty on the parameters (confidence intervals).
B.5 CLEAR ALGORITHM: DETAILS
Algorithm 2 Fast Conformal Implementation of Step 3 from Algorithm
1:
Input: Data (X,, Y,) for = I,... n, split into training Duainn calibration Deal, and validation Dval (we consider Deal Dval): grid of À values A; significance level a.
2: Step 3: Define prediction intervals for each À A.
First, define a preliminary (non-calibrated) interval:
=
i
cale la/2
eepi /22
f + gale 1/-0/2 + 101-
max { f(X.)-Y (x(x)- f(x.))-((x) where
Y- x-f((x) 1
Then, compute conformity
scores
S
(x),, ūx (z) are the lower and upper bounds of (z)). Let 71 be the [(1 ( + 1)
smallest score among { } (if [[1 ((aal + 1)] > (Deal take the largest score). Define the calibrated interval:
Cx cx - ma/ cale
Xm1ga/2 f + 7 + Am
3: Output: calibrated prediction intervals C2 for each X from the grid A.
B.6 ASYMPTOTIC OPTIMALITY OF JOINT CALIBRATION RELATIVE TO BASELINES
In this subsection, we formalize the intuition that CLEAR's joint calibration spans a strictly richer solution space than saggl--arameter baselines. We analyze the methods at the population level, which corresponds to the limit of infinitely large calibration and validation datasets.
Let (X, Y) P be the true data distribution. For any non-negative calibration parameters 71 2 0. define the prediction interval input x as:
C = [f(x) - 71 Gall (x) - - f(x) + 1 + a2
(6)
Let Cov(n, 72) 10: P(Y E C, (X)) be the population marginal coverage. We assume Cov(y), 72) continuous and strictly increasing in both arguments until it saturates at 1.
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Let A C [0, oc) be the search space of ratios À /2/71 evaluated by CLEAR. For any À E A, let (1) = in > 0 Cov(y, An) 2 1 - a} be the minimal population scaling factor required to achieve 1 - marginal coverage. Let L())
denote the
population expected quantile loss of the resulting interval.
We define the population-Ieva intervals produced by the three methods as follows:
CLEAR (CCLEAR): Selects A = arg mingex ((()) and outputs (,()))))))))))
2. Fixed-A Baseline (Cbase-2): Fixes a hyperparameter λ A and outputs Cy (Ao) (Ao)-
E.g., An 0 for methods that do not explicitly model epistemic uncertainty (see Ap-
pendix A.6 An 1 for methods that model both but do not rebalance them (see
Appendix A.2).
3. Fixed-71 Baseline (Cbase-r.): ): Fixes hyperparameter Y1,0 > 0 (e.g., 71,0 as in
Appendix A.1 and calibrates 7/2 to achieve exact coverage, finding "/2,huse = iin >> 0
Cov Cov(7,,0
7) 2 - a). It outputs C71,0.72, buse"
We assume its implicit ratio X'
12, is
71.0
contained in A.
Lemma B.9 (Population Optimality of CLEAR). Assume that the population marginal coverage function Cov(m, 122) is continuous and strictly increasing in both arguments. Furthermore, assume that the predefined search grid A contains both the hyperparameter Xo of the Fixed-A baseline, and the implicil optimal ratio X'
"Y2.wu of the Fixed-7i baseline. Then, the expected quantile loss of T/.
CLEAR is upper-bounded by the expected quantile loss of both single-parameter baselines:
EE ElQuantileLoss (ossciLLALLss(CCA) CCLEAR)] < [[QuantileLoss( Chase->)] EE [[QuuntileeLoss( (ostammillLooss(Cceeee) (CCLEAR) < [QuantileL.oss Chase-ra )]
Proof. By definition, CLEAR outputs the interval corresponding to X", which minimizes the popula- tion quantile loss L(A) over the set A. Thus, E[QuantileLosss (CCLEAR) L(X*).
Part 1: fixed-A baseline. The Fixed-A baseline evaluates the interval corresponding to a pre- specified Ag- Since An E A, it trivially holds that the minimum over A is bounded by the value at Ap:
L(>*) = min L(A) < L()o) = E[QuantileLLoss(unee)) xEA
Part 2: fixed-y1 baseline. The Fixed-71 baseline enforces 7/1 = Y1,0 and sets 7/2 = 72,ba such that exact marginal coverage is achieved: Cov(?1,0, "22. 1 - a. Let the implied ratio be X' 22,
m1.0
By definition, evaluating CLEAR's population subroutine at X' requires finding the minimal ; such Because Cov is strictly increasing, substituting y 71,0 yields exactly
that Cov(y,X) XY) 2 1 the required coverage:
Cov(Tn,, Xm,,) = Cov 71
Y2.base
n.o)
Cov(71,0, (2,771
(10)
71,0
Because 71,0 achieves the 1 - a coverage constraint exactly, the minimal scaling factor 7 (X) evaluated by CLEAR at X' is exactly 71,0-
Since X' E A by assumption, it follows that:
L(>") min L(A) s L(X) E[QuantileLosss (bbsse
(11)
xеλ
This concludes the proof.
Remark B.10 (Finite Sample Convergence). Lemma B.9 establishes that CLEAR is optimal among
this class of methods at the population level. In practice, Algorithm operates on finite calibration and validation sets Deal and D.a.- Because the quantile loss is Lipschitz continuous and the bounds of the intervals are linear combinations of fixed heuristic functions, the interval class has bounded Rademacher complexity. Standard results statistical learning theory guarantec that the empirical loss Lippa1( (A) converges uniformly to the population loss L(A) almost surely as (Deall |Dval + oc. Continuous mapping theorems for M-estimators therefore ensure that the limits of the empirical mini- mum and empirical minimizer converge to their true population counterparts, translating Lemma B.9 guarantees to the asymptotic limit of the finite-sample algorithm.
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Remark B.11 (Reuse of the Validation Dataset). In saitle regimes, reusing the same dataset Deal = val can lead to a violation of the theoretical marginal coverage guarantees, see Appendix [ However, in the limit of infinitely large dataset ((Deal vaviil oo), the empirical distribution perfectly converges to the true population distribution P. Consequently, empirical coverage and empirical expected quantile loss evaluated on this dataset become exactly identical their population counterparts Cov(71, 72) and L(A). Because the finite-sample noise vanishes entirely. "overfitting" mathematically impossible in this asymptotic regime. Thus, whether CLEAR uses independent infinite datasets for validation and calibration, or reuses the exact same infinite dataset (Deal Dvai), the resulting parameters (71, X) converge the identical optimal population parameters.
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C SIMULATIONS: DETAILS
For each simulation run, the coefficients $1. .......... Pa are drawn independently from Gaussian distri- bution (1, 0.5²). The mean function (xX) then defined μ(X) 5.0 E0++E111 (-1)*+1 (-=( [ [x," where the exponent = 1.5 if is odd, and e, = 1.25 if is even.
C.I HETEROSKEDASTIC CASE
In Section 3.1, we focused on the univariate homoskedastic setting (d and o(x) 1). Here, we briefly report results for heteroskedastic data, which show similar patterns. Figures 4 and illustrate the conditional coverage and interval width of the algorithms under two heteroskedastic noise structures: (2
[x] and (2) 1+ 1-22. As mentioned, the results analogous to the homoskedastic setting in Section 3.1
C.2 MULTIVARIATE CASE
C.2.1 TEST POINT GENERATION
Following setup in (Bodik & Chavez-Demoulin 2025 let P1 < <.... TK denote set of predetermined distances. For each radius Tk, we sample points uniformly from the surface of the unit d-dimensional sphere follows. First, draw a vector standard normal random variables. We then normalize v to obtain a unit vector u v/llv||2. Finally, Tk u. In the one-dimensional case, this procedure
Rd whose entries are independent
we scale by 7k to obtain the test point reduces to selecting z тж or z -Tk with equal probability. This mechanism ensures that, for each rk, the generated points are uniformly distributed on the surface of the sphere of radius rk- thereby allowing precise evaluation of prediction intervals function of distance from the origin. Finally, we generate with the same data- data-genen setiaaggneaatnnggg mechanism as in the train set.
C.2.2 RESULTS
Figure 6 iillustrates the conditional coverage and interval width of the evaluated algorithms in the multivariate setting, where X is drawn from an independent multivariate Gaussian distribution and Y xxX e, with E (0, 1). The regression function (xX), as previously defined, involves sum of randomly weighted power transformations of the absolute values of the input features.
Consistent with the univariate homoskedastic results in infSection 3.1 Figure 6/shws that both CQR and naive conformal prediction achieve reliable coverage in high-density regions but tend to under-cover low-density or extrapolation areas. In contrast, CLEAAR maintains valid conditional coverage across the entire input space by appropriately adjusting interval widths.
Conditiona Coverage
Average width
20
0.7
15
0.6
5
0.4
CLEAR
PCS-EPISTEMIC
ALEATORIC-R
Naive
Figure 4: Univariate conditional coverage and average width of the prediction intervals for het- croskedastic case where (2(2) = 1 + |z].
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1.0
Conditional Coverage
0.9
Average width
0.5
5
-6
0
2
CLEAR
PCS-EPISTEMIC
ALEATORIC-R
Naive
Figure 5: Univariate conditional coverage and average width of the prediction intervals for a het- eroskedastic case where a3(x) = 1 I+s
1.0
Conditional Coverage
Average Interval Width
25
0.9
20
8.8
0.7
10
0
Distance Origin (Radius)
Distance Origin (Radius)
CLEAR
PCS-EPISTEMIC
ALEATORIC-R
Naive
Figure 6: Multivariate conditional coverage and average width (distance from the origin is shown) of the prediction intervals for a homoskedastic case where E (1, 2, 3, 20) was randomly selected.
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D EXPERIMENTAL SETUP: DETAILS
D.I REAL-WORLD
To evaluate our method in real-world scenarios, we apply it 17 publicly available regression
datasets curated by AAgaral ct al. (2025) that includes various domains such as housing. energy,
materials science, and healthcare Kelley Pace & Barry (1997] Mayer et (2022); Tüfekei (2014);
Tsanas & Xifara 2012); H (2018); Camacho & Torgo (2014); Tsanas et al. 2009); Coraddu
al. (2016); Yeh| 1998); ) Cassotti al. (215)) Brooks et al. 1989) : PyCaret (yCaret(nndd)) Torgo (2014);
Computer Activity Allstate Claims Severity Grinsztajn et al. (2022): Ghahramani (199)] kin8nm
dataset sulfur dataset Description of the datasets can be found in Table 13
D.2 BASELINES FOR VARIANTS (A), (B), (C)
The following baselines provide further context and comparisons, omitted from the main paper for brevity, All compared methods (except UACQR) use the same median f(x) = fres(z) obtained from PCS. The results from Appendix F. use PCS variant (a), Appendix F22 uses PCS variant (b), and Appendix F3 uses PCS variant (c). This allows us to isolate the effect of different UQ methods.
1. Conformalized PCS median (Naive): For the PCS ensemble, we compute the median prediction f(x) fpcs(z) for each æ. On the calibration set, absolute residuals a, = y) - -x;))] are computed. The (1 - a-)-th quantile Yanive of these absolute residuals is then used to define the interval:
ONaive = [f()) Ynaive, f(x) + Ynaive]
This method applies a constant, symmetric width adjustment to the point predictions. The result of this baseline has been presented for the simulations. We omit it from the real-world data for brevity. However, all the baselines can found the supplementary code.
2. CLEAR with fixed À = 1: This is variant of the main CLEAR methodology where the ratio À 72/% is fixed to This implics V = 72. The prediction interval, based on Equation 3 becomes:
Cx-1(x) = [f(x) - -m (Tel)2(a)) (1 +00))))))2/(0)))
cepi (0)) f(x) + m (f==a//((x) + (T=a/((x))
In this configuration, the single parameter 71 is calibrated using the standard split conformal procedure on the validation set. It effectively learns single scaling factor for the sum of the pre-calibrated aleatoric and epistemic uncertainty widths, without adjusting their ratio.
3. CLEAR with fixed 71 then 7/ = X and the prediction interval from Equation ( reads as:
Another variant CLEAR where Y1 is fixed to 1. With 71
())=(() = [f(x) (x)) 2
f(xx) + gale ) + 101-a/2(z)) (x))
Here, À (or equivalently 7/2 the parameter calibrated on the calibration set through coverage- based adjustment. This approach fixes the contribution of the (pre-calibrated) aleatoric uncertainty component and adaptively scales the epistemic uncertainty component.
4. UACQR-S and UACQR-P For variant (c), instead of computing Cz,= 1 (z) and CA (z) with CLEAR, we directly use the implementation from Rossellini et al. (2024) to assess our performance against this alternative. Since this is only relevant for variant (c),
the exact same configuration as the aleatoric part for QRF (see (Appendix D.4 and Table 6).
and
D.3 DEEP ENSEMBLES AND SIMULTANEOUS QUANTILE REGRESSION IMPLEMENTATION
To demonstrate CLEAR's versatility beyond tree- rree-based methods, we implement mate--o-thhe-a deep learning approaches for uncertainty quantification.
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D.3.1 DEEP ENSEMBLES FOR EPISTEMIC UNCERTAINTY
Our deep ensemble implementation follows Lakshminarayanan (2017) with several enhance- ments for improved diversity and calibration. Each ensemble consists of M 5 neural networks
(adaptive based on dataset size), with the following specifications:
Architecture: Each network employs fully- connectdd architecture with hidden layers sizes (256, 128), ReLU activations, batch normalization, and dropout (rate=0.1). Skip
connections are incorporated stabilize training.
Diversity strategies: To promote ensemble diversity, we employ: (i) different random initializations, (ii) bootstrap sampling where each member trains on different subsamples, (iii) varied learning rates, (iv) different learning rate schedules, and (v) small input noise augmentation.
Training: Each network is trained for up 1500 epochs using Adam optimizer with learning rate 10-3, early stopping (patience=50), weight decay (10-9), and batch size 64.
Following training. we apply a multiplicative calibration factor computed on the validation set using grid search over c-values to ensure the ensemble achieves target coverage.
Calibration:
The epistemic uncertainty is quantified through the empirical quantiles of the ensemble predictions: gepi (a/2(() f(x) Qa/2( (=))))))))))))))))) and प.h o1-a/2
(mmmmmmmmm))))))mm))mm f(x), where f(x) the ensemble median and Q, denotes the empirical quantile function.
D.3.2 SIMULTANEOUS QUANTILE REGRESSION FOR ALEATORIC UNCERTAINTY
For aleatoric uncertainty, implement simultaneous quantile regression (SQR) following Tagasovska & Lopez-Pa7 2019), which directly models multiple conditional quantiles through single neural network with specialized architecture:
Architecture:
The network uses hidden layers of sizes (256, 256, 128) with LeakyReLU activations (negative slopc=0.01), layer normalization, and dropout (rate=0.2). The output layer produces three values corresponding the a/2, 0.5, and 1 a/2 quantiles.
Loss function: We minimize combined loss consisting of: (i) the pinball loss for each quantile, and (ii) a crossing penalty term that encourages monotonicity of quantiles: Lewossing = ReLUUgg,/2 qu.s) ReLUU(qo.5 41-a/2).
The model trained for up to 3000 epochs using Adam optimizer with learning rate 5 10-A, cosine annealing schedule, and early stopping (patience-200). Gradient clipping (max norm=1.0) prevents training instabilities.
Training:
Ensemble averaging: To improve stability, we train an ensemble of SQR models with different random seeds and average their quantile predictions.
The aleatoric uncertainty estimates are computed as Gal = (0.5(x) - 22/2(x)) and -aa/2(()) =
91 Q1-a/2 (2) (z)) ensuring consistency with the median prediction.
D.3.3 INTEGRATION WITH CLEAR
The DE and SQR components are integrated into CLEAR using the same calibration procedure as our PCS-UQ (model selection and tree-based methods). The epistemic estimates from DE and aleatoric estimates from SQR are combined using the optimized parameters À and 71 according to Equation (3). The conformal calibration step ensures approximate marginal coverage while the adaptive selection balances the relative contributions of epistemic and aleatoric uncertainties, accounting for potential scale differences between neural network and tree-based estimators.
D.4 PCS IMPLEMENTATION: DETAILS
This section outlines the specific implementation details for generating the PCS ensembles, which provide the point predictor f and the epistemic uncertainty estimates gepl used as input for the
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CLEAR method (Section 2.4). and also form the basis for the standalone calibrated PCS baseline intervals. We explicitly opted for experimenting with three variants of CLEAR to assess the frame- work's performance and robustmess across different modeling choices for epistemic and alcatoric uncertainty components, as discussed in the main text. The core methodology. involving model selection and model perturbations via bootstrapping capture epistemic uncertainty. follows the principles described in Sectiin 2)
The process begins with data partitioning and bootstrapping. For each dataset and unique random seed, the data is divided into training (60%), validation (20%), and test (20%) sets. Subsequently, Nboot 100 bootstrap resamples are drawn from this designated training set construct the PCS ensemble. Then, for our variants, two distinct pools of base models were developed to generate these PCS ensembles, catering to the different CLEAR variants. [Table 6 details the quantile estimators used for CLEAR variant (a). Variant (b) only uses QXGB from Table 6 estimators utilized for CLEAR variant (c).
Table 7 describes the mean
Table 6: Base models and key hyperparameters for the quantile models used in CLEAR variants (a). Variant (b) uses only QXGB from this table. All models target the conditional median (T = 0.5).
Model
Key Hyperparameters
Ref.
QRF
100 trees, min. leaf size: 10
Meinshausen (2006)
QXGB
100 trees, tree method: histogram, min. child weight: 10
Chen &
Guestrin
(0016)
Expectile GAM
P-splines (order 3), smoothing parameter optimized
Servén &
CV (5 min. timeout)"
Brummiti
(2018)
Included only if hyperparameter optimization converged successfully, otherwise using the default values.
Table 7: Base models and key hyperparameters for the mean estimators (used in CLEAR variant
c). All models target the conditional mean and are from Scikit-learn (Buitinck et al., 2013), except XGBoost, which is from (Chen & Guestrin 2016). All unspecified hyperparameters use Scikit-
learn defaults.
Category
Model
Key Hyperparameters
Ordinary Least Squares
Default
Ridge
Default alphas (CV)
Linear Models
Lasso
3-fold CV
FllastiNet
3-fold CV
Random Forest
100 trees, min. leaf size: 5, features: 0.33 Same hyperparameters as random forest Default
Extra Trees
Tree Ensembles
AdaBoost
XGBoost
Default
Neural Network
MLP
Single hidden layer (64 neurons)
Ensemble construction and model selection. For each of the Aboott = 100 bootstrap samples, all models within the relevant pool (median estimators for variants a/b, mean estimators for variant c) were trained. The single best-performing model type (k: = 1) was then identified based on the lowest RMSE achieved on the held-out validation set. Consequently, the final PCS ensemble for each random seed comprised 100 instances of this selected lop-performing model type. Specifically, CLEAR variant (a) considered all models from Table for this selection process, variant (b) was restricted selecting always QXGB from this pool, and variant (c) considered all the models from Table 7 insteead. All models used default parameters from their respective libraries unless otherwise specified in the tables, and random states were fixed ensure reproducibility.
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Derivation of f and qeΩi for CLEAR. The point predictor f(z) and the raw epistemic uncertainty contributions /-/ (x) and 41-0/2 (x) supplied the CLEAR method are derived from this final
ensemble of 100 model instances. As detailed in Section 2.2 f(x) the pointwise empirical median of the ensemble's s predictions. The epistemic uncertainty terms represent the pointwise distances from this median the ensemble's empirical a/2 and a/2 quantiles, respectively.
Calibration of the standalone PCS baseline. The standalone PCS bascline method involves distinct calibration process. From the ensemble's s raw pointwise a/22 and - a/2 quantile predictions (1))(()) and fi fi-a/2(z) a single, global multiplicative calibration factor, Yrcs, is computed. This 7PCs is the smallest value ensuring that prediction intervals, formed by scaling the raw epistemic uncertainty around f(x) (i.c., [f()) - "PCs (fcs(f(x) - fa/2(z)), f(x) + "Pcs(fiaa/2(
f()))), coverage on the validation set. incorporating the standard finite- correction. This "res is then applied to generate the PCS baseline intervals on the test set. It is important to reiterate that this ypcs is separate and computed independently from " and À parameters optimized within the CLEAR framework.
achieve the target
D.5 METRICS
This appendix provides detailed definitions for the evaluation metrics used in the main paper. We consider test data (X,, Yi) and prediction intervals [L, U.].
Prediction Interval Coverage Probability (PICP): Measures the proportion true values falling within the predicted intervals, calculated as:
PICP(L,
Σ (L,
where [[2,.ua (Y;) is the indicator function; it equals if Y, E [L,, U:] and otherwise.
Normalized Interval Width (NIW): Quantifies the average width of prediction intervals normalized by the range of the target variable:
EE (U, T4)
NIW(L,U)
max min
Quantile Loss (also known pinball loss): Evaluates the accuracy of predicted quantiles by penalizing both under- and overestimation. It reflects a trade-off between coverage (PICP) and interval width (NIW), rewarding narrow intervals that still maintain proper coverage and penalizing data points that are far outside the intervals (see Appendix B.3 f.3 more details). For a given quantile level т, the quantile loss function is:
QT. q) = (y - () - ((>>oq]())
where is the predicted r-qunntile. For prediction intervals at level a, we evaluate this at both a/2 and - = 2/2 using
Quantileloos((,.U)) Σ [Q ((x)) (1x /2.
Average Interval Score Loss (AISL) (Gneiting & Raftery 2007): This score balances
interval width with coverage penalties (with the same properties the quantile loss explained in Appendix B.3). defined as
AISL(L, U) 1 N
Σ (U (U.-
L.) [w.
2c. -L.-Y)) < 4) 2x, U.}
where [..) the indicator function.
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E CLEAR WITH DE AND SQR: RESULTS ON
DATA
This section presents the comprehensive experimental results of CLEAR when combined with deep learning- maaeig uncertainty estimators: deep ensembles (DE) for epistemic uncertainty and simultaneous quantile regression (SQR) for aleatoric uncertainty. The experiments further validate CLEAR's generality beyond the PCS and CQR methods that presented the body of our paper. The results demonstrate that CLEAR remains effective across different modeling paradigms. We evaluate both the neural baselines individually and their integration through CLEAR, comparing against conformalized versions to assess the added value of our -ual--arrmeter calibration approach. Results are reported across all datasets with 95% nominal coverage, using up to 10 random seeds for robustness.
Table 8: DE-SQR PICP 95% prediction intervals, aggregated across 10 seeds. CLEAR consists of the deep ensemble (DE) and the simultaneous quantile regressor (SQR). DE-conformal is the conformalized (calibrated) deep ensemble and SQR-conformal the conformalized simultancous quantile regressor.
Dataset
CLEAR
DE
SQR
DE-conformal
SOR-conformal
ailerons
0,95 + 0.00
0.95 = 0.00
0.94 + 0.01
0,95 + 0.00
0.95 = 0.00
airfoil
0.96 + 0.02
0.95 = 0.02
0.98 + 0.01
0.95 + 0.02
0.96 - 0.01
allstate
0.95 + 0.01
0.95 + 0.01
0.91
0.95 + 0.01
0.94 + 0.01
ca.housing
0.95 = 0.01
0.95=0000 0.00
0.00
0.95 = 0.00
computer
0.95 + 0.01
0.95 = 0.01
0.94 +.94+0011
0.95 +
0.95 - 0.01
concrete
0.94 0,02
0,95=00 0.02
0
0.96
0.95 = 0.03
elevator
0.95 0.00
0.95 = 0.00
0.94 1.04
0.00
0.95 0.00
energy efficiency
0.96 + 0.01
0.95 = 0.02
0.99 + 0.01
0.01
0.96 = 0.03
insurance
0.96 0.01
0.95 = 0.01
0.96
0.06
0.96 = 0.02
kin8nm
0.95 0.01
0.95 =
0.98
0.01
0.95 = 0.01
miami housing
0.95 + 0.00
0.95 = 0.01
0.95
0.95 = 0.00
naval_propulsion
0.95 0.01
0.95
1.00
0.05
0.95 0.01
parkinsons
0.95 + 0.01
0.95 0.01
0.00
0.01
0.95 = 0.01
powerplant
0.95 = 0.01
0.00
0.95
0.00
0.95 0.00
qsar sulfur
0.95 + 0.01
0.01
0.93
0.96
0.95 0.01
0.95 + 0.01
0.95 0.01
0.00
0.95
0.95 0.01
superconductor
0.95 0.00
0.95 = 0.00
0.96 0.00
0.95 0.00
0.95 0.00
Table 9: DE-SQR NIW 95% prediction intervals, aggregated across 10 seeds. CLEAR consists of the deep ensemble (DE) and the simultaneous quantile regressor (SQR). DE-conformal is the conformalized (calibrated) deep ensemble and SQR-conformal the conformalized simultaneous quantile regressor. Values 2 100 or < 0.01 are presented in scientific notation with decimal place. Bold values are the minimum (best) for that dataset, while underlined values indicate the second-best result.
Dutuset
CLEAR
DE
SQR
DE-conformal
SQR-conformal
ailerons
0.201
0.1% 0.014
0.025
airfoil
0.185 0.023
0.223 + 0.023
0.349 - 0.025
0.226 = 0.023
0.311 = 0.025
allstate
0.333 + 0.061
0.380 +: 0.073
0,283 + 0.043
0,383 + 0.075
0.324 + 0.058
ca housing
0.357 9.1e-03
0.502 0.020
0.378 1 8.0c-03
0.502 0.020
0.374 4.0c-03
computer
0.087 3.2e-03
0.121 + 6.3c-03
8.9-003
0.122 6.1c-03
0.106 5.76-03
concrete
0.264 0.033
0.365 + 0.068
0.441 +.044]
0.369 + 0.067
0.392 = 0.069
elevator
0.117 1177::00 6.6c-03
0.174 - 0.011
0.014
0.174 0.011
0.117 + 8.8-03
energy efficiency
0,085 0.012
0.115 0.020
0.480
0,120 0.020
0.368 - 0.036
insurance
0.393 0.044
0.523 0.135
0.397 0.003
0.558 0.141
0.406 - 0.085
kinšnm
0.197 1: 8.2c-03
0,260 + 9,5e-03
0.246 8.7e-0
0.261 = 9.6e-03
0.208 1 8.8-00
miami housing
0.093 1: 2.9c-03
0.131 + 5.6e-03
0.101 2.8--33
0.131 5.7e-01 5.7e-03
0.100 + 1.8e-03
naval propulsion
1.5e-03 4.2e-04
5.4e-03 1.1e-03
3.7e-03
5.4e-03 1.1e-03
1,7e-03 1.5e-04
parkinsons
0.378 0.014
0.476 0.027
0.19
0.479 0.027
0.403 + 0.017
powerplant
0.186 6.7e-03
0.266 + 0.012
0.204 + 6.8e-03
0.267 + 0.013
0.202 + 7.3e-03
qsar
0.410 0.136
0.508 +: 0.178
0.402 + 0.135
0,509 1: 0.179
0.456 + 0.153
sulfur
0.105 8.5e-03
= 0.015
0.120 7.4-03
0.200 0.015
0.120 8.3e-03
superconductor
0.208 0.024
0.295 + 0.038
0.247 + 0.027
0.295 0.038
0.245 + 0.028
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Table 10: DE-SQR Quantile Loss at 95% prediction intervals, aggregated across 10 seeds. CLEAR consists of the deep ensemble (DE) and the simultaneous quantile regressor (SQR). DE-conformal is the conformalized (calibrated) deep ensemble and SQR-conformal is the conformalized simultaneous quantile regressor. Values 2 100 or < 0.01 are presented in scientific notation with decimal place. Bold values are the minimum (best) for that dataset, while underlined values indicate the second-best result.
Dataset
CLEAR
DE
SQR
DE-conformal
SQR conformal
silerons
9.6e-06 3.1e-07
1,3e-05 3.5e-07
9.6e-06 +: 4.0c-07
1.3e05 3.5e-07
9.Se-06 =: 3.7e-07
airfoil
0.097 + 6.8c-03
0.117 0.010
0,162 0.017
0.117 + 0.010
0.153 = 0.018
allstate
14c+02 9.329
1.5e022 8.112
1.5e+02 9.210
1.5c+02 8.106
1.c+002 3.368
ca.housing
3,0c403 + 63,719
3,9e403 + 1.2c+02
3.0c+03 54,590
3,9e403 + 1.2c+02
3.0c+03 53.118
computer
0.146 + 5.5e-03
0.183 6.8c-33
0,162 = 8.3c-03
0.183 6.8e-03
0.162 + 7,9c-03
concrete
0.348 0.049
0.427 + 0.056
0.467 0.031
0.428 0.057
0.458 = 0.051
elevator
1.2c-04 = 1 1.9c-06
1.6e-04 1:04-:2 3.5c-06
L.c-04 + 2.5c-06
166-04 +
3.5e-06 3.5e-06
1.1e-04 = 2.4e-66
energy efficiency
+ 9.5c-03
0.067
+: 3,9e+02 37.867
0.077 +
0.188 + 0.020
insurance
3,9e402 42.155
5,3e402 97,011
5.4c402 = 93.793
3.9c+02 45.372
kin8nm
4.1e-03 4.4e-05
5.2c-03 8.2c-05
4.5e-03 + 5.7e-05
5.2c433 8.2e-05
4.2e-03 + 7.1e-05
miami housing
4.5e+03 2.1c+02
5.4e403 1.7e+02
4.4c403 1.2c+02
5.4e403 1.7c402
4.4c+03 = 1.2e+02
naval propulsion
3.6e-05 9.4e-166
1.3e-04 + 2.7e-05
8.2e-05 + 2.9c-06
1.3e-04 +: 2.7c-05
4.4c-05 + 4.3e-06
parkinsons
0.298 0.014
0,343 9.8--3
0,291 5.76-0
0.343 9.8603
0.289 + 5.8c-03
powerplant
0.227 0.013
0.301 + 0.014
0.234 4: 0.012
0.301 0.014
0.234 + 0.013
quar
0.055 + 3.2c-03
0.064 3.5c-03
0.061 = 3.7c-03
0.064 3.5e-03
0.060 + 3.6e-03
sulfuu
L.0e-44
18e
1.8c
supercondactor
0,545 0.024
0.707 + 0.024
0.560 +: 0.017
0.707 0.024
0,559 = 0.018
Table 11: DE-SQR NCIW at 95% prediction intervals, aggregated across 10 seeds. CLEAR consists of the deep ensemble (DE) and the simultaneous quantile regressor (SQR). DE-conformal is the conformalized (calibrated) deep ensemble and SQR-conformal is the conformalized simultaneous quantile regressor. Values 2 100 or < 0.01 are presented in scientific notation with decimal place. Bold values are the minimum (best) for that dataset, while underlined values indicate the second-best result.
Dataset
CLEAR
DE
SQR
DF-conformal
SQR-conformal
uilerons
0.205 0.017
0.292 1: 0.027
0.205 0.017
0.291 0.026
0.204 1: 0.017
uirfoil
0.173 0.016
0.220 + 0.025
0:290
0.217 + 0.025
0.293 + 0.026
allstate
0.328 0.048
0.384 +: 0.062
0.342 4: 0.051
0.051
0.381 0.060
0.330 + 0.045
ca housing
0.354 8.1e-03
0.501 + 0.021
0.372 7.1e-03
0.501 0.021
+:2 7.8c-03
computer
0,088 1.7e-03
0.120 + 5.0e-03
0.106 + 6.5e-03
0.120 5.1e-03
0.106 + 7.1e-03
concrete
0.274 0.026
0.345 + 0.053
0.371 + 0,036
0,339 +: 0.051
0.376 +: 0.040
elevator
0.117 7.0c-03
0.172 0.010
0.117
0.172 0.010
0.116 9.4e-33
energy efficiency
0.079 7.4e-03
0.114
0.014
0.046
insurance
0.364 = 0.060
0.465 = 0.125
0.351 0.064
0.460 - 0.122
0,350 = 0.061
kin8nm
0.194 6.6e-03
0.255 + 5.6e-03
0.205 6.3e-03
0.254 = 5.3e-03
0.208 + 6.3e-03
miami housing
0.092 3.5e-03
0.129 7.8-00
0..6033
0.129 7.7c-03
0.100 4: 2.9-03
naval propulsion
1.5e-03
5.4c-03 1.26-3
1.7c033 2400044
5.4-03 1.2c-03
1.6c-03 1.7e-04
parkinsons
0.382 0.015
0,473 0.025
0.398 0.013
0.470 + 0.022
0.398 1 0.013
powerplant
0.190 1: 6.0e-03
0.272 9.56-03
0.202 - 8.1e-03
0.272 9,5e-03
0.202 + & 1e-003
qsar
0.407 0.134
0.150
0.1551
0.476
0.441 + 0.146
sulfur
0.104 6.2c-03
0.198 0.016
0.120 7.1e-03
0.197 0.015
0.120 1 7.2c003
superconductor
0.209 0.024
0.303 0003+0000
0.242 0.027
0.302 + 0.036
0.245 + 0.027
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Table 12:
DE-SQR Interval Score Loss at 95% prediction intervals, aggregated across 10 seeds. CLEAR consists of the deep ensemble (DE) and the simultaneous quantile regressor (SQR). DE- conformal is the conformalized (calibrated) deep ensemble and SQR-conforms, is the conformalized simultaneous quantile regressor. Values 2 100 or 0.01 are presented in scientific notation with decimal place. Bold values are the minimum (best) for that dataset, while underlined values indicate the second-bes result.
Dataset
CLEAR
DE
SQR
DE-conformal
SQR-conformal
allerons
7.7c-04 2.5c-05
1.0e-03 + 2.8c-05
7.7e-04 + 3.2e-05
1.0e-03 2.8e-05
7.6e-04 :: 2.9c-05
airfoil
7.761 0,546
9.368 0,826
12.937 1.386
9,370 0.813
12.279 1.418
allstate
I.Ie00 7.5e+02
1.2e+04 + 6.5e+02
1.2c404 + 7.4c+02
1.2e104 6.5e402
1.2e004 # 6.7e+02
ca.housing
2.ce055 5.1c+03
3.1e405 9.2c+03
2.4c405 + 4.4e+03
3.1e405 9.2e403
2.4c+05 = 4.2e+03
computer
11,669 0.436
14,633 +: 0.543
12,938 0.662
14,634 + 0.545
12.926 0.634
concrete
27,80 + 3.946
34.179 + 4,507
37,343 2.464
34,203 = 4.564
36.678 4.051
2.0e-04
1.9e-04
energy energy.efficicee _efficiency
4.154 0.758
5.369 + 0.965
17.719 1.871
5.387 0.974
15:020 1. 1,595
insurance
3.le+04 3.4e+03
4.2e404 7.8e+03
3.1e404 t: 3.0e+03
4.3e+04
7.5e+03
3.ee+04 + 3.6e+03
kinšnm
0.327 3.5e-03
0.358 4.6e-03
0.417 6.6e-03
0.340 + 5.7c-03
miami miami.housing
3.6c+05 = 1.7e+04
4,3e+05 +: 1.4e044
3.5e+05 +: 9,5e+03
4.3e+05 1.4e+04
3.5e+05 : 9.5e+03
naval proppllion
7.5e-04
2.3e-04
3.5e-03
parkinsons
23,860 +: 1.124
27.437 +: 0.787
23,297 0.456
27.449 :: 0.784
23.159 0.461
powerplant
18.145 1.005
24.110 1.141
18.714 0.990
24.106 + 1.140
78. 1.002
qear
0.255
5.152
4.875 0.294
5.154
4,800 0.286
sulfur
0.134 +: 8.1e-03
0.213 + 0.015
0.144 + 0.010
0.213 0.015
0.144 :: 0.010
superconductor
43.610 1.377
56,594 + 1.43
44.763 1.388
56.591 + 1.943
44.731 1.406
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F CLEAR WITH PCS AND CQR: RESULTS ON REAL-WORLD DATA
This appendix presents the detailed quantitative results from the benchmark experiments conducted on the 17 real-world datasets, complementing the summary findings in the main paper (Section 4.2) As a reminder. we evaluate three distinct configurations (variants) of our proposed CLEAR method alongside the baseline approaches (PCS, ALEATORIC, ALEATORIC-R). The differences among these 3 variants are considered base models. For each variant, CLEAR is applied on the already trained PCS and uncalibrated ALEATORIC-R. To recap these variants:
Variant (a): Employs a quantile PCS approach using a diverse pool of quantile estimators (QRF, QXGB, Expectile GAM for estimating the conditional medians based the bootstraps. The top-performing model type (k = 1) on the validation set is selected. 1) Based on the slected model type, we compute the epistemic component q (via emprical quantiles over the estimated medians from the b m bootstraps) and the median f (as emprical median over the estimated medians). 2) Using the same selected quantile model type, the aleatoric component qale is derived from a bootstrapped ALEATORIC-R model.
Variant (b): Similar to (a), but restricts the model pool for both PCS and the paired ALEATORIC-R exclusively to QXGB.
Variant (c): Uses standard PCS approach with mean-based regression models (e.g., Random Forest, XGBoost) for the epistemic component (via empirical quantiles over the estimated medians) and median prediction f (as empirical median over the estimated means), selecting the top model (k- 1). The aleatoric component derived from a bootstrapped ALEATORIC-R model using QRF.
The subsequent subsections present the comprehensive results for each variant (a, b. c). All ex- periments were run across 10 different random seeds, and the results on the test set are presented. The subsequent tables report the average performance metric across these 10 seeds, along with the standard deviation (+o), to indicate the variability associated with data splitting. For each variant, we provide tables detailing the metrics presented in Appendix D.5 all methods across all datasets at a 95% nominal coverage level. Lower values are preferable for all metrics except PICP, which should
ideally be close to the target of 0.95. Additionally, summary plots (Figures 3 and g visualize the relative NCIW and Quantile Loss performance normalized against the respective CLEAR variant baseline. Additionally, only for variant (c), we provide Table 29 with the values of À and 7 to provide some insights into the aleatoric/epistemic allocations of CLEAR. Finally, Figure 9 Appendix F.4) provides a direct comparison of the NCIW and Quantile Loss between the three CLEAR variants themselves, using variant (a) the reference baseline.
Appendix F.4 shows that the primary configuration of CLEAR presented in the main text, variant (a), leverages a diverse set of quantile models for robust epistemic uncertainty estimation and generally delivers the most consistent and superior performance. As detailed in Figure D in the appendix, CLEAR in variant (a) tends to marginally outperform the model in variant (b), which is restricted to QXGB. and variant (c), which utilizes standard mean-baser PCS models. For instance, in variant (a), QXGB was selected in approximately 64% of cases, QRF in 24%, and ExpectileGA.M in the remainder of cases. In variant (c), XGBoost was chosen about 70% of the time, with other models (excluding Ridge) sharing the rest. The importance of this dynamic model selection evident in datasets like naval 1_p -propul 1 i sion, on, where variants (b) and (c) can struggle due to their fixed model choices (QXGB and QRF for the alcatoric part, respectively). whereas variant (a) can adapt by selecting, for example, ExpectileGA.M. This highlights the stability advantage of CLEAR (a) and the benefit of PCS's s dynamic model choice, especially when the aleatoric component can also adapt. Nevertheless, the CLEAR framework overall demonstrates considerable robustness even with these alternative model choices. For instance, variant (b) still provides NCIW reductions of approximately 17.5% and 4.9% over its corresponding PCS and ALEATORIC-R baselines, respectively (Figure 7]. A similar advantage is observed for variant (c) (Figure 8), which reduces NCTW by about 7.9% against ALEATORIC-R, the bert-perfo variant of CQR that uses our novel residual-based technique.
SStrictly speaking, Expectile GAM estimates expectiles rather than quantiles. While it is not consistent estimator for quantiles in theory, mixing expectiles and quantiles may still be practical in applications.
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Detailed motivation behind variant (a). Overall, variant (a) achieves the best results (see Figure 9). Therefore, we recommend to use CLEAR variant (a). The experiments in variant (a) are fair, since each of the competing methods uses the same base model, which is selected per data set based on the RMSE on the validation data set. See Appendix F. the results.
Detailed motivation behind variant (b). Variant (b) the simplest to understand, easiest imple- ment, and computationally cheapest variant and provides maximal fairness, making it particularly scientifically sound. Each method simply uses QXGB as base model without any model selection step. Strictly speaking, variant (b) is the only variant where ALEATORIC and ALEATORTO TORTC-RRR are fully conformal, since variant (b) uses the calibration data set only for calibration, while variant (a) and (c) reuse the validation set used for model selection calibration data set. However, in practice, we observe this re-usage does not hurt calibration in any significant way (Table 14 and Table 24). See Appendix F.2 for the results of variant (b).
Detailed motivation behind variant (c). Variant (c) uses the same set of base models as suggested by the authors of PCS uncertainty quantification as Agarwal al. (2025). They conducted extensive experiments showing that PCS uncertainty quantification substantially outperforms popular conformal baselines such as split conformal regression (Lei al. 2018). Studentized conformal regression (Lei et al.) 2018), and Majority Vote (Gasparin & Ramdas (024) (by more than 20% on average in terms of interval width). Our experiments showing that CLEAR (a) outperforms CLEAR (c) (Figure 9) and CLEAR (c) outperforms PCS uncertainty quantification (c) (Appendix F.3), together with the experiments by |Agarwal et al. (2025). strongly suggests that CLEAR clearly outperforms these popular conformal baselines. See Appendix F33 for the results of variant (c).
Overall, CLEAR shows the strongest performance across all variants (a), (b) and (c), demonstrating CLEAR's s stability across different settings, data sets, and metrics.
Table 13: Dataset statistics where d represents the number of variables, represents the number of observations, followed by the minimum, maximum, and range values for y.
Dataset
T
Vrii
Yinas
yrazae
ailerons airfoil
13,750
40
-0.0036
0.00e+00
0.0036
1,503
104.2040
150.
35.9540
allstate
5,000
1037
200,0000
3.31e+04
3.29e+04
ca housing
20,640
8
1.50e+04
5.00e+005
4.85e+05
computer
8,192
21
0.00e+00
99.0000
99, 99,0000
conerete
1,030
2,3300
81.7500
79.4200
clevator
16,599
18
0.0120
0.0780
0.0660
energy_ccficieeee efficienccy
768
10
6,0100
43.1000
37.0 $7.0900
insurance
1,338
8
1121.8739
6.26c+04
6.15c+04
kin8nm
8,192
0.0632
1.4585
1.3953
miami miami-housing
13,932
28
7.20e+04
2.65e+06
2.58e406
Laval--propllee
11,934
24
0.0690
1.8320
1.7630
parkinsons
5,875
18
7,0000
54,9920
47,9920
powerplant
9,568
420.2600
495.7600
75,5000
qsar
5.742
-6.2400
11.0000
17.2400
sulfur
10,081
0.00c+00
1.0000
1.0000
superconductor
21,263
79
3.25c-04
185.0000
184.9997
49

---

Published as a conference paper at ICLR 2026
F.1
VARIANT (A)
Table 14: Variant (a) PICP at 95% prediction intervals, aggregated across 10 seeds.
Dataset
CLEAR
ALEATORIC
ALEATORIC.R
PCS-EPISTEMIC
Naive
71
ailerons
0,95 = 0.00
0.95 + 0.00
0.95 + 0.00
0,95 + 0.01
0.95 4 0.01
0.95 i 0.00
0.95 1: 0.00
airfoil
0.95 0.01
0.95 1 0.01
0.95 1 0.01
0.95 1 0.01
0.95 1 0.02
0.95 1 0.02
0.95 1 0.01
allstate
0.95 0.01
0.95 + 0.01
0,95 0.01
0,95 0.01
0.95 + 0.01
0.95 + 0,01
0.95 4:01
ca ho.hining
0.95 0.01
0,95 1. 0.01
0,95 1 0.01
0.95 0.01
0.95 1 0.01
0.95 1. 0.01
0.95 1 0.01
computer
0.95 0.01
0.97 +: 0.01
0,95 + 0.01
+.00
0.96 + 0.01
0.95 0.01
0.96 0.01
concrcte
0.95 = 0.02
0.96 + 0.02
0.95 + 0.02
0.95
0.94 0.03
0.95 0.01
0.95 + 0.01
elevator
0.95 0.00
0.95 0.01
0,95 +
0.05
0.95 0.01
0.95 + 0,00
0.95 + 0.00
cnergy_efficieney
0.95 = 0.02
0.96 0.02
0.95 + 0.02
0.00
0.95 0.02
0.96 0.01
0.96 + 0.01
insurance
0.95
0.01
0.95 0.02
0.96
+ 0.95 + 0.01
kin8nm
0.95 = 0.01
0.95 0.01
0.95 0.01
0.95
0.95
0.95 0.01
housing
0.95
0,95
0.01
0.01
+ 0.00
naval. propulsion
0.95 1 0.01
0.96 1. 0.01
0.95 0.01
0.95
0.01 0.95
0.95 1. 0.01
0.95 1 0.01
parkinsons
0.95
0,95 0.01
0.01
0.95
1
powerplant
0.95 L 0.01
0.95 1 0.00
0.95 0.01
0.95
1.00
0.95 1 0.01
0.95 1 0.011
qear
+
0,95
0.01
0.00
sulfur
0.95 0.01
0.95 1 0.01
0.95 1 0.01
0 0.00
0.95 1 0.01
0.95 1 0.01
0.95 1 0.011
superconductor
0.95 0.00
+
0.95 0.00
0.95 + 0.00
0.95 +
+
Table 15: Variant (a) NIW at 95% prediction intervals, aggregated across 10 seeds. Values 2 100 or < 0.01 are presented in scientific notation with 1 decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second-best result. Red values are more than 33% worse than the best result.
CLEARR
ALFATORIC
ALFATORIC-R
PCS-RPISTEMIC
Naise
71
airfoll
0.012
0.013
0.018
0.012
0.024
0.013
0.012
albtate
0.007
0.045
0.015
0.042
Jussing
0.008
0.306
0.006
0.011
0.020
0.006
computer
0.001 0.008
0.009
0.001
0:091 0.010
18,360 ::
0.115 0.007
0.010
0.091 0.007
concette
0.0025
0.486 0.078
0:365 0.079
0.349 + 0.017
0.008
0.022
0.023
elevator
0.156 0.009
0.144 1 0.007
0,149 1 0.007
0,198 1 0.012
0,144 1 0.008
0,156 0.009
0.150 1. 0.008
eiegy miley-elleekeey
IJ047 0.005
0.212 0.013
0,000 = 0.107
0.062 + 0.105
0.IK2 + 0.013
0.151 + 0.004
0.053 + 01004
insurince
0.200
0.032
0.332 = 0.049
0.520 = 0.156
0,565 +: 0.167
0 +: 0J070
0 + 0.077
0.295 =: IL044
kindom
0.360 0.012
0 0.070
0.374 0.014
0.373 + 0.014
0.997 + 0.017
0,359 + 0.012
0 199 0.11
mismi, honeing
0.085 0.002
0.105 0.001
0:006 - 0.004
0 + 0.00
0.126 0.005
0J184 + 0J002
GTKS T. 0007
naval propulsion
06 6.3e 06
6.2c 04 1 6.5e 06
6.le 01 1 6e 06
7.2e.04 1.3e (6
6.ee-l 1 7.0c-06
6.2e01 5.7e
6.1e 1. 5.6e 06
parkinsans
0.227 0,010
0.321 =: 0.013
0.349 = 0.014
0,312 :: 0.021
0,303 :: 0.022
0.228 +: 0.008
0.225 = 0.00S
powerplant
0.370 0.007
0,396 0.009
0.180 0,009
0,178 2: 0.007
0.153 1 0.896
0,173 2: 0.007
0,172 #: 0.007
0.363 = 121
0.486 =
0:386 = 0.136
0.358 + 0.151
0.430 + 1.300
0.962 + 121
0,362 + 0.121
sulfur
0.009 6,010
0.112 0.010
0,307 0,010
0,131 + 0.017
0,122 + 0.015
0.100 COBIT
0.108 + 0.011
superoonductor
0.196 0,022
0,233 0.025
0,196 0.023
0,248 1 0.028
0,329 1.0042
111 0002
6,194 1. 0.021
Table 16: Variant (a) Quantile Loss 95% prediction intervals, aggregated across 10 seeds. Values 2 100 or < 0.01 are presented in scientific notation with decimal place. Bold values (desirable) are the minimum for that dataset and metrie, while the underlined values indicate the second-best result. Red values are more than 33% worse than the best result.
Dataac
CLEAR
ALEATORIC
ALEATORIC&&
ICS-EPISTMIIC
Naive
T:
Allerons
9:2005
2.3e-07
28c
2.
1.0e 2.2 07
11e 3.7.e
9.2e-06
2.3e
9.2c-66 d:
diriil
0,109
D183
0,123
1.0019
0.109
0011
0.109
allsian
1.le+02 7,835
1.ee02
1.3e002
12ee02
17eet2
1Jee002 7.563
1 7.61
ci.bossing
4.8-00
73.283
1.2e405
1.2e-00
1
computer
0,338 0.340
0.566
0.359
BJ136
0.055
0.077
0,331 0.087
1.5
29c-06
24
1.7e
5.1ee
L.5
2.4g
1.c
0,631 1: 0.004
0.102
0,03
0.038
DUKG
0.005
0033
0.000
0.004 0.005
Insurince
3.5e+11
61.616
3.fe+02
3.eel2
54,512
S.ee02
1.0evt
4.4e+122
540
3.0e+02
06.243
3.5e 62.851
kināam
7.2e-3
1.7e-04
9.5e-00
14e00
032e-03:
1.7e-Ml
7.6e-66
9.2e-05
8:
2.
7:2e400
14
7.3e-13 1 1.7--1
momi
1.9e-02
1ee00
5.3ee03:1
34ee02
16000
S.le-lt
1/fe-12
1.ee40
3.0e-13 1: Ke-02
tural propulsion
15a-05
2 0.
1.
2.6-07
1.
0.
276-07
05 3.
1.5e-05
2.1e
LS 4: 2.3e-07
pakinsons
0.175 2: 0.010
0.202
00057
0.15 2: 0.012
0.224 D.022
0.251 0.014
0.155 2: 0.011
0.176 + 0.010
powerplane
0,212 J: 0.012
0.23 0011
0,220 0.011
0.221 0.012
0.231 1: 0,010
0213 2 0.012
0:013
qar
0.049: 2 0.003
[100 0.00
0,03 0,003
0.053 D.0G
00857 1 0,003
0J849 2 0J03
0,349 2 0,00S
siltur
De-133 1: 1.4e-04
2.le-00
1.3e04
23-401-2 1.te-04
2.3e40
1.9e
315 2.0e00
200-01 14
2.0e-15- 1: 1.4e-11
siperooduazor
0,490 3: 0,015
0.511
0014
0,538
1:5882000 0,023
0.608
0.033
0.361 0.006
0,491 1: 0.015
50

---

Published as a conference paper at ICLR 2026
Table 17: Variant (a) NCIW at 95% prediction intervals, aggregated across 10 seeds. Values > 100 or < 0.01 are presented in scientific notation with I decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second-best result. Red values are more than 33% worse than the best result.
Dutanet
CLEAAR
ALEATORIC
ALEATORIC-R
PCS-FRSTEMIC
Naive
71 -==
nikroos
0.195 = 0.017
0.209 # 0.019
0.196 = 0.017
0.217 = 0.021
0,220 0.020
0.195 # 0.017
6.195 =: E.017
antoil
0.205 0,006
0.356 0.016
0.216 0.012
0 2: 0.007
0,246 4: 0J019
0.200 :: 0J007
0,200 2: (1005
allannt
0.152 0.037
0.264 0.042
0.263 + 0.042
0.366 +.005
0.128 + 0.055
0.245 + 0.94
0.245 0.01
on Junssinng
0.336 0.008
0.760 0.009
0.348 0.010
0.354 + 0.012
0.440 + 4.0116
0.331 + 0.007
6,13 1OR7
computer
0.091 0,008
0.099 1 0.001
0.091 0.011
0,112 0.028
0.113 1 0.008
G0V2 1 0.009
0.891 1. 0.007
concrcte
0.246 0.033
0.421 0.07%
0.259 0.015
0.341 0.025
0.264 1: 0.037
0.238 :: 0.027
0.240 :: 0.025
elevator
0.155 0.010
0.343 0,008
0,148 0.008
0,177 2: 0.011
0,145 1 0.010
0.155 1: 0.010
0.159 2: 0.010
offieiency
0.846 0,005
0. 0.196 0.013
04000 0.005
0.057 0.044
0.040 + 0.896
0.050 + 0.005
0.051 + 0.004
insurance
0.303 0,029
0.284 0.049
0,043
0 1 0.178
0,654 0.0.00
0.283 1 0.049
6.271 0.046
kinSam
0,354 0,008
0,482 0.014
0,371 0.005
0,351 1: 0,009
0.396 0.013
0.355 0100
0.355 1: 0.008
0,003
0,004
0.003
0.006
0.002
0U044 DOME
propulsion
6,1c
6,1c
6.1e
7.1:
5.4-44
6.1:044
6.1c.44
parkinsans
0.225
0.052
0.013
0.007
powerplant
gar
sultur
0.008
0.013
0.011
0.011
0.001
0.025
0.021
0.023
0.035
0.194 0.021
0.194 0.021
Table 18: Variant (a) Interval Score Loss at 95% prediction intervals, aggregated across 10 seeds. Values 2 100 or 0.01 are presented in scientific notation with decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second- best result. Red values are more than 33% worse than the best result.
ALEATORIC
ALBATORIC
PUS-UPI
T:
7:4:404 L.8-05
23:45
2.2.00
3.000
1.9u-6
shtill
8.717 0,870
14,60
1.007
9.543
11.761 1.548
8.727
(192
5.730 0.592
alleate
9.1eat3 6.3ed82
L0ca04 6.3ea02
133e04
1.lew03
9.2x-03
bousing
2.2v00 4.8e005
3des05
24ei05
5.5e00
2
7.5elt
3.2i05
2.20400
2.2c-05
5.6e-03
computer
1.723
12.033
12255
1.2+100
15.300 0.795
1.598
11.761
177000 1.00
43.60
4.842
18484 4.351
26.165
1887
30/54 4.600
28366 1.976
36.465 2.995
clentor
0.012 +: 0.000
tuilt (UI0)
0.012 +: 0.000
0014 n.00
0.013 0.000
Tr 0100
0.013 0.000
afliciencyy
2,465 :: 0.347
8.199 0.67
3,671 2:0.011 0.416
3.053
0217
3.253 0.385
2.667 0.250
2.721 0.270
insrnee
2.8ce44 2: 4.9c483
29e.04
3.2e403
2.8ex04 4.4ev63
4e7004 8.1440
3.6e-04
3.1e.03
2.9c334
5.5
2.5e.04 d: 3.0e-03
kinSam
0.577 d: 0.014
0.260 0.013
0.616 0.021
0.607 0.008
0.665 0.000
0.577 à 0.033
0,577 4: 0,013
miami housing
3.le+05 d: 1.5e-04
L.ee00
25c+04
4.2e-05
2.7ex04
3.3ev05
1.3e104
6,5e-05
1.
3.lert
1.5ev04
3.1ei05 4: 1.5e-03
roral goipuleson
1.c403 + 1.76-00
12-di
2.1:405
133-48 +.2.
1'4cl 2.2c405
1.3c41 2.5x088
1:--00 177-05
133-13 1.6-48
pelinens
14.23 0.792
16.142 0.582
17.185 0.985
17.947 + 0.957
20083 1:130
14.762 0.82
14.250 0.796
powcrplat
16.93 0.937
18.211 0.87
17.577 0.584
17:715 0.965
18.501 0.315
17.864 1: 0.951
17127 03551
3.951 :: 0.230
4.755 0.219
4.187 0.231
4.30 0.217
4.555 1: 0.216
3950 :: 0.225
1,933 4: 0.224
sillur
0 2: 0,011
0.172 8011
0.187 0,005
0.166 0.015
0,290 0,016
0.161 = 0011
0,161 1: 0.011
esperondaztor
1,41
40811 1.157
1.345
68 3.583
90182
19.766
51

---

Published as a conference paper at ICLR 2026
F.2
VARIANT (B)
3.50
PCS PCS-EPISTIRR EPISTEMIC
ALEATORIC
ALEATORIC-R
Quantile Loss (Norma lized for CLEAR 1)
2975
3.00
20775
Relative increase
2.50
10055
2,00
-9096
Volues show average CLEAR
1.50
1.00
0.65
CLEAR
PCS-EPISTEMIC
ALEATORIC
ALEATORIC-R
PCS-EPISTEMIC
ALEATORIC
ALEATORIC-R
197%
17.8%
19.1%
4.9%
125%
3.00
Relative increase
NCTW (Normalized Sonmalized for CLEAR
75%
2.50
3056
2,00
Values show the average increase CLEAR
1.50
1.00
Powerplan
0.63
Ailerons
Qiar
Kin8Nin
Allsrate
Elevvtor
Insurance
Computer
Parkinsons
Suparcondbetor
Ainfoil
Concrete
Sulffor
Miani Hausing
Ca Housing
Energy Elfieiency
Naval Propulsion
Figure 7: Quantile loss and NCIW performance of different methods (CLEAR, PCS, ALEATORIC, ALEATORIC-R) for variant (b), over 10 seeds normalized relative to CLEAR (baseline 1.0). Lower values indicate better performance. The inset boxplot shows the % improvement relative to the CLEAR baseline =1o. Values inside each subplot represent the mean improvement across all datasets.
52

---

Published as a conference paper at ICLR 2026
Table 19: Variant (b) PICP at 95% prediction intervals, aggregated across 10 seeds.
Dataset
CLEAR
ALFATORIC
ALEATTORIC.R
PCS-EPISTEMIC
Naive
7
A
allerons
0,5 0.00
0.95 = 0.00
0,95 + 0.00
0,95 + 0.01
0.95 0.01
0.95 =
0,95 + 0.00
airfoil
0.95 = 0.01
0.95 + 0.01
0.95 0.01
0.95 + 0.01
0.95 + 0.02
0.95 + 0.02
0.95 + 0.01
allstaae
0.95 0.01
0.95 = 0.01
0.95 = 0.01
0
0.95 0.01
0.95 =
0,95 0.01
housing
0.95 0.01
0.95 + 0.01
0.95 0.01
0,95 +:01
0.95 4: 0.01
0.95 - 0.01
0.95 0.01
computer
0.95 = 0.01
0.95 = 0.01
0.95 = 0.01
0
0.95 - 0.01
0.95
0,95 0.01
concrete
0,95 0.02
0.96 + 0.02
0.95 0.02
0,95 +:01
0.94 1 0.03
0.95 0.01
0.95 0.01
elevalor
0.95 0.00
0.95 1. 0.00
0.95 0.01
0.95 1 0.00
0.95 1 0.01
0.95 1. 0.00
0.95 1 0.00
energy efliciency
0,95 0.02
0.96 + 0.02
0.95 0.02
0,96 0.01
0.95 0.02
0.96 : 0.01
U.96 1 0.01
insurance
0.95 0.01
0.96 1. 0.01
0,96 0.02
0,95 0.01
0.95 1 0.02
0.96 1. 0.01
0.96 1 0.01
kin8am
0.95 0.01
0.95 + 0.01
0.95 0.01
0,95
0.95 0.01
0.95 0.01
U.95 1 0.01
miami, housing
0.95 0.01
0.95 1. 0.00
0.95 0.01
0.95 0.00
0.95 1 0.01
0.95 1. 0,00
0.95 1. 0.00
naval propulsion
0.95 0.01
0.95 + 0.00
0,95 0.01
0.00
0.95 0.00
0.95 0.01
0.95 + 0.00
parkinsons
0.95 0.01
0.95 1. 0.01
0.95 1.01
0.00
1.00
0.95 1.01
0.95 1 0.01
powerplant
0.95 0.01
0.95 +: 0.00
0.01
0.05
0.95
0.95
0.95 +
qear
0.95 = 0.01
0.95 0.01
0.95 0.95 0.01
0.95
0.00
0.01
0.95 0.01
sulfur
0.95 0.01
0.95 + 0.01
0,95 0.01
0.00
0.95
0.95
0.95 0:01
superconductor
0.95 = 0.00
0.95 + 0.01
0,95 0.00
0.00
0.95 0.00
0.95 0.00
0.95 + 0.00
Table 20: Variant (b) NIW at 95% prediction intervals, aggregated across 10 seeds. Values 2 100 or < 0.01 are presented in scientific notation with I decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second-best result. Red values are more than 33% worse than the best result.
Datesel
CLEAR
ALFATORIC
ALFATORIC-R
PCS-FPISTEMIC
Naive
A=1
aileroas
0.195 2: 0.015
0,208 0.017
I.195 4: 0.017
0.155 00016
0.220 :: 0.821
0.193 0U015
0.193
0,016
sirfeel
0.307 0.812
0.367 0.018
0.215 + 0.18
0215==000 0.022
0.346 + 0.074
0.205 0.012
0.206
6.022
allstate
0.256 0.045
0,265 0.047
0,250 0.047
0.325 1 0.055
0.243 0.044
0.345 0,446
ca.bousing
0.339
0.760
0.350
0,354 00011
0.434 0,000
0.333
0.007
computer
0.990
0,143
0:122 0.009
0.101
0.
0.005
concrese
0.249
0.486
0.299
0251
clevatr
0.007
0.171 0.010
B.146
0.193 0003
0 0.010
0.136 0.008
0:00
energy efficienccy
0.047 6.005
0.212 0.015
0,050
0.007
0.022 0.006
0,062 1: 0.013
0.51 0.004
0,053 0:004
insenance
0.329 0.045
0.407 0.064
0.381 1 0305
0.444 0.179
0,578 2: 0
0.339 0.104
0.306 0.058
kinknm
0.360 1360000.012
0.490 0.030
0.574 + 0.014
0.373 0.014
0 + 0.017
IL359 6
0. 360
0:011
miami housing
0L085 0.002
0,105
0,001
0,086 1 0.004
0.098 0.003
0,126 1 0.006
0,884 0.002
0,0KS 0.00
naval propulsion
19c413 = 7.3c-15
0.222
D:K0
1.9-03 + 9.6405
8.e-11
4.1244
2.42408 # 1.4c-04
2.1c-33 5.1-41
2.Ne-115 1.32-04
markinons
0.281 0.009
0.483 0.005
0,502 + 0.012
0.325 0011
0.345 + 0.011
07722 0.000
1.281 0.009
powerplant
0.170 0.007
0.195
0,130
0.178 0.007
0,183
0.175 0.007
0.007
0.366
0.486
0.367
0.393
0.362 0.121
0.121
sulfar
0.108
0,110
0.011 0.228
0.129
0,122
0.106
0. 0.009
supercenducsor
0219 0.023
0,309
0 0.027
0,354 2: 0,942
0.233 0.025
0,024
Table 21: Variant (b) Quantile I Loss at 95% prediction intervals, aggregated across 10 seeds. Values 100 or 0.01 are presented in scientific notation with decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second-best result. Red values are more than 33% worse than the best result.
Datnact
CLEAR
ALEATORIC
ALFATORIC-
ICS-EPISTEMIC
Naiee:
ailcross
9.26006 d: 2.3e-07
10e-05 28:07
9.7606 2.8c
L0c =: 2.26
1.e-05 4 3.7e
9.2e-06 : 2.3 07
9.2e066 d: 2.3e-07
ainbll
0,109 :: 0.011
0.13 0.03
0,23 2: 0.015
0.117 0012
0.147 2 0,019
01109 2 0011
0,109 1 0,011
allsine
1iee60 = 7.318
L4ee02 à 8,321
1.3x00 8.46
1.er02 4: 8,557
17eel2 4: 107782
1.2e-02 4. 7,533
1.2e1 7.762
ca.bnssing
2Me-013 54,70
4.e-00 25.753
30eell 4: 73,243
1.2ee03 4. 93.396
40e-03 1: 1.2e-02
255e05 - 41.414
2 1: 70545
computer
0.159 A 0.007
0214 0.011
0:206 0.015
0.207 0,000
0:234 0/011
0.159 2 0J87
0.161 d: 0,005
controkc
0.336 2: 0.340
0.566 B.051
0.359 0,003
0.327 8U136
0.376 0.058
0:330 0.37
0.331 +: 0.087
elevsor
14e-04 1: 2.0e-06
1.5-04 3.0c 06
16c 04 3.1c
1.9e04 6.26 06
2. 1: 3.3.5e
L.e-04 : 2.0g 05
144-04 4: 2. 06
eneryy efficlncyy
0,831 1: 2.0001
0.102 8.008
0,033 0,005
0.038 0,003
0,041 0,005
10033 2 0.03
0,004 1: 0.003
Insurance
3.2e+03 :: 31.593
3.9e+02
36.964
3.7e+02
1: 35.
4.9e002 1: 88.647
4.4e+112
1.7e+02 1: 62,390
3.4es11 40.457
kmnša
0:2e-05
1.7e-04
9.50 6G
146
7.7e
0.
7.6e 6G
9.7e
0:
2.5e
1:2e-00 à 01
7.3e43 1.7e-8i
momi housing
19e-01
1.9e-02
Seek
5.2e-11
1.0e40
Slle+01
3/te-00
(Me+61 1.9e+12
3 1: 52-02
suval propulsion
5.46-15
1.9c-06
4.3--0
7.06055
2.7c.06
9.1c-05 1.5c-04
8:8c-05 1.5e600
6:0e-05
Te-600
pakinons
0.209 0.010
0.319
0.209
powerplate
0.212 0.012
0.229-
0,220
0213 0.012
0.213 0.012
gar
0.049 0.303
0.000
0,003
0.003
0.003
siltur
9.6e-95
2.2e-00
04
0.
3.5e
sipercondszorr
0,525 0:015
0.573
0.611
0,523
53

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Published as a conference paper at ICLR 2026
Table 22: Variant (b) NCIW at 95% prediction intervals, aggregated across 10 seeds. Values 2 100
or < 0.01 are presented in scientific notation with I decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second-best result. Red values are more than 33% worse than the best result.
ALEATORIC-R
Naive
ailerons
0.195 0.017
0.209 0.019
0.196 0.017
0.217 0.02)
0.220 1. 0.020
1.99110001
0.195
0.017
airioal
02203 0.866
0.356
0.016
0.216 : 0.122
0211 = 0.007
0.246 2: 0.019
0L200 0.077
0.200
0:006
allatae
02500 0.043
0.265
0.041
0.266 + 0.044
0.20 0.022
0.329 + 0.053
0.245
0.065
0.244
0.044
Ca co.housing
0.336 0.008
0,760 0,009
0,348 1 0.010
0.35 0.012
0,540 1 0.016
03331 0,07
0.338
0.007
computer
0JP98 0JM05
0.144 0:034
I1103 + 0.003
0.142 0J00S
0.111 + 0,005
0.101 0004
0.105
0.004
concrete
0.346 1-0.333
0:421 0.079
0.259 1: 0.035
0,241 0.025
0,264 1: 0.017
0:238 0.027
0.340
0.036
elevator
0.137
0.173
0.147
0.012
0.137 0.007
0,007
eaergy cffieiency
(U36
0.057
0,500
0.482
0JIP
Jonsing
0.084 0.003
0.106
BA0S
0.098 0.002
0.883
maval. propuliio
18-003 3.6e-08
0.195
0.011
03 03
8.0e
03
1.2
2.4e-0
2.1e 7.26
2.8
pa kinsonnn
0.284 0.010
0.73 0.016
11300 + 0LONN
0.324 0.00S
0.349 + 0.019
01285 0.011
0.265 0:11
powerplant
0.173 0.007
0.197 0.009
0.182 + 0.007
0.882 0.007
0.167 : 0.008
0.176 2:76::000 0.007
0.174 0.007
qsar
0.263 0.121
0.480 0.157
0,390
1: 0.130
0.393 0.131
0,423 1 0.142
0.362 1 0,120
0.362 0.130
sulfiar
0.106 0.D07
0,109 0.006
10105 0.066
0.225 = 0.00P
0.127 + 0.008
0.105 0.006
0.104 0.006
0,18 0.022
0.308 0.032
0.226 0.002
0.211 0.028
0,347 1 0.006
0223 0.024
0.216 0.024
Table 23: Variant (b) Interval Score Loss at 95% prediction intervals, aggregated across 10 seeds. Values 2 100 or < 0.01 are presented in scientific notation with decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second-best result. Red values are more than 33% worse than the best result.
Donset
CLEAR
ALBATORIC
ALEATDRIC-K
PUS-IPISTEMIC
Nave
T:
A=
1.Re-0
22:-05
2.3x00
2.0005
3.4-041
airial
8.717 0.570
87227
8,730 0.892
9:2e003 59ei02
6.7402
9.3e.03 6.2e-02
ca-bousing
2.2e-05 4.8c-03
39ex00
5.9e133
9.7e.00
2
5
1.2e-05 03
computer
37300 3.302
43,600
1857
26.665 1.97
0.011
0.013
T1011 0100
0.012 0,000
2,465 0.347
8.199
3.253
0.200
2.721 0.270
insunee
2.6e004
3.0c
2.5ew05 0.616
7.1e40d
2.7ev03
3.0ev04
2.5ex04 3.2c
0,577 0.014
0.760
0,020
0.577 0.13
0.577 0.013
mimii Junsing
3. 1o-05 1. 1:5p004
4 Loelt
2.5o404
4 4.3o405
3.70004
Scett 1.3444
6.5p=l15
3.3p-04
3.fe48
1.54404
3. + Sp-04
toral grppulion
4.4e-13 1.5e-04
0.301 0.006
S.cc40355 2.2
0.044 0.001
7.1c4
4.7o44
4.5o-1 14:004
pelinsere
16,697 0.794
25.89 0.692
18,270 1.122
15.931 0,600
11.220 1.011
1668 0.71
16,99 0.750
powerplazz
16,993 0.837
18:211 0.897
17.577 0.384
17:715 0.965
18.501 0.855
17.064 1: 0.951
17,627 0.954
3.556 2: 0.215
4300 0.219
4.193 0.227
4.270 n.230
4.562 0.217
3,96 0.222
3,946 d: 0,223
siltur
0.155 2: 0,005
0.174 11009
0,186 0.012
0.150 0.007
0.277 0,000
0.156 0.007
0.155 à 0.007
spperndaator
41.301 1,407
$1.836 1.88
45.360-1 1.383
48.809 3.097
71,323 3.307
41.747 +: 1.460
41331 1:37)
54

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Published as a conference paper at ICLR 2026
F.3
VARIANT (C)
2.50
PCS-EPISTEMIC
ALEATORIC
ALEATORIC-R
179%
Quantile Loss (Normalized for CLEAR 1)
5,8%
38.5%
18.0%
1976
129%
2,00
Rolative incressc
13055
75%
975
25%
1.50
-256
Values show average increase over CLEAR
1.00
0.65
CLEAR
PCS-EPISTEMIC
ALEATORIC
ALEATORIC-R
PCS PCS-EPISEE EPISTEMIC
ALEATORIC
ALEATORIC-R
2976
8.4%
43.0%
7.9%
2005
NCTW (Normalizod for CLEAR 1)
2.50
Relative incevsse
107%
2.00
-906
average incrose CLEAR
1.50
1.00
Powerpiant
Concreto
Insurance
Ailerons
Computer
KmnlNi
Quar
Ca Housing
Allstato
Aisfii
Parkinsons
Elevator
Sulfur
Superomdocttor
Miami Housing
Energy Efficioncy
Naval Propalsion
Figure 8: Quantile loss and NCIW performance of different methods (CLEAR, PCS, ALEATORIC. ALEATORIC-R) for variant (c), over 10 seeds normalized relative CLEAR (baseline 1.0). Lower values indicate better performance. The inset boxplot shows the % improvement relative to CLEAR baseline 1lo. Values inside each subplot represent the mean improvement across datasets.
55

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Published as a conference paper at ICLR 2026
Table 24: Variant (c) PICP at 95% prediction intervals, aggregated across 10 seeds.
Datasct
CLEAR
ALEATORIC
ALEATORIC-R
PCS-EPISTEMIC
Naive
T1 == 1
À =
ailerons
0.95 0.01
0,95 1 0.01
0.95 1 0.01
0.95 1. 0.01
0.95 1 0.01
0.95 1 0.01
0.95 1 0.01
nirfoil
0.96 0.01
0.95 + 0.02
0.96 0.01
0,95 + 0.02
0.95 + 0.02
0.96 + 0.01
0.96 + 0.00
allstate
0.95 0.01
0.95 1 0.01
0.95 0.01
0.95 1 0.01
0.95 1 0.01
0.95 1. 0.00
0.95 1 0.01
housing
0.95 0.01
0.95 + 0.00
0,95 0.01
0,95 +
0.95 + 0.01
0.95 +: 0.01
0.95 +
computer
0.95 0.01
0.95 1. 0.01
0.95 1. 0.91
0.95 0.01
0.95 1 0.01
0.95 1.001
0.95 1 0.01
concrete
0.96 0.01
0.96 +: 0.02
0 0.02
0.96
0.95 0.02
0.96 +: 0.01
0.96 + 0.01
elevator
0.95 0 0.01
0.95 0.00
0.95 + 0.00
0.95
0.95
0.95 0.01
0.95 + 0.01
energy eefegyeffiieenyy
0.95 0.01
0.96 + 0.01
0.96 0.01
0.02
0.95
0.97 0.01
0.96 + 0.01
insurance
0.95 = 0.01
0.95 + 0.01
0.00
0.95
0.96 0.01
0.96 + 0.01
0.95
0,95 0.01
0.05
0.95 0.01
+
miami housing
0.95 = 0.01
0.95 + 0.01
0.95 0.01
0.00
0.05
0.95 0.01
0.95 + 0.01
naval propulsion
0.95
0,95
0.01
0.00
parkinsons
0.95 1 0.01
0.95 1 0.01
0.95 0.01
0.95
0.95 0.01
0.95 1 0.01
0.95 1 0.01
powerplant
0.95
0.95 0.01
0.01
0.95 L 0.01
0.95 1 0.01
0.95 0.01
0.95
0.95
0.95 0.00
0.95 1 0.011
sulfur
0.95
0.95 +:
0,95 0.01
0.05
0.95
0.95
0.95 0.01
superconductor
0.95 0.00
0.95 1 0.00
0.95 1 0.00
0,95 0.00
0.95 1 0.00
0.95 1 0.00
0.95 1 0.00
Table 25: Variant (c) NIW at 95% prediction intervals, aggregated across 10 seeds. Values 2 100 or < 0.01 are presented in scientific notation with I decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second-bess result. Red values are more than 33% worse than the best result.
Dutsset
CLEAR
ALEATORIC
ALEATORIC
PCS EPISTUMIC
Nuive
'1
A-1
0.017
0,018
0.034
+-0.30
0.17
0,013
0.016
0.019
0.034
0.018
0.014
allbaate
0.052
0.065
0.083
callussing
0.010
0:005
0.013
0.011
0.016
0.008
0.311
0.008
compater
0.004
0:001
0.034
0.034
0.006
0.004
0.079
0.019
0.027
0.031
0,276 0.028
0.029
0.029
elevator
0.127 0.008
0.192 0.010
0.1 0,008
0.152 0.00s
0.144 0.010
0,122 1 0.007
6.127 0.007
energ mergy-efficreccy
0,043
0.007
0.138 0.007
0:051
0.005
0,045 1 0.077
0.019 1 0.005
0,047 1 00077
0.065 0.007
insurince
0.297
0.001
0,421 :: 0.049
0,59 = 0.050
0,395 -: 0.0%%
0,423 4: 0.0K3
0,440 1: 0.092
0.475 =: 0.109
kinSam
0.325
0.012
0,415 1: 0.018
0,330 1 0.014
0,529 1: 0.011
0,344 1: 0.012
0,323 1: 0.014
0.325 1: 01013
mismi Jonsing
0.088 0.001
0.112 0.002
0.113 0.088
0 + 0.802
0,123 + 0.10S
0JI78 + 0J001
0.075 + 0.002
naval prepulsion
66-05 3:3e 05
0.1e-03: 3.9e 05
03 053e-05
1.7e-00 1 06
1.e-00 1 1.
1áe-83 1 1.9
66-001 30
parkinsons
02100 0.007
0,331 0.011
0 2 0.011
0,360 2: 0.008
0,265 1: 0.010
0,251 1 0.008
6,245 1: EJNI
powerplant
0.157 = 0.007
0.175 1: 0,008
0,161 2: 0.088
0,169 1: 0.005
0,164 1: 0.807
0,158 1: 0.007
0,158 :: 0.007
qur
0.344 0.117
0.464 = 0.155
0,353 = 0.117
0,309 + 0.158
0,386 + 0.131
0,353 + 1200
0.345 + 0118
sulfur
0.311 (6,09)
0.124 = 0.010
0.109 = 0.014
0.134 + 0.009
0.114 + 0.013
0,105 + 0.009
0,104 = ELM99
spperonducttrr
0,195 0.000
0,250 1 0.025
0,210 : 0.029
0,235 0.002
0,308 0.035
0,193 1 0.020
0.191 :: 0.000
Table 26: Variant (c) Quantile Loss at 95% prediction intervals, aggregated across 10 seeds. Values > 100 or 0.01 are presented in scientific notation with decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second-best result. Red values
than 33% worse than the best result.
Danset
CLEAR
ALEATORIC
ALEATORIC R
PCS EPISTEMIC
Nake
'p
A-1
ailerm
9.6e-06 :: 2.9-07
1.e-dt 2.2-07
995-05 3.6e-07
1.le4t 2.2 2
1-3e-15 E: 1.5e-7
9:be-06 is 2.9-47
9. d: 2.9-07
sirial
0,107 0.014
1.140 0,000
0.120 0.120++0.00 0.011
0 0013
0.130 0.019
011.0 t 0014
0.108 +t 0.014
alletsic
1.2e4822 = 6,774
1.p-02 : 4.960
1.22402 + 10-401
144-02 + +.7337
[ecel2 + 10.679
1.2e002 + 7.509
1"5c400 7
ca. co.iinn
23e+03 : 92.560
3.2+000 L 71.387
3.2e-03
1.0e+02
29ex00 1: $7.120
3.5e-03 1.2e-02
Ne5e00 2: 88.558
2.8e 87.820
computer
0.137 0.011
01152 9003
0.160 0.012
01400 0.002
0.164 1 1.0133
0.1.7 2 0011
0.138 d: 0:011
concrese
0,357 1: 0.002
0.14 8044
0:381 0.009
0.3% 0.032
0.584 0.06
0,551 1: 0.032
dlcasor
1.3-04
3.2e-06
1.7e44
2.2-06
1.5--04
1.3o-06
1.4e-4 3.e-05
6e-04
1e-06
Tech 24
1.3-04 2.0e-06
enery efficirncy
0.029 0.007
0.08 0,005
0.014 0.010
0030 0.00
0.004 0.000
00.1 0000
0.090 00007
imarmes
4.2:001-:
90.250
3.9p02 31.576
4
Sc400
67.994
knšam
6.76-13
1.5c-04
2.5e
6.7
momii housing
3.7e-033
1.4e-02
4.20400
1.26+00
3 1.2e400
7:te-015 4.3e-02
4.le+00
2.4e+00 4
400-01
nowal_gmpuloonn
9,5e-t
powcrplare
0.313 0.012
IUIT
gar
0.045
0.002
0,57
0451
0,00
0,055 0.003
0.000
1.5c-03
6.3e--85
1.2c
2.7 03 1.fe 04
6.6. 05
siperoonduzorr
0.089
0.020
0.553 0,188
0.568 0,042
0.508 0.025
0,792 0305
(1.90 01020
0,489 0.021
56

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Published as a conference paper at ICLR 2026
Table 27: Variant (c) NCIW at 95% prediction intervals, aggregated across 10 seeds. Values 2 100 or < 0.01 are presented in scientific notation with I decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second-best result. Red values are more than 33% worse than the best result.
CLEAR
ALEATORIC
ALEATURIC-R
PCS-EPISTEMIC
T1
0,014
albaae
0.365
0.041
0.070
0.051
hossing
0.010
0:014
0.010
0.009
0.018
0,009
compater
0.004
0:007
0.004
0.005
0.005
0.0044
concrete
0.250 0.002
0.303 0.054
0.263 0.025
0,257 6.031
0,265 0.027
0.258 0.033
1: 0033
elevztor
0.126 0.008
0. 1.0 0.008
0.138 0.008
1: 0.011
0.14 0.010
1127 0007
0.127 1: 0.007
energy eiticteccy
0,041 0.0906
0.116 0,015
b,0 :: 0,003
0.043 1: 0.005
0.819 1: 0.00
0,042 1: 0.005
0.042 1: (1005
invernnce
0.345 0.0778
0.413 + 0.043
DA24 0.074
0.343 + 8.087
0,385 + +01
367 + 180
0.417 + 0.137
kinSom
I,323 01,800
0.420 0.018
0.327 = 0J0X8
0.331 + 0.010
0.341 + 0.012
0.325 + 0.00
0.324 + 0.009
miami housing
0.088 0,002
0.113 0.004
0.111 0.007
0,094 1: 0.003
0.120 1 0.007
0.077 1 0.001
6.07 2: TUN0
macal grepulsion
6e-03
5.0e-85
4.0c-03 2 2.4e-04
66--00 2 le-05
1.7c000 2 4. 7--66
188-00 + 5.8c00
LAc-0 $2
fe-11 +: 6-26-05
partinsons
0,250 0,012
0.306 0.017
0.279 1: 0.013
0.259 1: 0,009
0.254 1: 0.013
0.251 1: 0.011
0.299 1: 0.011
powerptant
0.361 = 00,007
0.176 0.009
0.163 0.038
0,171 0.008
0.166 + 0.807
0.161 + 0.007
0.161 0.007
qua
0.348
0.118
0.465 =: 0.155
0.361 = 0.134
0,407 =: 0.142
0,395 + 0.134
0,359 + 0.124
0,354 = 0.122
unlfor
0.109 0,008
0.124 0.008
0,108 0.006
0,124 2: 0,008
0,115 1: 0.007
0.104 :: 0.007
6,102 2: 0.007
0,019
0.025
0.027
0.025
0.029
0.019
0.019
Table 28: Variant (c) Interval Score Loss at 95% prediction intervals, aggregated across 10 seeds. Values 100 or < 0.01 are presented in scientific notation with decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second-best result. Red values are more than 33% worse than the best result.
CLEAR
ALFATTORIC
HATORIICR
PUS-EMITPMIC
Th
2.3-00
22:055
2.0-05
2.2::05
3.0e05
7.7e:04
7.7c004
Aitall
8.569 1,099
0.499
1.548
1004
10.361
1,999
3.675
11116
8,524
1,105
allsuue
9.2cv03 5.5c-02
4.0e02
L.c-04
1.fe00
L.e-04
3.5e-02
9.2e100
5.9e+00
9.3e-03
sa.hnsing
2.2e-05
7.4e-05
25ex05
5.7el00
15e-05
3.1e+05
25e-05
9.7e-05
7eleet
2.2e-05
7.0e-03
computer
10.978 0,911
12.134 1.001
0.945
01996
13.030 1,55
10.98 0.775
11,030 0.366
cunrretc
26.928 2.529
29.055
3,544
30,449 4,603
26855
+ 2.535
50.720 1 4.505
26882 2: 2.568
26,962 2.581
cleutor
0,010 0.000
0.014 0.000
0.022 0.000
0.011 0.000
0.013 0.000
0.010 :: 0.000
0,010 2: 0,000
energy efficlecyy
2,345 1: 0.554
5.407 = 0.418
2.759 0.822
2.397 0,573
2.756 0.522
2.392 : 0.564
2,305 1: 0.595
Insurince
3.4e004 1: 6:4e-03
S.1e104
2.7
3.3e:05 1 4.3e105
3.5er0M
5.22+00
3.3e-04
1.3e-05
3.4e134
6.2ev10
3.5e-01 3.4e-
kinšam
0,534 1: 0,014
0.673 0/221
0.544 1 0,015
0.552
0.017
0,591 0.022
0.336
1 0.133
0,535 à 0.015
momii housing
2.9e-15
1.1e-04
5.3e+t0
9.5+00
3.9-0h
1.le+04
9.
6. 1: 3:4e-04
3.2e+th
3.2e-05 3: 199--5
sural propulsion
3.2c-113
9.4c-85
1.5e40
1.7--04
39:-03
1.3c-4
3.4c43 2 0.66505
4.6c-05 9.6e-25
3.2e-00
9.33-05
3 + 1.3e-04
pelinsoni
15.149 0.515
19,89 0.576
17.316 0.634
15:312 0.459
17.596 0.623
15,150 0.499
15.161 0.537
powerplane
16.243 0,985
17.062 1.128
16,673 0.987
16.86 :: 1.029
16.874 1: 0.975
16,231 0.984
16.231 a: 0:983
qaz
3,877 0.176
4.21 0.225
4,086 0,193
4.334 0.125
4,390 0.205
3.927 1: 0.146
3,892 1: 0.151
silfur
0,143 0.005
0.156 8.010
0.213 0.013
0.146 0.005
0.228 0.013
0.143 0.005
0,185 1. 00005
sipercandazoor
39,104 1,404
44,230 1.428
43.479 3,358
45:436 2.009
63337 1: 2.924
39.204
1.613
19.119 1.442
Table 29: Variant (c) CLEAR calibration parameters À and 71 for 95% prediction intervals across 10 seeds. Using all available variables. Showing median [min:max] values.
Dataset
?/1
ailerons
0.89 [0. (0.17:4.47]
0.98 [0.87:1.58]
airfoil
0.76 [0.23:5.25]
1.70 [0.25:4.30]
allstate
0.21 [0.00:14.06]
1.14 [0.16:1.93]
ca_housing
1.90 [1.14:5.08,
0.70 [0.33:0.93]
computer
2.21 [1.63:8.87]
0.74 [0.21:0.96]
concrete
1.33 |0.06:
1.65 [0.01:16.73]
clevator
0.94 [0.67:1.45
1.04 (0.82:1.22]
energy_efii efffcieecc
0.12 [0.01:100.00]
7.96 [0.02:24.25]
insurance
5.07 [0.14:100.00]
0.38 [0.02:27.45]
kin8nm
1.87 [1. (1.:8.2))
0.76 [0.58:0.83]
miami housing
7.06 14.84:19.45]
0.23 [0.09:0.33]
naval propulsion
16.16 [12.48:21.17]
0.63 [0.53:0.76]
parkinsons
1.16 [0.72:8.42]
1.26 [0.22:1.84]
powerplant
1,05 [0.73:3.26]
1.05 [0.51:1.36]
qsar
0.51 [0.32:1.70]
1.45 [0.96:2.37]
sulfur
2.41 [1.53:7.16]
0.71 [0.29:0.91]
superconductor
0.91 [0.34:1.39]
1.07 [0.80:1.34]
57

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Published as a conference paper at ICLR 2026
F.4
COMPARING VARIANTS OF CLEAR
CLEAR
CLEAR (6)
Quantile Loss (Nomalized for CLEAR (a) 1)
3005
2976
3.00
2075
1.0%
2.50
477
2,00
-91%
Values show the average increase ower CLEAR (o)
1.50
1.00
0.65
CLEAR (a)
CLEAR (b)
CLEAR (c)
3.00
CLEAR
CLEAR (c)
2976
NCTW (Normalized for CLEAR (a) 1)
2007
Relative incresse
1.0%%
10075
-976
Values the average increaze over CLEAR
1.50
1.00
0.65
Emerg Efficiency
Airfol
Elevator
Powerplant
KinSNin
Sulftr
Co Housing
Minmi Housing
Quar
Concrote
Ailarons
Computer
Allsat
Suporcundnette
Insuranee
Parkinsons
Naval Propulsion
Figure 9: Quantile loss and NCIW performance of different variants of CLEAR (a, b and c) over 10 seeds normalized relative to CLEAR (a) (baseline = 1.0). Lower values indicate better performance. The inset boxplot shows the % improvement relative to the CLEAR (a) baseline #lo. Values inside each subplot represent the mean improvement across all datasets.
F.5 RUNTIME
The grid search for finding the optimal parameter is extremely fast and negligible compared to the baselines, despite using grid with 4000 points, which is finer and larger than necessary. In Tables 30 given real-world dataset, computed on a machine with an Intel Core" 19- 13900KF CPU (with maximum 20+ threads used in parallel). For experiments involving SQR and DEs, we had access NVIDIA GeForce RTX 4090 GPU.
to 32 provide the average training time for each variant
58

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Published as a conference paper at ICLR 2026
All results computed using 32 GB of memory. The times are provided in seconds and have been averaged over 10 seeds. For CLEAR, the required computation the calibration time (computation of À and Y, denoted as Grid Search in the tables). Only the runtimes for the methods included in the paper are provided. The experiments were run exactly as described the paper (particularly for the 100 bootstraps).
As stated previously, CLEAR is highly modular, allowing the modules and their components (such the choice of base models within the PCS module) be replaced modified adhere to computational budget limitations. For example, if the dataset is very large, base models that scale well with the training data size can be used (such as deep learning models, trained via stochastic gradient descent). Alternatively, if training each model is expensive, one might want to use fewer than 100 different bootstraps per dataset, or maybe even use techniques (such as Monte-Carlo Dropout) to obtain an ensemble from single trained model (Kendall & Gal 2017 Gal & Ghahramani 2016 Wen et al. 2020 Havasi al. 2021 Rossellini et 2024 Chan et al. 2025 Agarwal et al. 2025).
Note that ALEATORIC-R cannot be computed without first computing PCS, it utilizes the residuals from PCS. In practice, the total computational costs ALEATORIC-R would be equal to the costs of PCS plus the additional costs of the residual approach. Thus, the minimal total computational costs necessary to obtain CLEAR results are equal to the minimal total computational costs required to obtain ALEATORIC-R results (up to few seconds of the grid search).
Remark F.1 (Parallelization and Scalability). Every step the entire CLEAR pipeline parallelizable and distributable. The most expensive part is fitting multiple models to multiple bootstraps of the data for PCS. However, these models can be trained perfectly parallel on different distributed nodes, as they do not have to communicate with each other. Similar parallelization (distributed) is also possible for ALEATORIC-R, where multiple models are fitted independently. The computational costs of CLEAR's ' calibration step distributing the exploration of the grid across multiple servers. The computations necessary for each grid point fully vectorized and are highly suited for GPUs (in the case of a large calibration dataset). In other words, regardless whether the user has access to many weak CPU servers, a powerful GPU, or multiple powerful GPU servers, the calibration step efficiently utilize all these different infrastructures (when tuning the implementation accordingly). For all these reasons, CLEAR highly scalable.
albeit negligible, could be further reduced by
Table 30: Variant (a) average runtime (over 10 seeds) seconds for base components and the CLEAR's grid search. The grid search includes any other overhead for CLEAR.
Dataset
PCS
ALEATORIC-R
Grid-Search
Total
ailerons
13.46
6.80
0,27
20.53
airfoil
3.10
1.92
0.17
5.19
allstate
1124.47
22.93
0.19
1147.59
ca_ houuhusing
8.51
7.79
0.54
16.84
computer
61.92
76.09
0.16
138.16
concrete
1.10
2.16
0.17
3.43
elevator
18.79
122.16
0.60
141.54
energy efficiency
0.81
1.65
0.16
2.62
insurance
3.48
17.23
0.11
20.82
kin8nm
3.25
5.43
0.21
8.89
miami haaninhuuing
7.31
7.66
0.72
15.68
naval propuusooniiion
19.83
81.32
0.29
101.44
parkinsons
39.53
60.99
0.14
100.66
powerplant
3.08
4.92
0.21
8.22
qsar
348.59
10.00
0.20
358.78
sulfur
21.55
16.56
0.21
38.32
superconductor
561.84
795.45
12.70
1369.99
Total
2240.60
1241.06
17.06
3498.71
59

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Published as a conference paper at ICLR 2026
Table 31: Variant (b) average runtime (over 10 seeds) seconds for base components and the CLEAR's grid search. The grid search includes any other overhead for CLEAR.
Dataset
PCS
ALEATORIC-R
Grid-Search
Total
ailerons
4.51
6,89
0.26
11.66
airfoil
0.78
1.64
0.16
2.59
allstate
27.73
14.22
0.20
42.15
ca ho.housing
5.42
7.69
0.39
13.50
computer
3.46
5.97
0.21
9.65
concrete
0.86
1.82
0.17
2.85
elevator
4.43
6.71
0.79
11.93
energy-efficiency
0.68
1.40
0.17
2.25
insurance
0.80
1.65
0.17
2.62
kin8nm
2.61
4.96
0.21
7.78
miami_housing
4.61
7.70
0.70
13.01
naval propuuloon
3.72
6.38
0.55
10.64
parkinsons
2.71
5.37
0.20
8.28
powerplant
2.73
4.32
0.22
7.26
qsar
5.00
9.67
0.20
14.86
sulfur
2.89
4.64
0.23
7.76
superconductor
13.81
22.74
1.00
37.55
Total
86.77
113.76
5.81
206.34
Table 32: Variant (c) average runtime (over 10 seeds) seconds for base components and the CLEAR's grid search. The grid search includes any other overhead for CLEAR.
Dataset
PCS
ALEATORIC-R
Grid-Search
Total
ailerons
10.20
104.32
0.65
115.17
airfoil
0.88
17.82
0.11
18.81
allstate
651.02
125.77
0.14
776.93
ca_housing
3.63
130.87
4.16
138.66
computer
4.55
65.28
0.16
69.98
concrete
1.16
17.49
0.11
18.76
elevator
3.25
98.73
1.97
103.95
energy efficciefy
0.95
15,42
0.10
16.47
insurance
0.90
19.80
0.11
20.81
kin8nm
16.89
69.17
0.16
86.22
miami housing
5.34
123.32
0.92
129.58
naval propulsion
1.14
115.41
0.74
117.28
parkinsons
3.71
53.43
0.14
57.27
powerplant
2.37
60.99
0.17
63.52
qsar
21.48
93,84
0.14
115,46
sulfur
2.43
67.41
0.17
70.01
superconductor
60.63
644.13
8.10
712.86
Total
790.52
1823.18
18.05
2631.75
60

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Published as a conference paper at ICLR 2026
G CONFORMALIZED CLEAR WITH PCS AND CQR: RESULTS ON REAL- WORLD DATA
This section presents results from conformalized experimental configuration, where split the 20% validation set into separate 10% validation and 10% calibration sets. This approach provides stronger finite-sample distribution-free marginal coverage guarantees following conformal prediction principles, as the calibration set remains completely unseen during model selection and hyperparameter optimization. While the conformalized approach may sacrifice some performance due to reduced data availability for validation, it offers theoretical rigor by ensuring the conformal calibration step operates on the held-out data. Similar more stringent experimental setting while maintaining its advantages over baseline methods across all three variants (a), (b), and (c).
standard results, CLEAR adapts this
G.1 VARIANT (A) CONFORMALIZED
Table 33: Conformalized Variant (a) PICP at 95% prediction intervals, aggregated across 10 seeds. Methods with suffix '-c' denote conformalized variants obtained using the validation divided into two parts, one for validation and one for calibration.
Dataset
CLEAR-c
PCS-EPISTEMIC.
ALEATORIC-R
CLEAR
ailerons airfoil
0.95 + 0.01
0.95 0.01
0.95 + 0.01
0.95 0.00
0.96 + 0.01
0.95 0.02
0.95 0.02
0.95 - 0.01
allstate
0.96 0.01
0.94 0.01
0.95
0.95 0.01
ca_housing
0.95 0.01
0.95 0.01
0.95
0.95 0.01
computer
0.01
0.95
0.96
0.95 0.01
concrete
0.95 0.02
0.94
elevator
0.95
0.95
0.95
0.00
energy -neegyeefficiency
0.97
0.95
0.06
0
insurance
0.96
0.95
0.06
0.95
kin8nm
0.95
0.95
0.95 0.01
miami_housing
0.95 0.01
0.95 0.01
0.95 0.01
0.95 0.01
naval propulsion
+
0.95 0.01
0.95
0.95 0.01
parkinsons
0.95 + 0.01
0.95 0.01
0.95
4.95
powerplant
0.95 0.01
0.95
0.95
0.95
qsar
0.95 + 0.01
0.95
0.95
0.05
sulfur
0.95 0.01
0.96 0.01
0.01
0.95
superconductor
0.95 0.01
0.95 + 0.01
+.01
0.95 0.00
61

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Published as a conference paper at ICLR 2026
Table 34: Conformalized Variant (a) NIW at 95% prediction intervals, aggregated across 10 seeds. Methods with suffix '-c' denote conformalized variants obtained using the validation set divided into two parts, one for validation and one for calibration. Values 2 100 or < 0.01 are presented in scientific notation with I decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second-best result. Red values are more than 33% % worse than the best result.
Dataset
CLEAR-c
PCS-EPISTEMIC
ALEATORIC-R
CLEAR
ailerons airfoil
0.193 1 0.016
0,217 0.017
0.195 1 0.018
0.193 1 0.016
0.215 1 0.019
0.217 1 0.022
0.227 1 0.031
0.207 1 0.012
allstate
0.267 1 0.041
0.263 0.060
0.259 1 0.046
0.260 1 0.037
housiiig
0.339 0.006
0.355
0.349 1 0.008
0.339 0.008
computer
0.091 = 0.006
8.020 11.958
0.094 = 0.016
0.091 = 0.008
concrete
0.272 = 0.030
0,260 + 0.050
0.266 = 0,038
0.249 = 0.025
clevator
0.157 = 0.009
0.177 0.014
0.149 = 0.008
0.156 = 0.009
energy energy_efficieney
0.053 0.009
0.058 0.009
0,053 0.009
0.047 = 0.005
insurance
0.338 + 0.063
0,535 1 0.315
0.379 = 0.089
0.309 = 0.032
kin8nm
0.362 + 0.014
0.373 0.013
0.370 + 0.011
0.360 + 0.012
miami niami_housing
0.085 + 0.003
0.097 + 0.002
0.087 + 0.005
0.085 + 0.002
naval propulsion
6.2c-04 + 5.9e-06
7.1c-04 1.5e-05
6.0e-04 9.6e-06
6. + 6.3e-06
parkinsons
0.227 + 0.017
0.315 0.034
0,253 + 0.020
0.227 + 0.010
powerplant
0.171 + 0.010
0.179 0.008
0.183 + 0.008
0.170 1 0.007
qsar
0.369 1 0.123
0.388 1 0.132
0.381 1 0.125
0.363 1 0.121
sulfur
0.112 + 0.009
0.136 0.014
0.108 0.010
0.109 1 0.010
superconductor
0.197 0.022
0,249 0.032
0.195 0.023
0.196 1 0.022
Table 35: Conformalized Variant (a) Quantile Loss 95% prediction intervals, aggregated across 10 seeds. Methods with suffix '-c' denote conformalized variants obtained using the validation divided into two parts, one for validation and one for calibration. Values 2 100 or 0.01 are presented in scientific notation with decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second-best result. Red values are more than 33% worse than the best result.
Datasel
CLEAR-c
PCS-EPISTEMIC
ALEATORIC-R-c
CLEAR
ailerons sirfoil
9.2e-06 0.3
1.0e-05 2.1e-07
9.7e-06 # 2.7e-07
9.2e-06
0.109 + 2.3e-07
0.110
0.119 0.012
0.126 + 0.015
allstate
1.le+02 + 8,530
1.3e+02 = 8.381
1,3++02 7,926
1.le+02 = 7.835
ca_hous houhhusing
2.8e+03 56.920
3.2e403 + 91.466
3.0e+033 + 73.272
2.8c+03 59.700
computer
0.142 + 0.013
9.961 + 14.797
0.157 = 0.021
0.147 = 0.022
concrete
+
0.337 + 0.047
0.355 + 0.047
0.338 + 0.040
elevator
1.Se-04 2.0e-06
1.7e-44 1 6.3e-06
1.5c-04 +:: 2.5e-06
1.5e-04 +: 2.0c-06
energy effficcnfcc
0.033 0.005
0.038 + 0.003
0.034 = 0.005
0.031 0.004
insurance
3.5e+02 55.800
5,8c402 1.7e+02
3.8c+02 49.983
3,5c+02 61.616
kin8nm
7.2e-03 = 1.7e-04
7.6e-03 + 1.1e-04
7.7e-03 + 2.7e-04
7.2e-03 = 1.7e-04
miami housing
3.9c+03 + 2.1e402
4,2c+03 + 1.7c+02
5.2c+03 + 3.4e+02
3,9c+03 = 1.9c+02
daval-propplsion
1.5e-05 + 1.9e-07
1.7e-05 + 2.9e-07
1.5e-05 = 2.7e-07
1.5e-05 + 2.1e-07
parkinsons
0.179 1 0.008
0,226 + 0,016
0,216 = 0.012
0.178 + 0.010
powerplant
0.213 + 0.011
0.222 + 0.011
0.220 + 0.011
0.212 = 0.012
qsar
0.050 + 0.003
0,053 1 0.003
0.052 1 0.003
0.049 + 0.003
sulfur
2.0e-03 1.7e-04
2.4e-03 2.2e-04
2.3e-03 + 1.6e-04
2.0e-03 1.4e-04
superconductor
0.490
0.609 + 0.021
0.538 + 0.023
0.490 + 0.018
62

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Published as a conference paper at ICLR 2026
Table 36: Conformalized Variant (a) NCIW 95% prediction intervals, aggregated across 10 seeds. Methods with suffix '-c' denote conformalized variants obtained using the validation set divided into two parts, one for validation and one for calibration. Values 2 100 or < 0.01 are presented in scientific notation with I decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second-best result. Red values are more than 33% % worse than the best result.
Dataset
CLEAR-c
PCS-EPISTEMIC
ALEATORIC-R
CLEAR
ailerons airfoil
0.195 1 0.018
0,217 0.021
0.196 1 0.017
0.195 1 0.017
0.203 1 0.007
0.211 1 0.007
0.216 1 0.011
0.203 1 0.006
allstate
0.255 1 0.040
0.272 1 0.057
0.263 1 0.043
0.252 1 0.037
housoiinng
0.336 0.006
0.354 0.012
0.348 0.011
0.336 0.008
computer
0.091 - 0.008
0.108 0.016
0.090 = 0.008
0.091 = 0.008
concrete
0.262 = 0.038
0.259 - 0.047
0.264 = 0,040
0.246 = 0.033
clevator
0.155 = 0.010
0.177 0.011
0.148 = 0.008
0.155 = 0.010
energy energy_efficieney
0.046 0.006
0.057 0.004
0,050 = 0.005
0.046 = 0.005
insurance
0.299 + 0.037
0.480 0.181
0.312 = 0.036
0.303 = 0.029
kin8nm
0.355 + 0.008
0.371 + 0.009
0.371 + 0.009
0.354 + 0.008
miami niami_housing
0.084 + 0.004
0.098 + 0.002
0.085 + 0.003
0.084 + 0.003
naval propulsion
6.Tc-04 + 7.3e-06
7.1c-04 1.5e-05
6.0e-04 5.4e-06
6. + 7.9e-06
parkinsons
0.226 0.012
0,316 0.032
0.248 + 0.007
0.225 + 0.012
powerplant
0.173 1 0.008
0.182 0.007
0.182 + 0.007
0.173 + 0.007
qsar
0.361 + 0.120
0.389 1 0.134
0.389 1 0.131
0.360 1 0.119
sulfur
0.110 + 0.007
0.127 0.008
0.105 0.005
0.109 1 0.010
superconductor
0.195 0.021
0.247 0.028
0.194 0.021
0.195 1 0.021
Table 37: Conformalized Variant (a) Interval Score Loss 95% prediction intervals, aggregated
across 10 seeds. Methods with suffix '-c' denote conformalized variants obtained using the validation divided into two parts, one for validation and one for calibration. Values 2 100 or 0.01 are presented in scientific notation with decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second-best result. Red values are more than 33% worse than the best result.
Datasel
CLEAR-c
PCS-EPISTEMIC
ALEATORIC-R-c
CLEAR
ailerons
7.4e-04 1.8c-05
8.1e-04 1.7e-05
7,8e-04 = 2.2e-05
7.4e-04 1.8e-05
sirfoil
8.794 1.047
9.523 + 0.959
10.075 1.227
8.717 + 0.870
allstate
9.2e+03 + 6.8e+02
1.0++04 1 6.7e+02
1.0+004 = 6.3e+02
9.1e+03 + 6.3e+02
ca_houe houhhusing
2.2e+05 +: 4.6e+03
2.6c+05 + 7.3c+03
2.4++05 + 5.9c+03
2.2c+05 + 4.8c+03
computer
11.388 + 1.051
8.0e+02 +: 1.2e+03
12.535 + 1.698
11.741 + 1.723
concrete
27.347 3.094
26.963 3.739
28.402 3.793
27.020 3.202
elevator
0.012 1 0.000
0.014 +:: 0.001
0.012 = 0.000
0.012 = 0.000
energy .ffficccfc
2.609 0.417
3,019 + 0.237
2.681 = 0.431
2,465 0.347
insurance
2.8c404 4.5e+03
4.6e404 +: 1.3e+04
3.0e404 + 4.0e403
2.8c+04 +: 4.9c+03
kin8nm
0.578 + 0.014
0.607 + 0.009
0.615 = 0.022
0.577 = 0.014
miami housing
3.2c+05 + 1.7e404
3.3e+05 + 1.4c+04
4.2c+05 + 2.7e+04
3.1c+05 + 1.5e+04
daval_propplsion
1.2e-03 + 1.6e-05
1.4e-03 + 2.3e-05
1.2e-03 = 2.1e-05
1.2e-03 = 1.7e-05
parkinsons
14.332 + 0.644
18.091 1.263
17.266 0.988
14.221 + 0.792
powerplant
17.066 0.919
17.724 0.919
17.567 + 0.899
16.993 0.937
qsar
3.967 + 0.222
4.251 1 0.212
4.194 + 0.234
3,951 = 0.230
sulfur
0.161 0.013
0.189 + 0.018
0.182 = 0.013
0.161 0.011
superconductor
39.234 1.418
48.702 1.695
43.062 1.835
39,217
63

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Published as a conference paper at ICLR 2026
G.2 VARIANT (B) CONFORMALIZED
Table 38: Conformalized Variant (b) PICP 95% prediction intervals, aggregated across 10 seeds. Methods with suffix '-c' denote conformalized variants obtained using the validation set divided into parts, one for validation and for calibration.
Dataset
CLEAR-c
PCS-EPISTEMIC.
ALEATORIC-R
CLEAR
ailerons airfoil
0.95 + 0.01
0.95 + 0.01
0,95 + 0.01
0.95 + 0.00
0,96 + 0.01
0.95 0.02
0.95 0.02
0.95 0.01
allstate
0.95 + 0.01
0.94 0.01
0.95
0.95 0.01
ca va.housing
0.95 + 0.01
0.95 0.01
0.95
0.95 0.01
computer
0.95 0.01
0.95 0.01
0.95
0.95 0.01
concrete
0.95 0.02
0.95 0.02
0.95
0.95 0.02
clevator
0
0.95
0.95
0.95 0.00
energy_efficiency
0.97
0.95 0.02
0.02
0.95 0.02
insurance
0.96
0.95 0.02
0.06
0.95 0.01
kin8nm
0.96
0.95 0.01
0,95
0.95 0.01
miami housing
0.95 0.01
0.95 0.01
0.95
0.95 0.01
naval propulsion
0.95 =
1.00
0.05
0.95 0.01
parkinsons
0.95 + 0.01
0.95
0.95
0
powerplant
0.95 0.01
0.95
0.05
0.95 0.01
qsar
0.95 = 0.01
0.95
0.95 0.01
0.95 0.01
sulfur
0.95 1 0.01
0.95 0.01
0.95 1 0.01
0.95 0.01
superconductor
0.95 + 0.01
0.95 0.00
0.95 + 0.00
0.95 0.00
Table 39: Conformalized Variant (b) NTW at 95% prediction intervals, aggregated across 10 seeds. Methods with suffix '-c" denote conformalized variants obtained using the validation set divided into two parts, one for validation and one for calibration. Values > 100 or < 0.01 are presented in scientific notation with I decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second result. Red values are more than 33% worse than the best result.
Dataset
CLEAR-C
PCS-EPISTEMIC-
ALEATORIC-R-e
CLEAR
ailerons airfoil
0.193 + 0.016
0.217 + 0.017
0.195 + 0.018
0.193 + 0.016
0.215 + 0.019
0.217 0.022
0.227 + 0.031
0.207 + 0.012
allstate
0.257 + 0.044
0,263 0.057
0.258 + 0.046
0.256 + 0.045
ca housing
0.339 + 0.006
0.355 + 0.012
0.349 + 0.008
0.339 + 0.008
computer
0,099 + 0.006
0.142 1: 0.010
0.104 1 0,005
0.099 + 0.006
concrete
0.268 + 0.023
0.247 0.040
0.271 1 0.044
0.249 1 0.025
elevator
0.137 0.009
0.194 0.012
0.145 0.008
0.137 1 0.007
energy -eregeeeffccencyy
0.053 1 0.009
0.058 0.009
0.053 1 0.009
0.047 1 0.005
insurance
0.357 1 0.070
0.461 1: 0.292
0.408 1 0.102
0.329 1: 0.045
kin8nm
0.362 = 0.014
0.373 0.013
0.370 = 0.011
0.360 = 0.012
miami niami_housing
0.085 0.003
0.097 0.002
0.087 0.005
0.085 = 0.002
naval- proppuloo
1.9c-03 9.1c-05
8.1c-03 2 4.1e-04
1.9c-03 = 1.0c-04
1.9e-03 - 7.3e-05
parkinsons
0.279 + 0.014
0.325 1 0.016
0.305 = 0.012
0,281 = 0.009
powerplant
0.171 + 0.010
0.179 4: 0.008
0.183 + 0.008
0.170 1 0.007
qsar
0.371 + 0.126
0.395 0.134
0.386 0.127
0.366 + 0.123
sulfur
0.111 0.010
0.129 0.016
0.106 + 0.011
0.108 + 0.009
superconductor
0.220 + 0.025
0,240 0.027
0.227 + 0.025
0.219 + 0.023
64

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Published as a conference paper at ICLR 2026
Table 40: Conformalized Variant (b) Quantile Loss at 95% prediction intervals, aggregated across 10 seeds. Methods with suffix "-c" denote conformalized variants obtained using the validation set divided into two parts, one for validation and one for calibration. Values 2 100 or 0.01 are presented in scientific notation with decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second-best result. Red values are more than 33% worse than the best result.
Dataset
CLEAR-C
PCS-EPISTEMIC
ALEATORIC-R
CLEAR
silerons
9.2e-06 2.3e-07
1.0e-05 + 2.1c-07
9.7e-06 +: 2.7c-07
9.2c-06 2.3c-07
airfoil
0.110 0.013
0.119 0.012
+ 1112600000
0.109 = 0.011
allstate
1.2c+02 + 7.662
1.3e402 = 8.290
1,3e+02 7.774
1.le+02 + 7.318
ca_housing
2.8e+03 + 56.920
3.2+003 + 91.466
3,0++03 + 73.272
2.8e+03 + 59.700
computer
0.159 + 0.007
0.207 + 0.009
0.206 0.015
0.159 + 0.007
concrete
0.340 = 0.037
0.327 + 0.039
0.356 = 0.048
0.338 = 0.040
elevator
1.4e-04 2.4e-166
1.9c-04 6.7e-06
1.6c-M + 3.0e-06
1.4e-04 2.0e-066
energy -efficeefy
0.033 0.005
0.038 + 0.003
0.034 +
0.031 0.004
insurance
3.3e+02 28.437
5.2e402 + 1.4e402
3.8c+02 43.883
3,2c+02 31.593
kin8nm
7.2e-03 + 1.7e-04
7.6e-03 + 1.1e-04
7.7e-03 + 2.7e-04
7.2e-03 = 1.7e-04
miami housing
3.9c+03 + 2.1e0022
4,2c+03 + 1.7c+02
5.2c+03 + 3.40402
3,9c+03 + 1.9c+02
naval propulsion
5.5e-05 + 1.9e-06
1.8e-04 +: 9.1e-06
6.4e-05 + 2.7e-06
5,4e-05 = 1.9e-06
parkinsons
0.2019 + 0.008
0.237 + 0.008
0.222 + 0.013
0.209 + 0.010
powerplant
0.213 + 0.011
0.222 1 0.011
0.220 + 0.011
0.212 = 0.012
qsar
0.050 0.003
0.053 0.003
0.052 + 0.003
0.049 0.003
sulfur
2.0c-03 1.1e-04
2.3e-03 + 8.5e-05
2.3e-03 + 1.5e-04
1.9c-03 9.6c-05
superconductor
0.523 =
0.611 + 0.026
0.573 + 0.024
0.523 = 0.018
Table 41: Conformalized Variant (b) NCTW 95% prediction intervals, aggregated across 10 seeds. Methods with suffix '-c' denote conformalized variants obtained using the validation set divided into two parts, one for validation and one for calibration. Values 2 100 or < 0.01 are presented in scientific notation with I decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second-best result. Red values are more than 33% worse than the best result.
Datasct
CLEAR-c
PCS-EPISTEMIC.
ALEATORIC-R
CLEAR
ailerons airfoil
0.195 1 0.018
0,217 0.021
0.196 1 0.017
0.195 1 0.017
0.203 1 0.007
0.211 1: 0.007
0.216 1 0,011
0.203 1 0.006
allstate
0.248 1 0.041
0.270 0.052
0.266 1 0.044
0.250 1 0.043
housing
0.336 0.006
0.354 0.012
0.348 1 0.011
0.336 + 0.008
computer
0.098 + 0.005
0.142 = 0.008
0.103 = 0.003
0.098 = 0.005
concrete
0.248 + 0.032
0.241 + 0.025
0.261 = 0,039
0.246 = 0.033
elevator
0.137 + 0.008
0.192 0.012
0.147 = 0.009
0.137 = 0.008
energy_efficiency
0.046 0.006
0.057 0.004
0,050 0.005
0.046 = 0.005
insurance
0.314 0.052
0.346 0.076
0.335 + 0.059
0.320 + 0.059
kin8nm
0.355 + 0.008
0.371 + 0.009
0.371 + 0.009
0.354 + 0.008
miami housing
0.084 + 0.004
0.098 0.002
0.085 + 0.003
0.084 + 0.003
naval propulsion
1.8c-03 + 4.7c-05
3.2c-03 1.2c-(M
1.9c-03 8.4e-05
1.8e-03 + 5.6e-05
parkinsons
0.284 + 0.011
0,324 0.008
0.309 + 0.018
0.284 + 0.010
powerplant
0.173 1 0.008
0.182 0.007
0.182 1 0.007
0.173 1 0.007
qsar
0.363 1 0.122
0.393 1 0.131
0.390 1 0.131
0.363 1 0.121
sulfur
0.108 1 0.008
0.125 0.009
0.105 0.006
0.106 1 0.007
superconductor
0.218 0.023
0.241 0.028
0.226 1 0.022
0.218 0.022
65

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Published as a conference paper at ICLR 2026
Table 42: Conformalized Variant (b) Interval Score Loss at 95% prediction intervals, aggregated across 10 seeds. Methods with suffix '-c' denote conformalized variants obtained using the validation set divided into two parts, one for validation and one for calibration. Values 2 100 or < 0.01 are presented in scientific notation with decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second-best result. Red values are more than 33% worse than the best result.
Datasel
CLEAR-c
PCS-EPISTEMIC
ALEATORIC-R+
CLEAR
ailerons
7.4e-04 + 1.8e-05
8.1e-04 +:2 1.7e-05
7.8e-04 + 2.2e-05
7.4c-04 + 1.8c-05
airfoil
8.794 + 1.047
9.523 + 0.959
10.075 = 1.227
8.717 = 0.870
allstate
9.2c+03 6.1e+02
1.0c404 + 6.6e+02
1.0+004 + 6.2e402
9.2c+03 + 5.9c+02
ca_housing
2.2e+05 4.6e+03
2.6e+05 7.3e+03
2.4e+05 = 5,9e+03
2.2++05 4.8e++3
computer
12.704 0.571
16.543 0.706
16.505
12.707 0.575
concrete
27.168 2.969
26.195 3.142
28.500 3.874
27.020 3.202
elevator
0.011 - 0.000
0.015 + 0.001
0.013 = 0.000
0.011 = 0.000
efeegceeficieecy
2,609 + 0.417
3,019 + 0.237
2.681 = 0.431
2.465 + 00.34
insurance
2.6e404 + 2.3e+03
4.2e+04 +: 1.1e+04
3.1e+04 + 3.5e+03
2.6c+04 + 2.5e+03
kin8nm
0.578 0.014
0.607 + 0.009
0.615 0.022
0.577 = 0.014
miami housing
3.2c+05 1.7e+04
3.3c+05 1.4c+04
4.2c+05 = 2.7e404
3.1c+05 1.5c+04
prvol.propnlsion
4.4e-03 1.5e-04
1.001
5. = 2.2e-44
4.4e-03 1.5e-04
parkinsons
16.712 0.669
18,967 0.645
+
16.697 0.794
powerplant
17.066 + 0.919
17.724 0.919
17.567 1 0.899
16.993 0.937
qsar
3.960 + 0.207
4.277 + 0.215
4.195 = 0.229
3,956 + 0.218
sulfur
0.157 + 0.008
0.180 1 0.007
0.186 + 0.012
0.155 + 0.008
superconductor
41.818 + 1.464
48.902 + 2.063
45.854 + 1.910
41.801 1.407
66

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Published as a conference paper at ICLR 2026
G.3 VARIANT (C) CONFORMALIZED
Table 43: Conformalized Variant (c) PICP 95% prediction intervals, aggregated across 10 seeds. Methods with suffix '-c' denote conformalized variants obtained using the validation set divided into two parts, one for validation and one for calibration.
Dataset
CLEAR-c
PCS-FMISTEMIC
ALEATORIC R-c
CLEAR
UACQR-P
UACQR-S
ailerons
0,95 # 0.01
0.95 + 0.01
0,95 # 0.01
0.95 0.01
0,95 = 0.01
0.97 + 0.00
airfoil
0.96 + 0.02
0.96 + 0.02
0.96 + 0.02
0.96 +
0.95 = 0.01
0.96 + 0.02
allstate
0.95 + 0.01
0,95
0,95
housing
0.95 1: 0.01
0.95 0.01
0,95 0.01
0.95 1.01
0,5
0,96 0.00
computer
0.95 +: 0.01
0.95 1 0.01
0.96 4.00
0.95 1.01
0.95 0
0.97 1 0.00
concrele
0.96 0.02
0.96 - 0.01
0.96
096 0.02
0.97 + 0.02
elevator
1
0.01
095 0.5 0.00
energy efficiency
0.97 0.01
0.95 0.03
1.00
0.00
0.97 0.02
0.01
insurance
0
0.00
0.00
0.95
0.96 = 0.01
0.96 0.01
kin8om
0.95 0.01
0.95 0.01
0.95 0.001
0.95
0.95 = 0.01
0.95 0.01
miami miami_housing
0.95 1.95+00001
0,95 +
0.9
0.95
0.95 0.01
0,95 + 0.01
naval propulsion
+
0,95
0.00
0.95
parkinsons
0.01
0,95
0.95
0.01
0.97 0.01
powerplant
0.95 0.01
0.00
0.001
0.90
0.01
0,95 0.01
qsar
0.95 0.01
0.01
0.94
0.95
0
0,96 0.01
sulfur
0.95 #: 0.01
0,95 0001
0.95 0001
0.95 #001
0.95 0.01
0,95 0.01
superconductor
0.95 0.01
0.95 0.00
0.95 0.00
0.95 0.00
0.95
0,95 1 0.00
Table 44: Conformalized Variant (c) NIW at 95% prediction intervals, aggregated across 10 seeds. Methods with suffix '-c" denote conformalized variants obtained using the validation set divided into two parts, one for validation and one for calibration. Values 2 100 or < 0.01 are presented in scientific notation with I decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second-best result. Red values more than 33% worse than the best result. 2/10 denotes that UACQR-P produced infinitely wide intervals for out of the 10 seeds, and the mean NIW of the remaining 10- seeds
Dataset
CLEAR-c
PCS.EPISTEMIC.
ALEATORIC-R
CLEAR
UACQR-P
UACQR-S
airfoil
0.210 = 0.021
0,212 1 0,00
D230
0.940
0.199 0.013
0.367 2: 0.022
0.395 :: 0.030
allsaae
0.282 0.052
0.292 0.061
0.279 0.037
0.279 0.052
0,284 0.050
0.306 0.053
housing
0.317
computer
0.079 = 0.00L
0,083 0000
D.88 0.004
0,000 D004
0.083 01003
0.102 0.01
concrete
0,290 0.037
0,273 1: 0.032
0287 0.037
0.78 1 0.029
0,378 0.020
0.405 0.022
energy efficiency
0.052 + 0.007
(LI43 0.008
0.053 0.007
0,043 0.007
0.157 + 0.006
0,303
0.328
0,327
0,327 0.015
0,325
0.017
miani housing
0.088 : 0.003
0,096 0.005
0.112 0.019
0.088 0.001
0,104 0.003
0.114 = 0.005
naval propulsion
16c-03 3.6e-05
1. 65
1.6e-03 4.9c-05
1.c-03 3.2c-05
1.7c.03 4.4c-05
3.5e-03 8.1e-05
parkinsons
0,250 0.007
0.271
powerplant
0.158 =: 0.011
0,170 0.007
0.162 0.008
0.157 2 0.007
0.195 1: 0.008
0.203 : 0.009
0.354 0.120
0,402 0.143
0.349 0.115
0.344 0.117
0.409 0.135
0,450 0.154
0.126
0.111
uppreonductor
0.196 = 0.022
0,234 0.026
0.208 0.025
0.195 0.620
0,219 1: 0.022
0.234 2: 0.025
67

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Published as a conference paper at ICLR 2026
Table 45: Conformalized Variant (c) Quantile Loss at 95% prediction intervals, aggregated across 10 seeds. Methods with suffix "-c' denote conformalized variants obtained using the validation set divided into two parts, one for validation and one for calibration. Values 2 100 or < 0.01 are presented in scientific notation with 1 decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second-best result. Red values are more than 33% worse than the best result. +o0/10 intervals for out of the 10 seeds, and the mean Quantile Loss of the remaining 10 a seeds was e.
a/10 denotes that UACQR-P produced infinitely wide
Dalasel
CLEAR-e
PCS-EPISTEMIC+
ALEATORIC-R+
CLEAR
UACQR-P
UACQR-S
ailerons
9.6e-6 2.9c-07
I.le45 + 29
9,8c.06 3.2c-07
9.6e-06 +: 2.9e-07
9.9c.06 3.8c-07
9.9e-06 3.Je-07
sirfoil
0,110 0.015
0.111
0:131 0.019
0.189 0,012
0.193 0.011
allsittt
1.2e+02 6.810
1/c+12 11.278
1.c+02 = 22.506
12z+02 1 6.74
Lle+02 7.290
1.2e+02 6.633
ca.housing
2.8e403 + 91,022
2.9c403 98.973
3.2e403 : 1.1e+02
2.Nesi13 : 92.560
2.9+03 1: 53,918
3.0k+003 +. 727
computer
0.138 + 0.012
0.140 + 0.012
0.163 0.013
0.137 + 0.011
0.147 0,006
0.152 0.004
concrete
0.343 0.033
0.029
0,426
clevator
L3e-04 + 3.3e-06
Ac-444 +: 36
1.5c-04 3.5e-66
1.3e-04 1: 3.3e-466
1.8c-04 4,2e-06
1.8e04 1 Ic-06
mnegy_efflcceng
0.31 + 0.006
0.830 + 0.007
0.035 0.010
I.019 + 0.007
0.078 + 0.005
incurance
4.1ex02 73.015
4.3ev112 4 342
3.9e
53.648
4.2sx012 80 80.209
3.4c
44,920
3.5e.02 49 833
6.7e.03 1.7e-04
2:
68
2.0c-04
8.8c
1.7e-04
9.1c-0
miani housing
3.8e+03
1.9e+02
3.8c+03 1: 2.2c+02
6.8++03
1: 1.2e++03
3.7e+03 1: L.le022
4.3e+03
1.3e+02
4,6c+03 1: 1e5+02
naval propulsion
4.0e005 + 1.0e-006
4.2e-415 + 1.3e-06
4.9e-05 1.6c-06
4.0e-115 + 1.0-466
9.7c.05
1.0e-05
8.3e-05 # 7e-06
parkinsons
0,190 0.007
0.192 0.006
0.217 0,008
0.189 1: 0.006
0,181 0,009
0.200 0.010
powerplant
0.201 0.012
0,209 0,013
0.226 0.011
0,235
qaa
0.019
0.002
0.854 0.001
0.051 0.003
0,048
0.002
0.051 0.003
0.054 0.003
sulfur
Nc-03 + X.
19-413 + Tc-05
2.6e-03 :: 2.6c-14
1.Ne413 1: 6.3e-115
1.9e-03
1.2o-14
2.0c-13 + 3e-04
supercondactor
0
11010
0,573 + 0.026
0.563 0.047
0.489 + 0.020
0.508 0,018
0,522 0.013
Table 46: Conformalized Variant (c) NCIW at 95% prediction intervals, aggregated across 10 seeds. Methods with suffix '-c" denote conformalized variants obtained using the validation set divided into two parts, one for validation and one for calibration. Values > 100 < 0.01 are presented in scientific notation with I decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second-bes result. Red values are more than 33% % worse than the best result. +010 denotes that UACQR-P produced infinitely wide intervals for out of the 10 seeds, and the mean NCIW of the remaining 10- seeds was v.
Dataset
CLEAR-c
PCS-EPITTMMIC
ALEATORIC-R
CLEAR
UACQR-P
UACQR-S
ailerons
0.209 + 0.021
0,242 0.021
0.203 = 0.021
02100 D,020
0.192 :: 0.830
0.207 : 0.018
sirfoil
0,189 : 0.016
0.199 1 0.013
0200 = 0.017
0,188 1 0.014
0,317 1: 0.025
0.315 2: 0.038
allstate
0.274 = 0.046
0.296 0.058
0.885 0.029
0.277 0.056
0.278 0.047
0.297 2 0.052
housing
0.312
0.329
0.442 0.011
0311
0.366
0.405
computer
0,080 :: 11.003
0.083 0.003
0.090 0.004
0,081 0.004
0.087 +: 0.004
0.102 :: 0.001
conerete
0.255 =: 0.032
0.257 0.031
0.263 0,028
0,259 1: 0.052
0.332 0.051
0,329 0.044
elevator
0.151 0.011
0.126 0.008
0.167 0.013
0.199
energy efficiencyy
0.043 :: 0.005
0.043 1: 0.005
0.048 0.005
0.041 1 0.006
0.135 4 0.015
0.35%
0.191
0.345 0.778
0.291
0.299
kinSmm
0.323 :: 0.008
0,331 0.010
0.327 0.008
0,323 0.008
0.436 1 0.011
0,411 : 0.011
miami miani.housing housing
0.087 :: 0.003
0.096 1 0.004
0.107 0.013
0.088 1 0.002
0,101 1: 0.004
0.109 4 0.004
noval propulsion
5.0c-155
1.7613 4.7c-05
1.6e-03 4.1c-05
66c-03 5.0e-08
1.7641 4.40-05
3.5e-03 1e1--
parkinsons
0.250
0,259
0.279
0,250 00012
0.274
powerplant
0.162 - 0.007
0.171 +: 0.008
D.163 0.108
0.161 0.007
0.190 :: 0.009
0.200 = 0.111
0.350 0.121
0.112 0,141
0.360 0.120
0.348 0.118
0.410 0.135
0,441 1 0.150
sulfur
0.111
0.124
0.107
0.109
0.115
0.126
superconducter
0.194 + 0.021
0.234 +: 0.025
0.204 0.024
0.192 0.619
0.219 + 0.023
0.333 + 0.025
68

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Published as a conference paper at ICLR 2026
Table 47: Conformalized Variant (c) Interval Score Loss 95% prediction intervals, aggregated
across 10 seeds. Methods with suffix "-c' denote conformalized variants obtained using the validation divided into two parts, one for validation and for calibration. Values 2 100 presented in scientific notation with I decimal place. Bold values (desirable) are the minimum for that dataset and metric, while the underlined values indicate the second-best result. Red values are more than 33% worse than the best result. +o0. +00/10 denotes that UACQR-P produced infinitely wide intervals for of the 10 seeds, and the mean AISL of the remaining 10 seeds was
0.01
Dataset
CLEAR
ALEATORIC-R PCS-EMSTEEMIC
R-c
CLEAR
UACOR-P
UACOR-S
allerons
7.7e-04 2.3e-86
2.3e-05
7.06-44
2.6e-15
7.7e-14
7.9e-04
7.9-044 2.5e-05
airfail
8.780 1,202
8,857 + 1.035
10:504 = 1.503
8.569 + 1,99
15:102 + 0.941
15.451 + 0.852
allisate
9.4e103
5.4e102
Lleid4 + 9.0e102
10ee04
1.8c003
9.2e.03 + 5.5ex02
8.9e.03 5.8e402
9.3cv03 4: 5.3er02
housing
2.2e+05
7.3c403
2.3e+05 7.1e+03
2.5ex05
8.6c+03
2.2+003 1. 7.4c+43
2.4c+05 4,3c+03
2.4e+05 4.0e+0S
computer
11.040
13.022
11
12,130
concrete
27,403 = 2.641
26,754 2.444
30.441 = 4.624
26.928 2: 2.529
32.895 4: 2.347
34,089 2.796
elevator
0.011 + 0.000
0.012 +: 0.000
0.012 2: 0.000
0,010 1: 0.000
0.014 1: 0,000
0.015 à: 0.000
energy elliciency
+000.0
insurance
3.3e+04
8o4
3-4404 1: 4.9403
3.1e404
4.3c+U3
3 1: 6.4003
2.7e+04
3.6e413
Beelll
Oc+03
kinkom
0.536 0.114
0.552 + 0.018
0.545 0.016
IL534 + 0.014
0.7144
0.013
0.729 + 0.114
miami housing
3,0+05
1.5e401
3.1e+45 1: 1.7c+01
5.5c405
9.6c-011
2.9e+415 1: L.let44
3.5c+05
1.0c+01
3.7e405 1: 2x4001
propulsion
7.8c-03
4c-04
parkinsons
0.476
17.375
0:518
14,4%
15.967
powerplant
16.287 + 0.959
16,899 1.005
16.687 1,010
16.243 0.983
18.062 + 0.917
18.771 + 0.800
3.503 + 0.190
4,360 +: 105
4.094 0.213
3,877 :: 0.176
4,101 0.233
4,324 2: 0.208
sulfur
0,145 0,006
0,148 1 0.007
0.209 0.021
0,143 0.005
0,154 0,010
0,161 0.011
superoonductorr
41,797
Table 48: Conformalized Variant (c) CLEAR calibration parameters À and 71 for 95% prediction intervals across 10 seeds. Using all available variables. Showing median [min:max] values.
Datasct
7/1
ailerons
1.01 [0.26:1.75]
1.01 [0.87:1.32]
airfoil
1.04 10.10:100.00]
1.45 [0.02:6.72)
allstate
0.72 10.00:100.00]
1.10 [0.02:1.43]
ca houhons
1.93 [1.11:13.99
0.67 [0.14:0.95]
computer
2.85 10..41
0.60 18.18:0.95]
concrete
1.18 [0.25:100.00]
1.79 [0.02:5.83]
elevator
0.73 [0.45:1.29
1.13 [0.86:1.43]
energy efficiency
0.22 [0.02:100.00]
6.87 19.02:19.92]
insurance
4.77 [0.37:100.00]
0.55 [0,03:5.83]
kin8nm
2.21 [0.39:3.27 2.2]
0.74 [0.61:1.02]
miami-housing
4.61 [1.23:100.00]
0.36 [0.02:1.09]
propl_pron
14,44 [10.01 :22.37)
0.69 [0.52:0.85]
parkinsons
1.11 [0.53:15.09
1.34 12:2.331
powerplant
0.96 [0.57:3.30]
1.12 [0.51:1.41]
qsar
0.59 [0.22:2.58]
1.45 [0.78:2.51]
sulfur
2.98 10.90:100.00
0.63 [0.02:1.25]
superconductor
0.73 [0.22:1.38
1.17 [0.85:1.33]
69

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Published as a conference paper at ICLR 2026
2.790
HATOR--P
BACOR-S
30.2%
32.0%
Quantile (Normalized for CLEAR (c)
2.255
Reluive incrase
200%
0%
Yalurs cheru the average incream ove CLEAR (9)
0.650
CLEAR (6)
UACQR-P
UACQR-S
3.500
VATURSS
250%
Relaiive incruset (%)
2.788
NCTW (Nommalized CLEAR
2.075
066
Values show the average increase over CLEAR (2)
1.363
0.650
Invurante
Ailerons
Alsuats
Sulfur
Supercanductor
Powarplas
Miani Housing
Computer
Parkinsoon
Char
Conorets
C's Housing
Kin&Nm
Elevator
Naval Propulaion
Airfoi
Energy Efficiency
Figure 10: Quantile loss and NCIW performance of UACQR-P and UACQR-S over 10 seeds normalized relative to conformalized CLEAR (c) (baseline 1.0). Higher values indicate worse performance. The inset boxplot shows the % increase relative to the CLEAR (c) baseline #lo. Values inside each subplot represent mean increase across all datasets.
70

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Published as a conference paper at ICLR 2026
H CASE STUDY: HOUSE PRICE PREDICTION WITH VARYING NUMBER OF PREDICTORS
We illustrate our method using the Ames Housing dataset, which contains data 2,930 residential properties sold in Ames, lowa, between 2006 and 2010. The target variable is the sale price of a house, and the full dataset includes around 80 predictor variables describing various aspects such as square footage, neighborhood, and building type. This dataset, originally collected by the Ames City Assessor's Office and curated by Cock (201T). was explored in Chapter 13 of Yu & Barter (2024).
The PCS framework Yu & Barter 2024) involves quantifying all sources of extended epistemic uncertainty stemming from the entire data- science cycle, such as uncertainty stemming from data processing. A pipeline of PCS uncertainty quantification applied to the Ames Housing data can be found in Chapter 13 in (Yu & Barter 2024). We follow the same steps for estimating f and .epi
estimating epistemic uncertainty bounds 40.05: 90.95-
To investigate how the amount of available information affects predictive uncertainty, we vary both the feature set and the sample size. Starting from approximately 80 features, we construct reduced version using only the top two predictors; this setup naturally increases aleatoric uncertainty (due to limited information) while decreasing epistemic uncertainty (due reduced model complexity). We also subsample the training data to 50% and 20% to isolate the effect of training sample size on epistemic uncertainty. each case, we applied the CLEAR procedure as described in |Section 2.4 This involved: (1) data cleaning and preprocessing (excluding irregular sales, imputing missing values, encoding categorical variables), resulting N₁ 438 cleaned datasets; (2) fitting predictive (3) estimating aleatoric uncertainty model ff and estimating epistemic uncertainty bounds T0.05: 90.05:
<ppi 00.05:90.051
bounds 0s 90.95 via quantile regression, as described in Section 2.3 (4) estimating À and calibrating prediction intervals on the validation set.
sale
Table 49 reveals that while PCS performs competitively with all features, CQR is stronger with fewer features but struggles severely with limited training data (20% subsample). In contrast, CLEAR adapts effectively to all scenarios through its calibrated uncertainty combination (calibration parameters from Table 4). CLEAR estimates À = 0.64 in the 2-variable case (prioritizing aleatoric uncertainty) versus A = 14.45 in the full-feature case (emphasizing epistemic uncertainty), with corresponding calibrated epistemiicto-aleatooicc ratios of 0.03 and 7.72. As training data is reduced 50% and 20%. ratio
CLEAR progressively increases À to 17.97 and finally to 100, driving the ppistemic-to-aleatee to 14.09 and 250.5, respectively. This demonstrates the model's 's ability to accurately quantify the increased epistemic uncertainty arising from limited training samples. Consequently, both in terms of NCIW and Quantile Loss, CLEAR never outperformed by more than 1%, but outperforms PCS by 20% for reduced features, and outperforms CQR by 20% for sub-sampled data.
Table 49: Ames Housing results for all four scenarios (90% coverage target).
Experiment
Method
NCIW
Quantile Loss
Average Width ($)
Coverage
PCS
0.214
3,818
107,880
0.87
2 features
CQR
0.186
3,448
104,741
0.90
CLEAR
0.171
3,131
95,177
0.89
PCS
0.105
1,922
57,594
0.89
All features
CQR
0.117
2,194
62,398
0.88
CLEAR
0.103
1,923
55,910
0.88
PCS
0.108
1,978
56,716
0.87
All features
CQR
0.124
2,438
68,855
0.90
50% data
CLEAR
0.108
1,975
57,125
0.88
PCS
0.110
2,026
58,574
0.87
All features 20% data
CQR
0.130
2,645
72,884
0.90
CLEAR
0.110
2,024
59,290
0.88
Minor differences arise due to: (a) implementation differences in base models between R and Python, and (b) manual calibration of PCS intervals, which was not part of the original implementation. Additionally, both CQR and CLEAR were trained using only linear quantile regressors as the model selection step of PCS selected linear model hased the RMSF on the validation dataset.
71

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Published as a conference paper at ICLR 2026
1 ON THE ROLE OF RELATIVE AND ABSOLUTE UNCERTAINTY IN COVERAGE GUARANTEES
In predictive inference, a fundamental distinction arises between conditional and marginal coverage- closely related to what has been termed "relative vs. absolute uncertainty" (Heiss et al. (002b)) or "adaptive vs. calibrated uncertainty." Conditional coverage requires that prediction intervals achieve the target coverage level at every individual input, thereby capturing local or relative uncertainty. Formally, prediction interval C(Xn+1) satisfies conditional coverage level 1 - af
Vz E supp(X) : P [ E C(Xn+1) Xn+1 = x] 2 1 a. By contrast, marginal coverage only guarantees coverage on averuge over the distribution of inputs:
P [Y,+1 C(Xn+1)) 2 1-a
While conditional coverage implies marginal coverage, the reverse does not hold. This distinction is especially important in settings with heteroskedasticity, where the variability of Y X changes across the input space, and under distribution shift, where the test distribution of X differs from the training distribution. Distribution shift-such as covariate shift or domain adaptation can render marginal guarantees umreliable since they depend on the marginal Px. In contrast, conditional coverage ensures that prediction intervals remain valid even when Px changes, provided the conditional distribution P(Y )) remains stable. In what follows, we explore the implications of these distinctions and how they shape both the evaluation and design of uncertainty quantification methods.
I.1 METRICS: RELEVANCE
To assess the quality of predictive uncertainty, various metrics capture different aspects of coverage and adaptivity:
Quantile Loss CRPS, and AISL combine both relative and absolute components incen- tivizing conditional coverage. These metrics penalize both poor ranking and miscalibration, rewarding methods that adapt well to heteroskcdassicity.
NCTW is invariant to the overall scale of the uncertainty but evaluates the ranking- method assigns wider intervals more uncertain points. In practice, low NCTW often correlates with good relative uncertainty. However, a minimal NCIW theoretically encourages suboptimal conditional coverage, as can lead under-coverage high- uncertainty regions and over-coverage in low-uncertainty regions. Consequently, good conditional coverage typically requires slightly higher NCTW than the minimum. Similar to NCIW, Minimum Negative Log- Likelioodd (NLLmin) (Heiss et al., (022b) also assesses relative uncertainty by focusing on whether a method correctly ranks more versus less uncertain inputs, independent of the predicted scale.
PICP and NIW each provide only partial information: PICP measures calibration but is blind to adaptivity, while NIW captures the average scale of uncertainty.
These metrics reflect different priorities in uncertainty quantification. Choosing among them (or combining them) depends on whether the primary goal is calibration, adaptivity, or both.
1.2 APPLICATIONS
Understanding whether a method captures relative or absolute uncertainty has practical implications across range of applications. In active learning, the primary objective is to identify inputs for which the model is most uncertain, guiding efficient data acquisition. Here, only the ranking of uncertainty matters- matters-selecting the point with the highest epistemic uncertainty. Methods that preserve good relative uncertainty, even if miscalibrated. often suffice. In Bayesian optimization. many acquisition functions (such as upper confidence bound or entropy search) depend more on relative than absolute uncertainty (De Ath et al. 2021: Weissteiner et al. 2023). Using upper bounds of the form f(x) + with constant across all æ does not improve over exploiting f(x) alone, highlighting the centrality of uncertainty ranking over calibration in this setting. In human-in-the- guide prioritization-Ba for instance, flagging uncertain cases for expert review. While calibrated intervals may not always be necessary, correct ordering of confidence improve decision efficiency and safety.
loop automation, relative uncertainty
72

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Published as a conference paper at ICLR 2026
1.3 METHODS
Different uncertainty quantification methods prioritize and estimate relative and absolute uncertainty varying degrees:
NOMU (Heiss al. 2022b is explicitly designed estimate only relative uncertainty. It does not attempt to calibrate the absolute scale, making it suitable for applications where ranking matters but calibrated intervals are unnecessary.
Deep ensembles (Lakshminarrayaaan al., 2017) typically yield strong performance in capturing relative uncertainty, particularly through diversity in predictions across ensemble members. However, they often suffer from miscalibration in the absolute scale of uncertainty unless explicitly corrected.
CLEAR separately estimates relative epistemic and aleatoric uncertainty and combines them using a tunable parameter A, which also refines the ranking of uncertainty-offerin an alternative to standard calibration techniques. The absolute scale is then calibrated using second parameter, 71, allowing for flexible control over both adaptivity and calibration.
These methods highlight the spectrum of approaches to uncertainty quantification, from ranking-only models to fully calibrated systems.
1.4 CALIBRATION TECHNIQUES
The absolute scale of uncertainty is critical in applications where the expected risk or cost over a population matters (e.g., in probabilistic risk management, climate forecasting, decision-making under uncertainty). However, as emphasized by pathological cases that achieve perfect PICP with no adaptivity, relative uncertainty remains essential for practicality. Several approaches can be taken to adjust the scale of uncertainty estimates while preserving different structures:
Multiplicative calibration scales all predictions by a constant factor, preserving the multi- plicative structure of uncertainty. This is appropriate when the model's ranking reliable.
Additive calibration shifts all intervals uniformly, preserving additive differences but potentially distorting proportional uncertainty.
Isotonic calibration applies a nonparametric monotonic transformation that preserves the ranking of uncertainty, suitable when only order is trusted. E.g.. (Kuleshov al. 2018) uses isotonic calibration for distributional uncertainty quantification.
CLEAR calibrates aleatoric and epistemic uncertainty separately, allowing independent yet coherent control of both components. This facilitates calibrated estimates of total predictive uncertainty while preserving the relative structure.
1.5 ACHIEVING CONDITIONAL COVERAGE
Achieving conditional coverage requires addressing both components of predictive uncertainty:
1. Estimating relative uncertainty accurately-capturee how uncertainty varies across inputs.
2. Calibrating the absolute scale ensure the desired coverage level holds each input.
Even when conditional coverage is required, improving relative uncertainty tends reduce average interval width and improve marginal calibration under distribution shifts. Thus, adaptivity and calibration are not mutually exclusive but can reinforce one another. CLEAR is explicitly designed to target both forms of uncertainty-nne moddelngg and calibrating relative and absolute epistemic, aleatoric, and total predictive uncertainty. As such, it provides a flexible and principled framework for applications demanding both adaptivity and reliability.
73


Reviews:
[{"review_id": "ryd1JotqLE", "reviewer": "Reviewer_WNoK", "summary": "Existing methods typically address either aleatoric uncertainty due to measurement noise or epistemic uncertainty resulting from limited data, but not both in a balanced manner. In this manuscript, however, the authors propose a calibration method that combines both aleatoric (data noise) and epistemic (model/data limitation) uncertainties to improve the conditional coverage of predictive intervals for regression tasks. Dubbed CLEAR, the framework uses two learnable calibration parameters, $(\\gamma_1, \\gamma_2)$, to combine the two uncertainty components. CLEAR is compatible with any pair of aleatoric and epistemic estimators, enabling adaptive weighting based on data characteristics; unlike prior methods that fix their ratio (e.g., $\\gamma = 1).", "questions": "1. Would it possible to test CLEAR on toy classification problems?\n2. Could the authors provide some asymptotic compute cost for larger datasets? Put differently, how do the authors see CLEAR scale for larger datasets?\n3. The paper focuses mainly on I.I.D datasets, without providing clear evidence on or discussing how CLEAR would perform on non-i.i.d data. Could the authors provide some insights for non i.i.d data?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "CLEAR presents a principled, practical, and empirically effective framework for calibrated uncertainty quantification by adaptively fusing epistemic and aleatoric components. Its main innovation lies in the dual-parameter calibration, which yields sharper, better-calibrated intervals without sacrificing coverage. In particular, its key strengths are:\n\n**Strengths:**\n\n1. **Balanced Uncertainty Integration**: CLEAR uniquely combines both aleatoric (data noise) and epistemic (model/data limitation) uncertainties using two learnable calibration parameters $\\gamma_1 \\text{ and } \\gamma_2, \\text{ with } \\lambda = \\frac{\\gamma_2}{\\gamma_1}$, enabling adaptive weighting based on data characteristics; unlike prior methods that fix their ratio (e.g., $\\lambda = 1$).\n\n2. **Improved Performance**: Across 17 real-world regression datasets, CLEAR consistently achieves narrower prediction intervals (e.g., 28.2% and 17.4% average width reduction vs. baselines) **while maintaining nominal 95% coverage**, outperforming its components (CQR and PCS ensembles) and other strong baselines.\n\n3. **Flexibility & Generality**: CLEAR is model-agnostic. It works with various uncertainty estimators (e.g., PCS ensembles + quantile regression on residuals, or Deep Ensembles + Simultaneous Quantile Regression), demonstrating broad applicability.\n\n4. **Theoretical Justification**: The paper provides asymptotic conditional coverage guarantees under mild consistency assumptions (Lemma 2.1), addressing a known weakness of standard conformal methods like CQR, which often under-cover in low-density or extrapolation regions.\n\n5. **Practical Design Choices**: Estimating aleatoric uncertainty on residuals (rather than raw targets) improves stability; using quantile loss for $\\lambda$ selection ensures proper scoring and incentivizes conditional calibration.\n\n6. **Interpretability**: The learned $\\lambda$ offers insight into whether aleatoric or epistemic uncertainty dominates in a given problem (e.g., $\\lambda \\approx$ 0.6 vs. 14.5 in the Ames Housing case study with 2 vs. 80 features).", "weaknesses": "Although CLEAR enjoys some key benefits over existing methods, it does have some limitations. In particular, its key weaknesses are:\n\n**Weaknesses:**\n\n1. **Dependence on Base Estimators**: CLEAR’s performance hinges on the quality of the underlying aleatoric and epistemic estimators. Poor base models may limit gains or require careful tuning.\n\n2. **Calibration Data Requirements**: The dual-parameter calibration ($\\gamma_1, \\lambda$) uses the validation set for both model selection and calibration. While empirically effective, this lacks finite-sample marginal coverage guarantees unless a separate calibration split is used (as shown in Appendix G).\n\n3. **Computational Overhead**: While the grid search for $\\lambda$ is fast, the full pipeline requires training ensembles (e.g., 100 bootstraps in PCS), which can be expensive on large datasets, although the authors note this is modular and parallelizable.\n\n4. **Regression-Only Focus**: The method is developed and evaluated only for regression; extension to classification or structured prediction is left for future work.\n\n5. **Limited Theoretical Scope**: The asymptotic guarantees assume i.i.d. data and consistent estimators—conditions that may not hold in complex real-world settings with distribution shifts or model misspecification.\n\n6. **Empirical Results Interpretation**: While CLEAR achieved improved results over existing methods, the paper failed to properly discuss what those improvements concretely mean in terms of choosing CLEAR over existing methods, and if such improvements can translate to more complex regression tasks.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1KnidGckxy", "reviewer": "Reviewer_bUL2", "summary": "The authors propose the CLEAR algorithm for UQ in regression, with the following steps.\n1. An epistemic model is built to learn the mean and the boundaries of a confidence interval.\n2. The predicted mean is subtracted from the target and an aleatoric model is trained on the residual. The boundaries of the confidence interval is kept.\n3. Two coefficients $\\gamma_1$ and $\\lambda$ are determined, from which $\\gamma_2 = \\lambda * \\gamma_1$. For all possible $\\lambda$, separately $\\gamma_1$ are computed by calibration, and $\\lambda$ that optimizes the evaluation metric is selected.\n4. The aleatoric/epistemic intervals around mean are scaled symmetrically with the coefficients.\nI find the contribution is the combination of known methods with a grid search, with limited theoretical justification and experimentation.", "questions": "How can you handle the following limitations of the proposed method:\n - confidence interval is built around the mean $\\hat{f}$, which is subtracted from the target for training the aleatoric model. Why not, for example, the median?\n- Confidence intervals are treated symmetrically, how bout skewed distributions, or bounded target domain?\n- How about shape parameters beyond mean and confidence intervals?\n - It is assumed that if you subtract mean from the target, what is left is aleatoric uncertainty. How about e.g. bimodal?\n\nWhat is the relation of the proposed method to  https://en.wikipedia.org/wiki/Nonhomogeneous_Gaussian_regression ?  There, also an ensemble is used to calibrate the prediction for the mean (deep ensemble as an epistemic model in the paper). Gaussian variance corresponds to the aleatoric distribution, which actually needs to be assumed at several places in the proposed method.\n\nHow do you measure whether the two types of uncertainties are properly balanced? Is there ground truth?\n\nThe introduction claims \"However, they may suffer from poor conditional coverage, meaning well-calibrated coverage at the individual or subgroup level\", i.e. in the literature, marginal calibration is insufficiently solved. How does the proposed method solve the question?", "limitations": "", "rating": 2, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "+ Learning the combination of aleatoric and epistrmic uncretainties in one model is an important practical question\n+ Reproducibility in the supplementary material\n+ Improvement in experiments", "weaknesses": "- Contribution is the combination of known methods via grid search \n- If the distribution is non-Gaussian, the proposed method is limited due to its strong focus on the mean, and its symmetry. Why not median or another statistic? How about skewed distributions? Bimodal?\n-  Theoretical justification is limited", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Y8C3pkMKEC", "reviewer": "Reviewer_MNyo", "summary": "This paper introduces CLEAR (Calibrated Learning for Epistemic and Aleatoric Risk), a novel framework for constructing regression prediction intervals by adaptively balancing epistemic uncertainty and aleatoric uncertainty. Theoretically, it guarantees asymptotic conditional validity; empirically, on 17 real-world datasets and synthetic data with distribution shifts, CLEAR outperforms baselines by reducing interval width and quantile loss while maintaining nominal coverage.", "questions": "1. Regarding computational efficiency, CLEAR’s grid search for lambda (over 4000 points) consumes significant resources, especially for large datasets. Have you explored adaptive search strategies? If yes, what was the reduction in computational time while maintaining performance?\n2. This paper verifies CLEAR’s performance on 17 regression datasets, but most of these datasets have relatively balanced feature distributions. For high-dimensional sparse datasets (e.g., tabular data with hundreds of features where most are irrelevant), how does CLEAR’s performance change?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Unlike methods that use a fixed ratio to combine the two uncertainties, CLEAR selects the balance parameter based on data characteristics. For example, it emphasizes aleatoric uncertainty when working with data with few features and epistemic uncertainty when using data with many features, making it flexible across different scenarios.\n2. It works with various uncertainty estimation methods (including tree-based PCS and deep learning-based Deep Ensembles or Simultaneous Quantile Regression) and maintains reliable coverage even for data points outside the training data distribution or in extrapolation regions—areas where many baseline methods struggle.", "weaknesses": "1. The proof requires that \"at least k base models in the PCS ensemble are consistent with the true function,\" but it does not define specific criteria for determining consistency (such as error convergence thresholds) nor explain the basis for selecting k. In practical experiments, the consistency of different models varies significantly, yet the paper fails to analyze the risk of theoretical guarantees failing in such scenarios.\n\n2. The additivity of the two types of uncertainty has not been proven — epistemic uncertainty and aleatoric uncertainty essentially belong to risks of different dimensions. The additive combination implies the assumption that the two can be directly superimposed on the numerical scale, but the paper does not verify this assumption. For example, it does not compare the performance differences between additive, multiplicative, and nonlinear combinations.\n\n3. In some datasets, there is a significant correlation between the two types of uncertainties. The additive combination may amplify the uncertainty superposition effect, leading to overly wide intervals. In low-correlation datasets, however, the additive combination may underestimate risks due to improper weight allocation. The paper does not analyze the impact of uncertainty correlation on the combination structure, which limits the generality of the method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "aAyrJi09iv", "reviewer": "Reviewer_yexK", "summary": "The paper proposes CLEAR, a method for constructing calibrated prediction intervals in regression by explicitly combining aleatoric and epistemic uncertainties. It estimates aleatoric uncertainty through quantile regression on residuals and epistemic uncertainty via bootstrapped ensemble variation, then forms intervals as a linear combination of both components.", "questions": "- In the default implementation, the same dataset is used both for tuning \\lambda and conformal calibration. How do the authors reconcile this with the independence requirement of split-conformal prediction? Can they provide empirical or theoretical evidence that nominal coverage is still preserved under this data reuse?\n- Since CLEAR is presented as an evolution of CQR-derived conformal methods, why are direct comparisons against UACQR and related interval-construction baselines relegated to Appendix D rather than integrated into the main results? Would including these baselines in the principal tables change the strength of the empirical conclusions?\n- Much of the substantive discussion—literature positioning, theoretical justification, and broader comparisons—appears only in the appendix. Could the authors explain the rationale for this organisation and whether key arguments could be moved into the main text to improve clarity and self-containment?", "limitations": "", "rating": 2, "confidence": 5, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The strengths of this paper are:\n- The method builds on an intuitive and practically relevant distinction between aleatoric and epistemic uncertainty, and operationalises it in a simple, interpretable way through a weighted combination of the two components.\n- I usually dislike arbitrary \\lambda weight params, but in this paper, it is justified in the method and offers an interpretable measure of how epistemic and aleatoric components contribute to overall predictive uncertainty.", "weaknesses": "The weaknesses of this paper are:\n- CLEAR builds directly on existing CQR-derived approaches that incorporate uncertainty decomposition, particularly Uncertainty-Aware CQR (UACQR; Rossellini et al., 2024). As acknowledged by the authors themselves, UACQR can be viewed as a special case of CLEAR with \\gamma = 1. CLEAR’s main extension is to calibrate both parameters, allowing it to adjust for miscalibration in the aleatoric component. While this generalisation is reasonable and practically useful, it represents an incremental refinement. Thus, novelity is very low in my opinion.\n- Correct me if I am wrong, but in the experimental setup, the same data is used to tune \\lambda and calibrate \\gamma, which should break the independence assumption required for coverage guarantees.\n- Since this is derivative to CQR and other prior methods, not comparing against interval-creation baselines in the main paper seems like a missed step. Why leave competitive results for Appendix D? This has left me confused.\n- After spending much time reading the appendix, I am left confused by the structure of the paper. It seems like the strong literature comparisons, justifications, good results are all in the Appendix and the main body of the paper is given less thought. This seems backwards to me?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)