CUPID: A Plug-in Framework for Joint Aleatoric and Epistemic Uncertainty Estimation with a Single Model
Authors: Xinran Xu
Xiuyi Fan
Venue: ICLR 2026 Poster (2026)
Keywords: uncertainty estimation
model interpretability
trustworthy AI
Primary area: interpretability and explainable AI

Abstract: Accurate estimation of uncertainty in deep learning is critical for deploying models in high-stakes domains such as medical diagnosis and autonomous decision-making, where overconfident predictions can lead to harmful outcomes. In practice, understanding the reason behind a model’s uncertainty and the type of uncertainty it represents can support risk-aware decisions, enhance user trust, and guide additional data collection. However, many existing methods only address a single type of uncertainty or require modifications and retraining of the base model, making them difficult to adopt in real-world systems. We introduce CUPID (Comprehensive Uncertainty Plug-in estImation moDel), a general-purpose module that jointly estimates aleatoric and epistemic uncertainty without modifying or retraining the base model. CUPID can be flexibly inserted into any layer of a pretrained network. It models aleatoric uncertainty through a learned Bayesian identity mapping and captures epistemic uncertainty by analyzing the model’s internal responses to structured perturbations. We evaluate CUPID across a range of tasks, including classification, regression, and out-of-distribution detection. The results show that it consistently delivers competitive performance while offering layer-wise insights into the origins of uncertainty. By making uncertainty estimation modular, interpretable, and model-agnostic, CUPID supports more transparent and trustworthy AI.

Published as a conference paper at ICLR 2026
CUPID: A PLUG-IN FRAMEWORK FOR JOINT ALEATORIC AND EPISTEMIC UNCERTAINTY ESTI- MATION WITH A SINGLE MODEL
Xinran Xu², Xiuyi F.23.2.3
LLe Kong Chian School of Medicine, Nanyang Technological University, Singapore Cooleee of Computing and Data Science, Nanyang Technological University, Singapore "Centre for Medical Technologies & Innovations, National Health Group, Singapore xinran007@e. ntu. edu. ed, xyfanentu.edu
ABSTRACT
Accurate estimation of uncertainty in deep learning is critical for deploying mod- els in high-stakes domains such as medical diagnosis and autonomous decision- making, where overconfident predictions can lead to harmful outcomes. In prac- tice, understanding the reason behind model's uncertainty and the type of un- certainty it represents can support risk-aware decisions, enhance user trust, and guide additional data collection. However, many existing methods only address a single type of uncertainty or require modifications and retraining of the base model, making them difficult adopt in real-world systems. We introduce CU- PID (Comprehensive Uncertainty Plug-in estImation moDel), a gencral-purpose module that jointly estimates aleatoric and epistemic uncertainty without modify- ing or retraining the base model. CUPID can be flexibly inserted into any layer of a pretrained network. It models aleatoric uncertainty through a learned Bayesian identity mapping and captures epistemic uncertainty by analyzing the model's in- ternal responses to structured perturbations. We evaluate CUPID across a range tasks, including classification, regression, and out-of-distributtin detection. The results show that it consistently delivers competitive performance while offering layer-wiss wise insights into the origins of uncertainty. By making uncertainty estima- tion modular, interpretable, and model-agnostic. CUPID supports more transpar- ent and trustworthy AI. Related code and data available https://githubb.co. com/a- Fomalhaut-//CUPD..
INTRODUCTION
Deep neural networks have achieved impressive performance across many domains, yet they of- ten lack reliable mechanisms for expressing uncertainty, leading to overconfident predictions and reduced trustworthiness (Li al. 2023, Gawlikowski al.) 2023). Robust uncertainty estimation is essential for identifying misclassifications, detecting of distriibbttio inputs, and facilitating human involvement in decision making within safety critical environments (Yu et al. 2024).
Uncertainty in deep learning is generally divided into two types: aleatoric uncertainty, which arises from inherent noise or ambiguity in the data, and epistemic uncertainty. which reflects limitations in the model training data (Der Kiureghian & Ditlevsen, 2009 Zou et al.) (023). Some studies further refine epistemic uncertainty into distributional uncertainty, caused by domain shifts, and model uncertainty, due to insufficient training architectural constraints (Ulmer 2021
Numerous methods have been proposed to estimate uncertainty in deep learning models (Franchi al.) 2022; Zhang al. 2024), but most focus only type fail to clearly distinguish be- tween aleatoric and epistemic components. This distinction is essential for decision-making in high- stakes domains like medical imaging (Hüllermeier & Waegeman 2021). For instance, in diabetic retinopathy screening, high aleatoric uncertainty may signal poor image quality due to noise or blur,
"Corresoonding author.

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Data and Model Porformance
15.0
0.
Data Model Performance
0.4
P'Akankk Uhonrttry
CUPID: Uncertainty
Geead Trch
Epistemic Uncortainty
Epistemic Uncertainty
Fradikan
10,0-
iquuri Unomerty
0.3
7.5
02
5.0
0.0
0.0
0.0
Figure I: CUPID uncertainty estimation on ID regression toy problem. CUPID is inserted into an MLP-based predictive model. CUPID captures both aleatoric (blue) and epistemic (red) uncertainty.
while high epistemic uncertainty suggests the model is unfamiliar with certain pathological patterns. Disentangling these sources guides appropriate actions such as image reacquisition, expert review, or model refinement, ultimately improving system reliability. While some joint estimation methods exist, they often rely on specialized architectures such as Bayesian neural networks (Kendall & Gal, 2017) or diffusion models (Chan al. 2024), and typically require retraining from scratch. This results in high computational cost and limits compatibility with existing systems.
In this work, we propose CUPID (Comprehensive Uncertainty Plug-in estImation moDel), lightweight and versatile module that estimates both aleatoric and epistemic uncertainty with single model, without requiring any alterations to model structure or retraining. Much like how Cupid's arrows unveil hidden affections, CUPID model disentangles uncertainties within pre- dictive models. Specifically, CUPID estimates aleatoric uncertainty by learning Bayesian identity mapping while quantifying epistemic uncertainty by analyzing the model's internal responses under structured perturbations. Consider the simple 1D regression task shown in Figure [ The model is trained noisy samples with varying density and continuity. Based on the CUPID results, regions with high observation noise yield higher aleatoric uncertainty, while regions with little or no training coverage, such as edges and discontinuities, exhibit high epistemic uncertainty. This demonstrates the good uncertainty discouple performance of CUPID and the value of distinguishing between uncertainty types- not only for identifying prediction confidence, but also for understanding the underlying causes of model doubt.
By inserting CUPID at various intermediate layers, we are able analyze how uncertainty evolves throughout the network, offering insight into where and how different types of uncertainty emerge during inference. Our experiments reveal that epistemic uncertainty tends to accumulate in the deeper parts of the network, where the model's representations become more abstract and task- specific. While integrating information from multiple layers can refine the estimates, the final layers particularly informative for identifying epistemic uncertainty. In parallel, aleatoric uncertainty more effectively captured from deeper feature representations, where variability the input data more prominently encoded. Beyond its simplicity. CUPID is broadly applicable both classifi- cation and regression tasks. In summary, our key contributions are:
Propose CUPID, a plug-in uncertainty estimation module capable of jointly estimating alcatoric and epistemic uncertainty without retraining the base model.
Demonstrate CUPID's effectiveness across misclassification detection, out-of-distribute (OOD) detection, and regression tasks, achieving state-of-the-art performance on estab- lished uncertainty-aware benchmarks.
Investigate how uncertainty evolves through network layers by inserting CUPID different depths, offering a new perspective on the dynamics of uncertainty propagation.
2 RELATED WORKS
Uncertainty estimation plays critical role in enhancing the reliability, safety, and interpretability of deep learning systems (Abdar et al. 2021] Liang et al.) (202)). Broadly, existing methods can be grouped into two categories based whether they require modifications the predictive model's
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parameters: model-preserving approaches, which estimate uncertainty without altering retrain- ing the base model, and model-redefining approaches, which involve architectural changes or full retraining to capture uncertainty within probabilistic framework.
Model-redefiingg approaches These methods require modifying or retraining the predictive model integrate uncertainty estimation. Bayesian Neural Networks (BNNs) treat model weights as distributions, capturing both aleatoric and epistemic uncertainty, but are computationally inten- sive due the need for retraining and posterior sampling (Blundell et al., 2015 Kendall & Gal, 2017 Maddox et al., 2019). Evidential Deep Learning (EDL) models predictive distributions via Dirichlet framework, interpreting output logits as evidence and distinguishes uncertainty types using distributional properties (Sensoy et al. 2018 Ye et al. 2024). Deep ensembles aggregate predictions from multiple independently trained models to estimate uncertainty through predictive variance (Lakshminarayanan et al., 2017 Durasov et al. 2021 Wen et al. 2020). While effective, these approaches incur high training overhead and are less practical for large-scale applications.
To address these challenges, HyperDM (Chan et al. 2024) integrates Bayesian hyper-networks with conditional diffusion models. It approximates the benefits of deep ensembles at a fraction of the computational cost. HyperDM highlights the potential of model-redefining approaches for extend- ing uncertainty estimation to complex, high-dimensional problems, though its reliance on diffusion architectures may limit applicability where other model families are preferred.
Model-preserving approaches These methods estimate uncertainty without altering the original model architecture. Test-time augmentation strategies estimate uncertainty by measuring prediction variability across transformed inputs (input rotation or noise perturbation) (Wang et al. 201 Sa Mi al. 2022). MC Dropout applies dropout at inference to approximate Bayesian sampling, but suf- fers from increased inference time due to multiple forward passes (Gal & Ghahramani 2016 Leibig al. (017). Gradient-basee based methods have proven effective in approximating epistemic uncertainty by using gradient norms proxy (Riedlinger et al. 2023 Wang & Ji, 2024). More recently, uncer- tainty has also been estimated by designing auxiliary loss functions that enable gradient computation without requiring ground truth labels (Hornauer et al. 2025).
Alternatively, training an additional model offers a practical solution. BayesCap (Upadhyay et al, 2022) learns to estimate uncertainty top of frozen pre-trained outputs, enabling efficient uncer- tainty quantification. Rate-In Zeevi et al. (2025) extends MC Dropout by adding dropout layers and treating dropout as a tunable component inference time. By quantifying information loss in feature maps, it adaptively adjusts dropout rates per layer and per input, making dropout behave like trainable model rather than fixed regularizer. RUE (Wang et al.) (023) estimates distributional shift via reconstruction error, while other works (Yu et al. 2024) explicitly separate aleatoric and
epistemic uncertainty estimation with two dedicated modules. These approaches balance efficiency and flexibility, making them suitable for deployment in real-world settings.
3 METHOD
3.1 PROBLEM FORMULATION
We consider a supervised learning setting in which neural network model M : X W is trained to map input data x e X to corresponding targets E Y. The training dataset is defined as a finite set of N labeled examples:
D (()n) X Y.
Given a new input sample x" E X, the predictive model M, parameterized by weights 0, produces a prediction y M (x*; 0). , In a Bayesian formulation, the predictive distribution over the target output is given by marginalizing over the posterior distribution of model parameters:
/ ply" x 0 ((0 D de.
p(y" x),
(2)
Epistemic
This decomposition reveals two sources of uncertainty: aleatoric uncertainty, which arises from inherent noise in the data, and epistemic uncertainty, which reflects the model's uncertainty about its own parameters. Our goal is to estimate both types of uncertainty using unified framework.
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Deep Learning Model M
Joint Training
m
x
9
Epistemic Uncertainty
Max
y
Min
Feature
Reconstruct Branch
Extractor
Uncertainty Branch
Aleatoric Uncertainty
Hoteroscodastic Gaussian Distribution
CUPID Model C
Figure 2:
The CUPID pipeline. Alcatoric uncertainty estimated using a dedicated Uncertainty Branch, while epistemic uncertainty is captured by measuring the variance between the original model output y and the perturbed output ý".
To this end, we introduce CUPID, plug-in uncertainty estimation module that can be flexibly inserted at any intermediate layer 1 of the predictive model M. CUPID consists of three main components: Feature Extractor, Reconstruction Branch, and an Uncertainty Branch. It operates on the intermediate feature representation at a selected layer and outputs both perturbed feature and uncertainty estimates. Formally, consider a predictive model M decomposed as a composition of two sub-networks:
(() = (x)(),
(3)
where Bt : % > Rd extracts the intermediate feature m Bi(xn) at layer with dimension, and Fj Rd Y maps it to the final prediction.
The CUPID module IRd IRd TRt parameterized by w, operates on m and outputs reconstructed feature m/.n E Rd and an aleatoric uncertainty estimate on E R²:
(m/, ).) C(m,.ii)
(4)
The reconstructed feature mf... is forwarded through the remainder the network to produce a perturbed prediction:
(mf,n))
(5)
Epistemic uncertainty is quantified the discrepancy between the original prediction ý, Fi(mi,)) and the perturbed prediction Y':
(x) = 1y Knll.
By explicitly modeling both the reconstruction of features and predictive variation, CUPID enables interpretable estimation of both aleatoric and epistemic uncertainties at any specified internal layer of the model.
3.2 ALEATORIC UNCERTAINTY ESTIMATION WITH CUPID
Aleatoric uncertainty refers to the inherent noise present in the data, arising from factors such as measurement error, sensor limitations, or ambiguous inputs. This type of uncertainty irreducible and persists even with unlimited training data. A common strategy to model alcatoric uncertainty is to assume that the network's 's output is corrupted by observation noise, which follows heteroscedas- tic Gaussian distribution with inpet-dependent variance (Upadhyay et al. 2022).
Specifically, for each input x,,, the predictive distribution over the target yn is modeled as:
plyn X,, 0, w) = N n> 2.),
where 62 E * the predicted detandependent variance output by the Uncertainty Branch of CUPID. equals the output dimension.
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Under this probabilistic modeling assumption, the optimal parameters of the Uncertainty Branch, w, are obtained by maximizing the log- likelllldood over the dataset:
arg max
log plyn Xy: 0,w)
Vnl
arg max
log(
(8)
20
2
The predicted variance o then serves as an estimate of the aleatoric uncertainty for sample n:
Uaten :- o
To improve numerical stability during optimization, we follow the standard approach of predicting
the log-variance 8, log(@²)) ) rather than the variance itself (Kendall & Gal 2017). The resulting loss function for the Uncertainty Branch becomes:
Calen
Σ n=1
1 2 sp)(l- - + 2 7,
(10)
While the formulation above is presented in regression setting, the same likelihood principle ex- tends naturally to classification. normalized evidence for each class; applying the Softmax yields the predictive probability vector y Softmax(ž,,) Both Y, and the one-hot label yn can be viewed as continuous distributions. This allows defining Brier-style heteroscedastic objective over ly,, 8 122-
this case, the model produces logits L. which represent un-
3.3 EPISTEMIC UNCERTAINTY ESTIMATION WITH CUPID
Epistemic uncertainty captures the model's lack of knowledge, often attributed to limited training data or uncertainty in model parameters. This type of uncertainty can be reduced with more data and typically increases in regions of the input space that are underrepresented during training. Addition ally, epistemic uncertainty is closely associated with distributional shifts, arising when test samples deviate from the training distribution. CUPID estimates epistemic uncertainty by encouraging the Reconstruction Branch to produce feature perturbation that is maximally different from the origi- nal intermediate feature Mi,s. while maintaining the same output prediction. Formally, seek to find a reconstructed feature mí., that satisfies:
maximize |lmi,n - mr,al1 and minimize I|8 - -ynll1- mi,-
(11)
mi..
The loss function to train the Reconstruction Branch therefore balances a differential feature term that promotes large deviations with prediction consistency constraint:
N
1 Σ N
fapis
[1
Valli X1 A1 m.,.
(12)
where A 0 is a hyperparameter that controls the trade-off between prediction invariance and feature perturbation magnitude. To avoid trivial solutions where the perturbation grows arbitrarily, we initialize CUPID close to the identity mapping. The epistemic uncertainty is then quantified by:
Uepic (x) == [m....) - Fimm(,))
(13)
To further interpret this measure, we consider first-order Taylor expansion of F around m assuming local differentiability, then we obtain the approximation:
Uepis(X) |/ mi, Fi(m.,.. (m/, mi,
(14)
This formulation reveals two key components that jointly determine the magnitude of epistemic uncertainty:
Uepis(X x Sensitivity x Deviation,
(15)
where the Jacobian V Vm,.. (imm))) reflects the local sensitivity of the model's output to perturba- tions in feature space. The perturbation ||m mı,all captures the extent to which the input
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deviates from the training manifold. In-distribution misclassified samples often exhibit high sensi- tivity, while OOD samples induce abnormally large deviation. CUPID therefore provides a unified estimate of epistemic uncertainty that responds to both failure modes. For classification tasks where softmax activation is used to produce probability distributions over discrete classes, the output dis- crepancy is computed in the softmax space.
3.4 Loss FUNCTION
To jointly estimate epistemic and aleatoric uncertainty, the total loss is defined as:
LCUPID Lepis + X2 falea:
(16)
where A2 is weighting hyperparameter balancing the aleatoric loss Latea against the epistemic loss Lapis- Both the epistemic and aleatoric estimation are optimized simultaneously under this unified loss, ensuring that CUPID learns both uncertainty types within single model.
4 EXPERIMENTS
In this section, we systematically evaluate CUPID's effectiveness in estimating both aleatoric and epistemic uncertainty across three distinet tasks: medical image misclassification detection, out-of- distribution detection, and image super-resolution. These tasks selected to highlight the general- izability of CUPID across classification and regression problems, as well as across high-stakes and general- purpose domains. We also perform an ablation study assess the impact of placing CUPID at different locations within the model architecture and to analyze the influence of internal hyper- parameters on its performance. Each experiment was repeated three times, and report the mean and standard deviation for all evaluation metrics. The detailed model architectures, implementation specifies, and main task performance metrics for all experiments provided in the appendix.
4.1 MEDICAL IMAGE MISCLASSIFICATION DETECTION
Performance of misclassification detection (misclassified samples as positive). The best model for each metric is in bold, and the second best is underlined. CUPID Alcatoric achieved the best performance on GLV2, while CUPID Epistemic performed best on HAM10000, suggesting different dominant sources of uncertainty across datasets.
Table I:
GLV2 AURC (L)
HAM10000
Method
AUC (f)
Spearman (1)
AUC (1)
AURC ())
Spearman (t)
CUPID Alea.
0.870 = 0.002
0.018 = 0.001
0.941 0.004
0.769 = 0.023
0.067 0.007
0.722 + 0.014
CUPID Epis.
0.769 + 0.015
0.034 + 0.002
0.701 0.051
0.855 # 0.006
0.047 # 0.001
0.907 + 0.001
MC Dropout
0.768 + 0.006
0.027 +: 0.001
0,888 0.005
0,829 0.001
0.076 0.001
0.861 + 0.002
Rate-in
0,815 + 0.006
0.024 0.001
0.816 =
0.846 + 0.001
0.048 + 0.000
0.915 + 0.000
IGRUE
0.642 = 0.007
0.058 0.002
0.199
0.004
0.548 = 0.004
0.157 0.002
0.027 = 0.018
PostNet Alea.
0.671 = 0.006
0.004
0.011
0.793 = 0.007
0.142 0.003
0.764 = 0.006
PostNet Epis.
0,559 + 0.031
0.238 0238+0.01
0.054
0.751 0.017
0.158 0.010
0.698 + 0.033
BNN
0.829 + 0.018
0.025 + 0.003
0.954 0.007
0.793 = 0.006
0.096 # 0.004
0.821 + 0.009
DEC
0.503 = 0.012
0.192 0.006
0.803 = 0.139
0.837 = 0.017
0.082 = 0.004
0.874 = 0.007
Experiment setting We begin our evaluation with medical image classification, domain where reliability, interpretability, and calibrated uncertainty are critical for deployment. clinical settings, model's ability to identify its mispredictions can directly impact diagnostic decisions and patient safety. To this end, we evaluate CUPID on misclassification detection using two medical imaging benchmarks: GLV2 (glaucoma detection) (Gulshan et al. 2016 Kiefer et al. 2022 and HAM10000 (skin lesion classification) (Tschandl et al. 20188).
Baselines include Rate-in (Zeevi et al.) 2025), MC Dropout (Folgoc et al. 202 1, PostNet (Charp- entier et al. 2020). IGRUE (Korte et al. 2024), DEC (Sensoy et al. 2018). and BNN Kendall & Gal. 2017), all implemented with ResNet18 (He et al. 2016) for consistency. CUPID is integrated after the final residual block of ResNet18. We report AUC, AURC (Ding et al., 2020). and Spear- man's rank correlation (Rasmussen al. 2023). AUC measures the ability to separate correct from

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incorrect predictions for misclassification detection; AURC assesses the confidence-error trade-off; and Spearman quantifies the correlation between uncertainty and error.
Results The results of misclassification detection on the GLV2 and HAM10000 datasets are pre- sented in Table Il CUPID Alcatoric achieves the highest AUC (0.870) and lowest AURC (0.018) on GLV2. Its Spearman score (0.941) is also competitive with BNN (0.954). These results suggest that data-driven noise is the dominant uncertainty source in the GL-V--rriied model.
On HAM10000, CUPID Epistemic achieves the best performance (AUC: 0.855, AURC: 0.047, Spearman: 0.907), highlighting the significance of model- uncertainty in more diverse skin lesion data. Rate-in, DEC and MC Dropout also perform better on HAM1000, reflecting their sen- sitivity epistemic uncertainty. These results confirm CUPID's ability disentangle and capture different uncertainty sources, with the dominant type varying by dataset. This underscores the need for clear uncertainty modeling domain-specifft applications.
4.2 OOD DETECTION
Table 2:
Performance of OOD detection (OOD samples as positive). PAPILA and ACRIMA share the same research problem (glaucoma detection) with the ID dataset while CIFARIO is a general classification dataset.
PAPILA
ACRIMA
CIFAR10
Method
AUC(f)
AUPR(1)
AUC(1)
AUPR(1)
AUC(1)
AUPR(1)
CUPID Alea.
0.379 + 0.027
0.333 0,007
0.717 0.029
0.661 # 0.027
0.983 + 0.005
0.998 = 0.001
CUPID Epis.
0,877 0.032
0.027
0.978 # 0.010
0.984 + 0.007
0.898 + 0.054
0.991 = 0.005
MC Dropout
0.733 0.002
0.869 0.007
0.816 0.009
0.887 # 0.004
0.986 0.001
Rate-in
0.328 0.005
2332900000
0.363 ::: 0.003
0.390 + 0.003
0.620 + 0.001
0.927 0.002
IGRUE
0.636 : 0.114
0.097
0
0.944
0.978 0.005
0.998 0.001
PostNet Alea.
0.638 0.060
+
0.549 # 0.040
0.487 + 0.040
0.657 + 0.032
0.952 0.005
PostNet Epis.
0.577 0.097
0.425
0.088
0.685 + 0.154
0.654 0.151
0.773 a 0.082
0.976 + 0.011
BNN
0.707 0.040
0.612 0.050
0.708 = 0.073
0.699
0.643 + 0.108
0.959
DEC
0,515 # 0.024
0.457
0.680
0.685
0.660 + 0.015
0.963 = 0.003
Experiment setting We evaluate OOD detection using GLV2 as the in-distribution (ID) dataset with ACRIMA PDiaz-Pinto et al. 2 PAPILA (Kovalyk et al. 2022), and CIFAR- 10 (Krizhevsky & Hinton 2009) as out-of-distribution (OOD) datasets. ACRIMA and PAPILA, though also related glaucoma detection, differ in image quality and focus: PAPILA has lower contrast and reddish tones, while ACRIMA highlights optic disc regions through cropping. CIFAR-I0 --0 serves as general OOD benchmark due to domain dissimilarity. Baselines follow those in misclassification detection. AUC and AUPR are used to measure performance, treating OOD samples as positive (a (Techapanurak & Okatani 2021). AUPR highlights robustness under class imbalance.
The results are summarized in Table 2 Our proposed CUPID model demonstrates strong OOD detection performance across all datasets. CUPID Epistemic achieves the best AUPR and AUC on ACRIMA and PAPILA, highlighting its sensitivity subtle distribution shifts within the same clinical task. Interestingly, CUPID Aleatoric performs best on CIFAR-10 (AUC 0.983, AUPR 0.998). CUPID Aleatoric models input-dependent (heteroscedastic) uncertainty through a learned variance term. It assigns high uncertainty when the input lies in feature space regions that are both underrepresented and unpredictable. This enables CUPID Alcatoric to respond robustly to extreme domain mismatches, explaining its superior performance on CIFAR-10.
Results
Among baselines, IGRUE perform well on CIFAR-10 10 and ACRIMA but struggle with PAPILA. Rate-in and MC Dropout, despite strong misclassification detection results, underperform in OOD detection, likely due to overconfidence. Overall, CUPID adapts effectively to both in-task and cross- task shifts, with aleatoric and epistemic branches complementing each other across OOD types.
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4.3 IMAGE SUPER-RESOUEE RESOLUTION REGRESSION TASK
Experiment setting We evaluate uncertainty estimation in super-resolution (SR) using pre- trained ESRGAN model (Wang et al. 2018h) trained on DIV2K (Agustsson & Timofte, 2017). CUPID is integrated before the upsampling module of ESRGAN, For testing, we utilize three stan- dard benchmarks: Set5 (Bevilacqua et al. 2012), Set14 Zeyde et al. 2010). and BSDS ISDS100 (Martin al. 2 To assess generalization across modalities, additionally include the IXI dataset (Biomedical Image Analysis Group. Imperial College London 2022), brain MRI dataset that dif- fers substantially appearance and domain. Specifically, we use Ti Te-weigtted MRI seans, which are grayscale and structurally different from the natural images in DIV2K. We compare CUPID with five baselines: BayesCap (Upadhyay al. (022), which reconstructs output distributions and learns Bayesian identity mapping, in-rotate and in-noise, which measure output variation from input per- turbations (Mi et al. 2022, Wang et al. 2019); and med-noise and med-dropout (Mi et al. 2022). which inject randomness into intermediate features.
Table 3: Performance natural image datasets (Set5, Set14, BSDS100) and medical imaging dataset IXI (MRI scans). CUPID Aleatoric achieves the best results on the natural image bench- marks, while CUPID Epistemic performs best on the IXI dataset.
Set5
Set14
Method
Pearson (1)
AUSE ())
UCE ())
Pearson (1)
AUSE (J)
UCE ())
CUPID Alea.
0.528 = 0.006
0.010 0.000
0.045 0,018
0.527 0,002
0.012 = 0.000
0.049 0.005
CUPID Epis.
0.416 0.004
0.018 0.001
0.266 0.007
0.449 = 0.005
0.019 = 0.000
0.226 = 0.003
BayesCap
0.485 0.038
0.010 0.000
0.098 0.001
0.422
0.012 0.000
0.100 0.000
in-rotate
0.493 0.000
0.010 0.000
0.071 0.000
0.490 0.000
0.013 0.000
0.072 0.000
in-noise
0.370 0.006
0.000
0.051 0.035
0.354
0.022 = 0.000
0.826 0.006
med-dropout
0.219 0.023
0.030
0.680 0.043
0.271 0.012
0.024 0.000
0.292 0.022
med-noise
0.312 = 0.003
0.022 + 0.000
0.826 0.006
0.293 0.002
0.022 0.000
0.826 0.006
BSDS100
IXI
Method
Pearson (1)
AUSE ())
(L)
Pearson (1)
AUSE (L)
(L)
CUPID Alca.
0.536 = 0.001
0.012 = 0.000
0.042 = 0.012
0.677 = 0.008
0.004 = 0.000
0.021 = 0.004
CUPID Epis.
0.464 = 0.007
0.018 = 0.000
0.185 0.007
0.734 # 0.018
0.004 = 0.000
0.298 = 0.013
BayesCap
0.427 = 0.034
0.011 0.000
0.100 0.000
0.447 0.034
0.004 = 0.000
0.100 = 0.000
in-rotate
0.465 = 0.000
=.01
0.077 = 0.000
0.000
0.004 = 0.000
0.093 = 0.000
in-noise
0.353 = 0.001
=.02
0.826 0.006
0.461 0.001
0.005 = 0.000
0.091 0.002
med- med-cropout
0.397 = 0.002
0.000
0.136 = 0.008
0.000
0.007 = 0.000
0.337 0.026
med-noise
0.293 0.000
0.024
0.700 0.002
0.439 = 0.000
0.006 = 0.000
0.859 0.002
map
Epis.
BayesCup
Med-dropout
Pearsen: 0.626
0.513
0.535
0.536
0.475
00.007
0,318
Pearson: IL.37
0.425
0.35K
0.481
0.547
0.15%
0.276
Figure 3: Comparison of visual results between error and uncertainty maps. CUPID Alcatoric shows the best texture alignment and highest correlation with error maps.
To evaluate the quality of uncertainty estimation in the regression problem, we adopt three com- plementary metrics. Pearson's correlation coefficient measures the linear relationship between pre- dicted uncertainty and eггor. The Area Under the Sparsification Error Curve (AUSE) (Ilg et al.) 2018) quantifies how well uncertainty identifies inaccurate predictions by evaluating deviation from an ideal sparsification curve. Finally, Uncertainty Calibration Error (UCE) (Laves et al. 2020) sesses alignment between predicted uncertainty and enror across confidence intervals, reflecting how well the estimates calibrated. The Ta loss map is used
compute these metrics.
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Results Table B reports quantitative results. CUPID Aleatoric achieves superior performance across all natural image datasets (Pearson > 0.52. AUSE < 0.13, UCE < 0.05). BayesCap and in-rotate show moderate performance, while CUPID Epistemic and med-dropout perform poorly on the first three datasets. Methods relying on noise injection degrade performance, likely due to pertur- bation sensitivity. These results suggest aleatoric uncertainty is the dominant contributor overall uncertainty in super-resolution tasks, rather than model uncertainty. On the IXI dataset, which differs significantly from the training distribution, CUPID Epistemic outperforms its aleatoric counterpart on Pearson, showing that epistemic uncertainty becomes more informative under domain shifts and highlighting CUPID's 's capacity to adapt unfamiliar distributions. Figure S provides a visual com- parison of uncertainty maps generated by different methods. CUPID's maps exhibit clearer structure and better alignment with actual error regions, reinforcing its advantage in uncertainty estimation.
4.4 HYPERPARAMETER EXPERIMENTS
To investigate how uncertainty evolves and originates during forward propaga- tion, we conducted experiments by inserting CUPID at different intermediate layers of the predictive model, summarized in Figure
CUPID location
CUPID Alcatori GLV2
Eyistemic GLY2
Aleaturis
CUPID Set5
AUC/ Squarrmen
AUC Spotrmmn
AURC:
Fa
Afer
CUPID Lecotion (Shallev
Deep)
CUPID Lacation (Shatiow Deep)
CUPID Lecation (Shallow
Deep)
CUPID
Lecation (Shallow
Deep)
AUC(1)
Spearman(1)
AURC(1)
Prarsen())
AUSF(-)
UCF(.)
Figure& 4: Performance of CUPID inserted at varying locations: misclassification detection (Left) and super-resolution (Right). Aleatoric uncertainty estimation improves when CUPID is placed closer to the output, while epistemic uncertainty benefits from carlier insertion points.
For the medical image classification task, CUPID was integrated after the 2nd, 3rd, and 4th stages of residual blocks in the ResNet- 18 model. The results demonstrate clear trend: aleatoric uncertainty is more accurately estimated when CUPID is placed closer to the output layer, while epistemic uncertainty benefits from earlier placements within the network. This observation aligns with the conceptual distinction between the two types of uncertainty. Aleatoric uncertainty, which originates from inherent noise in the input data, tends to manifest more prominently in high-level features near the prediction layer, where semantic decisions are made. The results show that estimating alcatoric uncertainty directly from input features is insufficient, whereas using deeper activations, especially those near the output, provides more reliable signal. Conversely, epistemic uncertainty reflects the model's 's internal representation and parameter uncertainty. Placing CUPID in earlier layers enables it to better observe how uncertain representations propagate and interact throughout the model's s depth. Notably, the strong epistemic performance observed with CUPID positioned near the output highlights that model uncertainty predominantly accumulates in the final layers. A similar trend is observed in the super-resolution setting. Specifically, when CUPID is inserted before (B) and after (A) the upsampling module in the ESRGAN, observe that epistemic uncertainty is better captured in earlier layers, while aleatoric uncertainty estimation improves post-upsampling.
Table 4: Performance of differential feature loss on OOD task. "No max" means remove - |m
mi,,.l lI in the loss function. Best-performing results for each metric are highlighted in bold.
PAPILA
ACRIMA
CIFARIO
Method
AUC(1)
AUPR(1)
AUC(t)
AUPR(?)
AUC(t)
AUPR(1)
Max
Alea.
0.379 + 0.027
0.333 + 0.007
0.717 = 0.029
0.661 # 0.027
0.983 # 0.005
0.998 0.001
No max
Alea.
0,389 + 0.026
0.338 = 0.009
0.739 = 0.042
0.696 # 0.055
0.988 # 0.003
0.999 = 0.000
Max
Epis.
0.877 + 0.032
0.854 + 0.027
0.978 + 0.010
0.984 # 0.007
0.898 + 0.054
0.991 0.005
No max
Epis.
0.839 = 0.017
0.790 = 0.054
0.977 = 0.006
0.982 + 0.005
0.875 # 0.024
0.989 0.002
Loss function We further conduct an ablation study on the loss function. On the HAM10000 dataset, incorporating the differential feature loss (-Ilmm,. - mi.. (I) yields slight improvement in aleatoric uncertainty estimation (Spearman 1 0.004) while maintaining comparable performance

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for epistemic uncertainty. The benefits of this loss term become more evident in the OOD de- tection as shown in Table 4 On the PAPILA dataset, the addition of the differential feature loss leads to a substantial improvement in CUPID Epistemic's performance (AUC: 0.839-0.877. AUPR: 0. 0.790-0.854). These results suggest that the differential feature loss enhances CUPID's sensitivity to distributional shifts and improves its ability under OOD conditions. Details and further studies are provided in the appendix.
Joint vs. Separate Training. Both the reconstruction branch (epistemic) and the uncertainty branch (aleatoric) in CUPID are present and jointly optimized. To evaluate whether this two-branch architecture provides mutual benefit, conduct ablation study in which we remove entirely and train the remaining branch in isolation. Specifically, (1) Alea. separate denotes a model where the epistemic branch is removed and only the aleatoric branch is trained, and (2) Epis. sepa- rate denotes a model where the aleatoric branch is removed and only the epistemic branch is trained.
branch
On the GLV2 misclassification detection task (Table 5). the fully joint model outperforms both single-branch variants across all metrics, indicating that each type of uncertainty estimation bene- fits from the presence of the other branch during training. In OOD detection (Table (6. the epis- temic uncertainty from the joint model also achieves substantially higher AUC and AUPR than the epistemic- only variant (PAPILA AUC: 0.877-0.771), demonstrating that the joint formulation yields a more distribution-awee aware and robust representation.
This improvement arises from the complementary objectives of the two branches. For alcatoric uncertainty, the prdiction-consistenn constraint used in the epistemic loss, minm; 11% yn Il,, regularizes the shared feature extractor by discouraging intturbatiio-sensiitvv tations. This yields better-conditioned intermediate features for variance regression. Conversely, the aleatoric branch's calibrated modeling of data-dependent variability provides an additional normal- ization signal to the backbone, helping the epistemic branch distinguish meaningful distributional deviations from sample-specifee noise. Overall, these results confirm that CUPID's two-branch de- sign forms synergistic training mechanism, with the joint model consistently producing more reli- able and discriminative uncertainty estimates than either branch trained in isolation.
unstable represen-
Table 5: Misclassification detection performance on GLV2 (Joint vs. separate branches).
Aleatoric AURC (L)
Epistemic
Model
AUC (1)
Spearman (1)
AUC (f)
AURC (1)
Spearman (1)
Joint
0.870 0.002
0.018 = 0.001
0.941 0.004
0.769 # 0.015
0.034 0.002
0.701 # 0.051
Seperate
0.863 0.003
0.019 0.001
0.899 0.035
0.744 0.017
0.043 0.005
0.699 0.014
Table 6: OOD detection performance (Joint vs. separate branches).
PAPILA
ACRIMA
CIFARIO
Model
AUC(1)
AUPR(1)
AUC(1)
AUPR(1)
AUC(1)
AUPR(1)
Alea.
Joint
0.379 + 0.027
0.333 0.007
0.717 + 0.022
0.661 # 0.027
0,983 # 0,005
0.998 0.001
Alca.
Seperate
0.508 # 0.097
0.385 0.052
0.739
0.661 = 0.066
0.969 = 0.027
0.995 = 0.005
Lpis.
Joint
0.877 + 0.032
0,854 0.027
0.978 0.010
0.984 0.007
0.898 0.054
0.991 0.005
Epis.
Seperate
0.771 + 0.051
0.707 0.073
0.010
0.978 + 0.009
0.844 0.049
0.986 0.005
5 CONCLUSION
CUPID acts a versatile and lightweight plug-in module for joint aleatoric and epistemic uncertainty estima- tion. Without moditying or retraining the base model, CUPID can be inserted at any intermediate layer to reveal hidden sources of uncertainty across wide range of tasks and datascts. Through comprchensive experiments on image classification and super-resolution. we show that CUPID consistently produces reliable uncertainty estimates. Beyond performance, analysis of uncertainty propagation offers new insights into the internal behavior of neural networks. CUPID thus contributes both a practical tool for trustworthy Al and a conceptual lens for understanding model confidence.
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ACKNOWLEDGMENTS
This research is supported by the Ministry Education, Singapore (Grant IDs RG15/23 and L.CMedicine Start up Grant) and the Centre of AT in Medicine (C-ATM), Nanyang Technological University.
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A THE USE OF LARGE LANGUAGE MODELS (LLMS)
We used large language models to aid in polishing the manuscript. Specifically, an LLM was em- ployed refine grammar when necessary. All conceptual contributions, experiment design, data analysis, and interpretation were performed by the authors, with LLM support limited to language refinement described in the paper.
B THEORETICAL ANALYSIS OF CUPID EPISTEMIC UNCERTAINTY
In this section, provide theoretical analysis of the CUPID Epistemic Uncertainty, focusing its sensitivity to perturbations and the magnitude of deviation. We begin by reviewing the relevant notation and definitions.
Let my = (x E Rd be the feature representation at the I-th layer of a sample x, and let Fj : Rd - Y denote the downstream sub-network from this layer. The CUPID Reconstruction Branch produces perturbed feature m² ('m)),, constrained to leave the final output nearly unchanged:
(m,)) (im))) e. The epistemic uncertainty is defined as the output deviation induced by
this transformation:
Uepis (x) == (m/)) (F(m))))(mm)) Fi(m))l.-
(17)
Theorem 1 (Sensitivity & Deviation Driven Approximation of Epistemic Uncertainty). Assume that Fi is locally differentiable at my, and let Am, = m
10/-m; my be the reconstruction perturbation. Then,
under first-order Taylor approximation:
Uxpin(X) |pp (mi) Amills
(18)
where J (mi) is the Jacobian of Fi evaluated at mp
This result implies that the epistemic uncertainty estimated by CUPID is determined by two crit- ical factors: the local sensitivity of the network (captured by the Jacobian) and the feature-space deviation introduced by the reconstruction:
Uepis(x) Sensitivity x Deviation.
(19)
In the following sections, we further analyze the reliability of CUPID's epistemic uncertainty esti- mation, with detailed discussion the roles of sensitivity and deviation.
B.1 DEVIATION -ASSED ESTIMATION OF EPISTEMIC UNCERTAINTY
Epistemic uncertainty arises from a model's incomplete knowledge of its parameters, typically due to limited insufficient training data. A common approach quantifying this type of uncertainty is to approximate the posterior distribution over model parameters and evaluate the variability in
predictions induced by sampling from this distribution. Proposition 3.1 in Wang Ji (2024) formal- izes this idea by showing that, under regularity conditions and in the large-data limit, the posterior distribution p(@ D) converges to a multivariate Gaussian centered at the maximum a posteriori estimate 0":
p(0 D) R NO I
(20)
This Gaussian approximation justifies modeling epistemic uncertainty through perturbations around model's 's parameter 0*. Building upon this idca, we propose an alternative formulation that character- izes epistemic uncertainty through structured deviations in the input space. Rather than introducing randomness in the parameter space, we identify directions in the input domain along which the model output remains stable under the current parameterization. This provides deterministic and geometrically interpretable estimate of epistemic uncertainty, formalized in the following proposi- tion:
Proposition 1. Let fix, 0*) be a neural network with fixed parameters O*, Define the deviation Ax" as the solution to the following optimization problem:
Ax
arg max Ax
||
subject
[| Ax, 0²)
f(x, S
(21)
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for small tolerance 8 >> 0. Then, there exists parameter perturbation A such that:
f(x + Ax" 0*) f(x. 0 + A0).
(22)
This result shows that the deviation Ax", which is constrained to preserve the output, can approx- imate the effect of parameter perturbation. Hence, the deviation acts proxy for epistemic uncertainty, enabling its deterministic and input-dependent estimation.
Proof of Proposition 1. of the model with respect to both the input and the parameters:
To justify the equivalence, consider first-order Taylor approximations
f(x + Ax", (*) R f(x,0' + Jx Ax", f(x, 0r + + == f(x, 0*) + J A0,
(23)
where Jx 0f(x.0") and Jg 0f(x00))
Since Ax is chosen such that f(x + Ax", 1²) = f(x, 0*), we have:
Jx Ax" ~ O.
(24)
We now seek AO such that:
f(x, A0) = f(x Ax 0') ~ f(x,0") Jo A0 ~ 0.
(25)
Thus, any AO in the null space of Jg satisfies this condition. In particular, can construct such AO by perturbing the first-layer weights.
Let 0₃ denote the weights of the first layer, and consider the linear transformation:
(x, 01) (0, x),
(26)
where a is the activation function. To preserve the first-layer output under the input deviation Ax", we require:
(01 + A0 = 0, (x Ax")
A0,x x 0, Ax"
(27)
Assuming 24 + 0, this condition satisfied by:
Art
A = 01.41
(28)
I
This construction ensures that:
(x Ax", 01) f() 01 + A01).
(29)
and under mild smoothness conditions and subsequent layers, this local equivalence propagates to the full network output.
Extension Intermediate Features. Although the preceding formulation is derived based on input-level perturbations, the underlying reasoning naturally extends to internal feature representa- tions within the network. Specifically, consider intermediate feature vector m; (x) E layer I, where B, denotes the sub-network up to layer I, and let Fi Rd + Y be the sub-network from layer to the output.
We define the deviation Am; in the feature space as the solution to the following optimization problem:
Am = arg max Ami
|mmil
subject
[[ Am:) (( (m(m)) s
(30)
for a small tolerance > This formulation mirrors the input-level case and enables direct manip- ulation of internal activations while preserving output consistency.
By optimizing deviations in intermediate layers, our method generalizes naturally to feature- based or modular architectures. It is particularly useful in scenarios where inputs are fixed or uninterpretable, but internal representations can be explicitly perturbed and interpreted. This makes our epistemic uncertainty estimation applicable across broader range of neural architectures and analysis settings.
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B.2 SENSITIVITY AS AN INDICATOR OF EPISTEMIC UNCERTAINTY
While the previous subsection establishes a connection between input perturbations and equivalent parameter shifts, this section explores how sensitivity (quantified via gradients) can serve direct and meaningful indicator of epistemic uncertainty. Specifically, we argue that the magnitude of the gradient of the model output with respect its parameters reflects the model's familiarity with given input.
Proposition 3.4 in Wang Ji 2024) shows that, for inputs close to the training distribution, sufficiently trained model exhibits vanishing gradients with respect to its parameters. More precisely, neighborhood N (xa) around an in distributioon point Xu, the gradient satisfies:
Vef(x, 0*) O, Vx
Wxxo).
(31)
This result motivates the use of the gradient norm as a proxy for epistemic uncertainty: for inputs the model is confident about, small vanishing gradients imply stability under parameter perturbations; conversely, large gradients indicate potential epistemic mismatch.
Building on this, we propose alternative formulation in which the perturbation Ax" is not sam- pled randomly but is instead optimized under a constraint that the output remains stable. Rather than exploring the entire local input neighborhood indiscriminately, we restrict the perturbation to directions that preserve the output, leading to a boundary- aware sensitivity measure. This motivates the following proposition:
Proposition 2. Lel f(x, 0) be neural network with fxxe parameters 0", and define the perturba- tion Ax [x((x) x|| by:
Ax
arg max Ax
||Ax||
subject to
I[f(x Ax, 0-) f(x, 0')l S 6,
(32)
for a small tolerance 8 >> 0. Then, if is in-distribution and the model is well-trained, we have:
Vof(x,0') = 0, and |V Ax', 00)
(33)
This result can be interpreted as extension of Proposition 3.4 from Wang & Ji (024)). While the original proposition attributes vanishing parameter gradients to the convergence of the posterior around in-distribution inputs, our formulation adds an output-preservant restricts the perturbation to lie within a local iso-response surface-ia the subspace where the output remains stable under the fixed model parameters. If large perturbation Ax" still maintains the output within 8, the model is considered epistemically confident around x. For in-distribution data x, a well-trained model satisfies the first-order optimality condition:
constraint. This constraint
(e 0*) = O.
(34)
Now, consider the perturbation Ax that maximizes | while ensuring f(x + Ax, 0*) se f(x, 0*). Since the output does not significantly change, we infer that the model's prediction re- mains on the same confidence surface. Assuming smoothness of f, we can expand f(x + Ax, 0) in Taylor series around and observe that for sufficiently small 6, the leading-order change in the parameter gradient at x + Ax" also small:
Vof(x Ax", 0*) as 8+00 0.
(35)
This implies that the model remains insensitive to parameter perturbations in the vicinity x confirming the epistemic confidence around that region.
C TABLAR EXAMPLE
Dataset We evaluate our method on the Covertype dataset, classical structured-date benchmark provided by the UCI Machine Learning Repository (Blackard, 1998). The dataset contains 581012 samples with 54 continuous and binary features, representing cartographic variables such as ele- vation, slope, and soil type. Each sample corresponds to a 30m x 30m cell of forest cover in the Roosevelt National Forest of northern Colorado. The classification task is to predict the forest cover type, which falls into one of seven possible categories (multi-clat setting). We randomly split the dataset into 80% training and 20% testing sets. Since this dataset is tabular rather than image-based, it provides a complementary evaluation to the image-centric experiments in the main paper, high- lighting the generality of our proposed uncertainty estimation framework.
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Table 7: Hyperparameters for tabular examples.
Hyperparameters
Predicted Model
CUPID
Epoch Batch size
50
256
256
Learning rate
0.001
0.0001
0.001
A2
0.01
Table 8: Performance of tabular examples.
Method
CUPID Alea. CUPID Epis. MC Dropout
AUC(t)
0.965 = 0.000
Accuracy(1)
0.837 = 0.000
AUC (1)
0.769 + 0.006 0.688 # 0.005 0.563 0.001
AURC (L)
0.060 = 0.001 0.088 # 0.002 0.138 0.000
Spearman (1)
0.812 = 0.017 0.627 0.012
0.365 0.001
Model For the classification task, we adopt Multi-Layer Perceptron (MLP) consisting of four fully connected layers. Each layer is followed by a Sigmoid activation function, and dropout layers applied between hidden layers to prevent overfitting. The CUPID model follows the MLP archi- tecture as the base classifier and is placed before the final Linear layer. The hyperparameters of the classification model and CUPID model toy examples shown in Table
Results We conduct misclassification detection experiments on the dataset and compare the per- formance of CUPID with MC Dropout, using the same MLP baseline model. The results are pre- sented in Table As shown, CUPID achieves significantly better uncertainty estimation perfor- mance across all metrics. In particular, CUPID's aleatoric uncertainty yields an AUC of 0.769 and Spearman correlation of 0.812, indicating a strong correlation between uncertainty and prediction errors. In contrast, MC Dropout achieves lower AUC (0.563) and Spearman (0.365), reflecting less effective uncertainty estimates. Morcover, CUPID's epistemic uncertainty also outperforms MC Dropout, achieving Spearman correlation of 0.627 versus 0.365 and demonstrating its ability to capture model uncertainty. These results validate the effectiveness of CUPID in estimating both aleatoric and epistemic uncertainties for misclassification detection in tabular data.
D SUPPLEMENTS FOR THE EXPERIMENTAL SETTING
All experiments are conducted on a workstation equipped with an NVIDIA GeForce RTX 4090 GPU. The software environment includes Python 3.9 and PyTorch 2.0.1.
D.1 DETAILS OF THE TOY EXAMPLES
Datasets We constructed
synthetic datasets evaluate method under controlled settings.
The dataset depicted in Fig. 1 (Left) of the main paper was generated using the following formulation for the target variable y:
sin((0.8z) +.3 + where NO0, 0.7),
[5, 8) [12, 14)
(36)
3 s0.(0.) +.7 where ~ (0, 0.3).
x E [8. 12)
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Table 9: Hyperparameters for toy examples.
Hyperparameters
Predicted Model
CUPID
Max epoch
50
Batch size
16
8
Learning rate
0.001
0.001
0.001
A2
0.01
And the dataset demonstrated in Fig. (Right) formulated as:
3 sin((..8z) + sin(2x) + 1.3 +, where ~ (O,, 0.7), E [5, 9)
(37)
sin + sin( 1.8 + where ~ ((0, 0.2). I E [11, 13)
Model The regression model used in the toy experiments a three-layer MLP with sigmoid acti- vation functions. To align with the structure of the predictive model, the CUPID model employed in this setting is also implemented as an MLP, following similar yet simplified architecture compared the version used our main experiments. Specifically, the feature extractor component in this variant consists of only two blocks, reducing complexity while preserving core functionality. The CUPID model is integrated into the regression network by inserting it immediately before the final linear layer.
The hyperparameters of the regression and CUPID model on toy examples are shown in Table 9
D.2 DETAILS MEDICAL IMAGE CLASSIFICATION EXPERIMENTS
Datasets We evaluate our method on two widely used medical imaging datasets designed for classification tasks involving different modalities and diseases: Glaucoma-Light (GLV2) and HAM10000. Visual examples are shown in Figure
GLV2 (Gulshan et al. 2016 Kiefer et al. 2022) is large-scale fundus image dataset comprising 4,770 referable glaucoma (RG) and 4,770 non-referable glaucoma (NRG) images. The data is di- vided into training, validation, and test sets, each containing 4,000, 385, and 385 samples per class, respectively. This structure ensures well-controlled evaluation setting with equal representation of both classes.
HAM10000 (Tschandl al. 2018) is dermoscopic image dataset developed for the classification of pigmented skin lesions. It includes a diverse range of skin conditions and imaging settings. For this study, follow the official split: 10,015 images for training. 193 for validation, and 1,512 for testing. Each image is centered on a lesion and captured under standardized lighting to minimize acquisition bias.
We quantify acquisition-related randomness using dataset-level pixel variance ("Noise") and signal- to-noise ratio (SNR), computed from the first- and second-order moments of all images (Sahoo) et al. 2025). Higher noise and lower SNR indicate stronger imaging variability, which is typi- cal driver of aleatoric uncertainty. 0.022) and markedly lower SNR (5,44 dB vs. 12.78 dB) than HAM10000. confirming that GLV2 contains more acquisition-inducer variability and therefore more AU-dominant, EU arises from limited or uneven data coverage. HAM10000 contains diagnostic classes with substantial im- balance (142-6705 samples), forming a heterogeneous and sparsely supported feature space. In contrast, GLV2 consists of only 2 well-balanced classes. This structural difference implies that models trained on HAM10000 must learn more complex and uneven decision boundaries, resulting in higher epistemic uncertainty, while GLV2's simpler and balanced distribution yields lower EU.
shown in Table 10 GLV2 exhibits higher Noise (0.024 vs.
Predictive Base Model M The predictive model M implemented as a ResNet18 (He et al. 2016) architecture, widely-used convolutional neural network known for its residual learning
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Glaucoma-Light V2
HAM10000
Figure 5: Data samples from GLV2 and HAM010000.
Table 10: Dataset characteristics indicating different sources of uncertainty.
Dataset
Noise (t)
SNR (4)
Classes
Class balanced
GLV2
0.024
5.44 dB
True False
HAM10000
0.022
12.78 dB
pabilities. The network consists 18 layers, including a x convolutional layer and max-pooling at the input, followed by four stages of residual blocks with increasing filter dimensions (64, 128, 256, and 512). Each residual block contains two 3 convolutions with batch normalization and ReLU activation, facilitating stable gradient flow in deeper networks. The model concludes with global average pooling layer and a fully connected layer to output classification logits.
Uncertainty Branch
Conv
PReLu
Conv
ReL.u
Linear
Linear
Feature Extractor
Trunk Block
Batch NarchNorm orrm
Conv
PRcLu
Cony
m
Cony
ReLa
ReLu
Reconstruct Branch
Figure The structure of CUPID used in ResNet18 for the classification problem.
CUPID Model C The architecture of CUPID used in these experiments is illustrated in Figure 6 The CUPID model consists of three functional components designed to estimate uncertainty and perturb latent representations in a structured way:
Feature Extractor: This module processes the intermediate feature map mj using a series of trunk blocks, which adopt the Seiidual-in-Reidduan Dense Block (RRDB) structure (Wang et al. (0188). batch normalization layer follows the trunk blocks, yielding refined latent representation that serves shared input to the subsequent branches.
Uncertainty Branch: This branch estimates aleatoric uncertainty by learning an uncertainty score. begins with convolutional and PReLU layers to maintain spatial structure and introduce nonlinearity, followed by fully connected layers and ReLU activations to project the features into uncertainty value. This score captures the noise-related variability in the data that affects model predictions. Linear layer is chosen as the last layer because we model the log variance s, = log( instead of variance directly.
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Table 11: Hyperparameters for medical image classification experiments.
GLV2
HAM10000
Hyperparameters
Predicted
CUPID
Predicted
CUPID
Max epoch
10
10
15
Batch size
4
4
8
4
Learning rate
0.00001
0.00001
0.0001
0.00001
Decay step size
1
4.5
1
Decay
0.8
0.9
0.75
0.9
X1
0.01
/
0.01
X2
0.009
/
0.009
Table 12: Mean accuracy and AUC for each classification model on GLV2 and HAM10000 datasets.
GLV2
HAM10000
Method
AUC(1)
Acc(t)
AUC(f)
Acc(f)
CUPID/IGRUE/aate
0.970 + 0.000
0.909 = 0.000
0.952 = 0.000
0.821 = 0.000
MC Dropout
0.979 0.000
0.921 0.000
0.946 = 0.000
0.765 0.002
PostNet
0.789 : 0.004
0.010
0.865 = 0.002
0.664 + 0.008
BNN
0.966 + 0.005
0,914 0.004
0.929 = 0.004
0.742 - 0.011
DEC
0.875 0.002
0.878 = 0.003
0.922 = 0.014
0.768 = 0.009
Reconstruction Branch: This module reconstructs a perturbed version m² of the original intermedi- ate feature map. It consists of two convolutional layers and ReLU activations, aiming modify the feature representation while preserving its overall structure. The reconstructed feature then fed back into the remaining layers of the predictive model M measure epistemic uncertainty through output deviation.
Empirically, observe that the number of layers in each component has limited impact overall performance, CUPID is inserted between residual blocks of ResNet18. Since ResNet employs ReLU activations after each residual block, it is critical that the final layer of CUPID's Reconstruction Branch also uses a compatible activation. Failing to match activation functions can hinder training due to mismatched feature distributions.
Training and Classification Results
The hyperparameters for both the predictive model and the CUPID module were optimized through random search and are detailed in Table T1
We benchmark our approach against several widely adopted uncertainty estimation techniques: Rate-in (Zeevi et al.] (025), Monte Carlo Dropout (MC Dropout) (Gal & Ghahramani 2016), Pos- terior Network (PostNet) (Charpentier et al., 2020 Deep Evidential Classification (DEC) (Sensoy et al. 2018), Bayesian Neural Network (BNN), and IGRUE (Wang et al. 2023). All models are trained under identical data splits for GLV2 and HAM10000 enable robust comparisons. All experiments were conducted using ResNet18 as the backbone to ensure a fair and consistent com- parison across all methods.
Table 12 reports the AUC and accuracy results. All models were trained with early stopping based on the validation loss to ensure optimal generalization performance. CUPID, Rate-in and IGRUE are applied on the same base model. They both operate on the intermediate features of the pretrained model without modifying the parameters of the original classifier. This design ensures that high classification performance can be maintained while gaining reliable uncertainty estimates.
MC Dropout was implemented by adding a dropout layer with a drop rate of 0.03 at the end of the ResNet18 model. During inference, uncertainty was estimated using ten stochastic forward passes. PostNet, BNN, and DEC required training from scratch and were implemented using the ResNet18 backbone for consistency. These models were trained for up to 200 epochs with early stopping (patience of 10 epochs), and hyperparameters were tuned to achieve optimal classification performance.
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Glaucoma-Light V2
PAPILA
ACRIMA
CIFAR-10
Figure Data samples of ID and OOD datasets.
D.3 DETAILS OF THE OOD DETECTION EXPERIMENTS
Datasets We employ four datasets evaluate the performance of out-of-distribution (OOD) de- tection. GLV2 serves as the in-distribution (ID) dataset, selected for its large scale, high quality, and balanced class distribution. Its diversity and size make it well-suited for learning the target distribution in a supervised setting. The remaining three datasets (ACRIMA (Kovalyk et al. 2022). PAPILA (Kovalyk al., 2022). and CIFAR-I0 (Krizhevsky & Hinton, 2009)) are treated as OOD datasets. These datasets differ from GLV2 in varying degrees, allowing us to assess how well each model estimates uncertainty under different levels of domain shift. The data samples of ID and OOD datasets are shown in Figure 7 A detailed description of each dataset is provided below:
PAPILA. This dataset contains high-resolution (2,576 x 1,934) fundus images from both eyes of 244 subjects, totaling 488 images. It includes three categories: healthy, glaucoma, and suspicious. Compared to GLV2, PAPILA images tend to be darker, redder, and lower in contrast. Furthermore, due to the fixed input size requirement (512 512), the images undergo deformation during resizing, adding an additional distributional shift.
ACRIMA. ACRIMA includes 705 labeled fundus images (396 glaucomatous and 309 normal). Un- like GLV2, the images in ACRIMA are preprocessed by cropping around the optic disc to emphasize clinically relevant features. This alteration introduces distributional shift focused spatial and contextual features.
CIFAR-1O. CIFAR-10 is natural image dataset designed for object classification across 10 distinct categories: airplane, automobile, bird, cat, deer, dog, frog, horse, ship, and truck. It contains 50,000 training images and 10,000 test images with resolution of 32 x 32. Its unrelated domain makes strong test for extreme OOD scenarios.
Model and Testing Protocol The models evaluated in this experiment the in the misclassification detection task, including our proposed CUPID, Rate-in, MC Dropout, Post- Net, BNN, DEC, and IGRUE, All models were trained exclusively on the GLV2 dataset and then tested on the ID (GLV2) and three OOD datasets (PAPILA, ACRIMA, and CIFAR-10). To ensure consistency, all input images were resized to 512 512 pixels. For CIFAR- 10, we retained their original training, validation, and test splits. Since PAPILA and ACRIMA lack predefined splits and are relatively small in size, the entire datasets were used for testing.
those used
D.4 DETAILS THE SINGLE IMAGE SUPER RESOLUTION EXPERIMENTS
Datasets for Super-Resolution We trained the ESRGAN and CUPID on DIV2K and evaluated method on four widely-used benchmark datasets for single image super-resolution under x4 scale setting with bicubic downsampling.
DIV2K (Agusssson & Timofte 2017) serves as the primary training dataset and consists of 800 high-resolution (2K) images across diverse scenes, offering rich textures and fine details suitable for learning robust SR models.
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Table 13: Hyperparameters for super-resolution.
Hyperparameters
Predicted Model
CUPID
Max iteration
300000
300000
Batch size
16
16
Learning rate
0.0001
0.00001
X1
/
0,0001
22
/
0.01
Se15 (Bevilacqua et al., 2012) includes classical natural images commonly used for SR bench- marking. The dataset covers various object categories such animals, architecture, and people, allowing for controlled evaluation in small-scale settings.
Set14 (Zeyde al. 2010) comprises 14 images with greater diversity in scene content and degrada- tion types compared to Set5, including urban structures, facial portraits, and natural landscapes.
BSDSI00 [arrtin al. 2001), subset of the Berkeley Segmentation Dataset, contains 100 test im- ages with a wide variety of textures and fine structures. It is widely adopted to assess generalization and edge reconstruction quality in SR tasks.
IXI (Biomedical Image Analysis Group, Imperial College London 2022) is publicly available brain MRI dataset collected from three hospitals London: Guy's s Hospital, Hammersmith Hos- pital, and the Institute of Psychiatry. It contains over 600 subjects and includes various imaging modalities such as T1-eeihhted, T2-weighted, proton density (PD), and diffusion wweeuhte--wwiggeedd images. The dataset covers broad age range and provides valuable anatomical diversity, making it a widely used resource for developing and evaluating medical image processing and machine learning algo- rithms, particularly in brain structure analysis and image reconstruction tasks.
ESRGAN Backbone The Enhanced Super-Resolution Generative Adversarial Network (ESR- GAN) widely adopted architecture for perceptual single-image super-resolution, known for its ability to produce high-fidehi fighlfiyelity reconstructions with visually realistic textures. It builds upon the original SRGAN framework by incorporating several architectural improvements aimed at enhane- ing perceptual quality and training stability.
ESRGAN comprises three primary components: a generator, a discriminator, and perceptual loss module. The generator is constructed using deep convolutional architecture based on Residual- in-Residual Dense Blocks (RRDBs), which synergize the advantages of residual learning and dense connectivity. Unlike traditional residual blocks, RRDBs omit batch normalization layers to avoid artifacts and enable more stable training. These blocks allow for efficient feature propagation and better gradient flow, enabling the network to preserve fine- ane-grained textures across deep layers. Mul- tiple RRDBs are stacked to form the feature extraction backbone, followed by a series of upsampling blocks that progressively increase the spatial resolution of the image. These upsampling layers placed toward the end of the network to reduce computational cost during earlier stages. final convolutional layer refines the output produce the high-resolution image.
In our experiments, we evaluate the placement of the CUPID module by inserting it either before or after the upsampling blocks. This setup allows study how the model's uncertainty estimation capabilities vary depending on its position relative to the reconstruction pipeline.
CUPID Model C' for Super-Resolution The CUPID architecture for image super-resolution largely follows the classification variant but adapts to the demands of pixel-level precision. Batch normalization omitted in the feature extractor to preserve local image statistics, while the Un- certainty Branch incorporates convolutional layers with Leaky ReLU and multi-stage upsampling module, producing a spatial uncertainty map aligned with the super-resolved image for aleatoric un- certainty estimation. In parallel, the Reconstruction Branch employs convolution and Leaky ReLU to reconstruct intermediate features, enhancing compatibility with the ESRGAN backbone and sup- porting epistemic uncertainty estimation.
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Table 14: PSNR and SSIM for ESRGAN model.
Dataset
SSIM(t)
PSNR(t)
Set5
0.828
28.533
Set14
0.684
24.865
BSDS100
0.631
23.539
IXI
0.806
33.142
Training and Resel--esolution Results The hyperparameters for both the predictive model and the proposed CUPID module were optimized using random search and are summarized Table [3] All models were trained using early stopping strategy based validation loss to prevent overfit- ting and ensure optimal generalization. We compare CUPID against five representative uncertainty estimation baselines. BayesCap (Upadhyay al. 2022) explicitly models the uncertainty by constructing an output distribution. shares the same training protocol and predictive backbone (ESRGAN) CUPID, allowing for fair comparison.
In-rotate and in-noise estimate uncertainty by analyzing output variation under input transforma- tions. In-rotate (Mi et al. 2022) applies four 90-degree rotations to the input image and computes the output variance as an uncertainty measure. In-noise injects 0.005% Gaussian noise into the input image and repeats the inference ten times estimate uncertainty based on output variation (Wang et al. 2018a).
Med-noise and med-dropout (Mi et al. 2022) assess uncertainty through perturbations in the latent space. Med-noise adds 0.005% Gaussian noise to the intermediate feature map and repeats inference ten times. Med-dropout introduces an additional dropout layer with a drop probability of 0.3, also using ten repeated forward passes during testing.
The super-resolution performance of the baseline ESRGAN modelare reported in Table
E OTHER EXPERIMENTS RESULTS
E.1 HYPERPARAMETEN EXPERIMENTS ON CUPID LOCATION
Table 15 shows the results of inserting CUPID at different intermediate layers. The results demon- strate clear trend: aleatoric uncertainty is more accurately estimated when CUPID is placed closer to the output layer, while epistemic uncertainty benefits from earlier placements within the network.
Table 15: Results of inserting CUPID at different intermediate layers. Best-performing results for each metric are highlighted in bold.
GLV2 AURC ())
HAM10000
Class Position
AUC (1)
Spearman (1)
AUC (1)
AURC ())
Spearman (1)
Aleatoric (4)
0,870 + 0,002
0.018 0.001
0.941 + 0.004
0.769
0.023
0.067 + 0.007
0.722 + 0.014
Aleatoric (3)
0.843 +: 0.008
0.022 +: 0.001
0.851 + 0.006
0.751 +: 0.004
0.072 + 0.003
0.624 4 0.024
Alcatoric (2)
0.805 # 0.014
0.026 = 0.001
0.772 = 0.019
0.749 + 0.011
0.103 = 0.006
0.675 + 0.021
Epistemic
0.769 0.015
0.034 0.002
0.701 0.051
0.855
0.006
0.047 + 0.001
0.907 0.001
Lpistemic
0.786 0.003
0.033 0.004
0.696 0.015
0.869 0.005
0.045 + 0.000
0.898 0.004
Epistemic (2)
0.789 +: 0.017
0.031 0.001
0.717 0.006
0.901 0.004
0.058 0.001
0.888 + 0.005
Set5
Set14
SR Position
Pearson (1)
AUSE (L)
UCE (1)
Pearson (1)
AUSE (L)
UCE (L)
Alcatoric (A)
0.560 + 0.001
0.009 0.000
0.034 = 0.007
0.569 0.001
0.011 = 0.000
0.017 + 0.008
Alcatoric (B)
0.528 # 0.006
0.010 = 0.000
0.045 = 0.018
0.527 + 0.002
0.012 = 0.000
0.049 + 0.005
Epistemic (A)
0.258 4: 0.005
0.026 0.000
0.320 4: 0.004
0.327 +: 0.005
0.026 + 0.000
0.231 + 0.005
Epistemic (B)
0.416 + 0.004
0.018 + 0.001
0.266 # 0.007
0.449 0.005
0.019 + 0.000
0.226 + 0.003
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Table 16: Uncertainty calibration results for misclassification (GLV2) and OOD detection (PAPILA,
ACRIMA, CIFAR-10)
GLV2
PAPILA
ACRIMA OOD-UCE ())
CIFARIO
Method
rAULC (1)
UCE (1)
CUPID Alea.
0.840 0.003
0.042 0.004
0.308 = 0.026
0.226 + 0.079
0,273 + 0.004
CUPID Epis.
0.649 0.031
0.033 = 0.008
0.115 + 0.016
0.240 + 0.006
0.633 + 0.017
MC Dropout
0.682 0.009
0,052 # 0.019
0.257 + 0.001
0.307 + 0.045
0.699 + 0.021
Rate-in
0.762 = 0.011
0.038 # 0.009
0,309 # 0.009
0,411 # 0.008
0.026
IGRUE
0.375 0.024
0.145
0.154 = 0.046
+ 0.252 0.016
0.596
PostNet Alea.
0.419 = 0.019
0.166
0.025
0.285 0.036
0.500
PostNet Epis.
0.184 0.066
0.050 = 0.014
0.182 0.104
0.726 0.289 0.089
BNN
0.747 = 0.017
0.268 = 0.010
0.332 0.012
0.705
DEC
-0.627 # 0.056
0.417 # 0.039
0.256 + 0.027
0.198 # 0.010
0.248 + 0.008
E.2
UNCERTAINTY CALIBRATION RESULTS FOR MISCLASSIFICATION AND OOD
Table 16 summarizes uncertainty calibration performance for both misclassification and OOD set- tings. CUPID consistently ranks first or second across all metrics, demonstrating strong uncer- tainty-error correlation and stable calibration. On GLV2, CUPID Aleatoric achieves the highest rAULC (0.840), while CUPID Epistemic obtains the best UCE (0.033), indicating excellent un- certainty ranking and calibration. For OOD detection, CUPID achieves the lowest OOD-UCE on PAPILA (0.115) and competitive results on ACRIMA (0.226 and 0.240). Notably, DEC attains the best OOD-UCE on ACRIMA (0. 198), but this comes at the cost of severely degraded misclassifica- tion calibration (rAULC = -0.227, UCE = 0.417), suggesting that DEC improves OOD calibration only by sacrificing its ability to reflect in-distribution prediction errors. This contrast highlights CUPID's balanced and reliable uncertainty modeling across both ID and OOD scenarios.
E.3 ABLATION AND HYPERPARAMETER EXPERIMENTS ON CUPID FOR CLASSIFICATION MODEL
Table 17: Performance of trunk block depth variants CUPID for the classification task on HAM10000 dataset. pest-performing results for each metric are highlighted bold.
Aleatoric AURC (L)
Epistemic AURC (L)
Method
AUC (t)
Spearman (1)
AUC (1)
Spearman (f)
12
0.725
0.028
0.098 0.010
0.614 0.027
0.890 4 0.003
0.052 = 0.001
0.889 + 0.004
14
0.725 0.013
0.107=0.000
0.604 0.035
0.902 0.004
0.055 0.003
0.886 0.001
Block
16
0.749 0.011
0
0.675 0.021
0.901 0.004
0.058 0.001
0.888 0.005
18
0.735 0.024
0
0.628 0.049
0.895 0.004
0.054 0.002
0.886 0.001
Table 18: Performance of loss function CUPID for the classification task on HAM10000 dataset. Best-performing results for each metric are highlighted in bold.
Alcatoric AURC ())
Epistemic
Loss
AUC (t)
Spearman (t)
|
AUC (t)
AURC (L)
Spearman (1)
Max
0.749 + 0.011
0.103 + 0.006
0.675 + 0.021
0.901 + 0.004
0.058 + 0.001
0.888 0.005
No
0.748 + 0.005
0.098 0.006
0.671 + 0.025
0.898 0.002
0.056 0.003
0.888 0.002
Effeet of Trunk Block Depth Table 17 presents the impact of varying the number of Trunk Blocks in the Feature Extracior module of CUPID on HAM10000 dataset. CUPID is placed after the 2nd stage of residual blocks. For aleatoric uncertainty estimation, CUPID shows sensitivity to the depth of the trunk. The best performance is achieved when using 16 blocks, yielding the highest AUC of 0.749 and the highest Spearman correlation of 0.675. In contrast, the performance of epistemic uncertainty estimation is relatively stable across different depths, with only minor fluctuations in
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AUC and AURC. This suggests that the epistemic branch reaches its representational capacity with fewer blocks, while aleatoric estimation benefits more from deeper feature extraction.
Effect of Differential Feature Loss We conduct ablation study to evaluate the contribution of the differential feature loss term - || - milli i. which enforces the difference between the inter- mediate feature my and the CUPID-generated reconstruction mi. Table 18 summarizes the results for the misclassification detection task on the HAM10000 dataset. Including the differential feature loss slightly improves aleatoric performance in terms of AUC (from 0.748 to 0.749) and Spearman correlation (from 0.671 to 0.675, while maintaining comparable epistemic uncertainty performance.
For the out-of-distribution (OOD) detection task, the impact of the differential feature loss is more pronounced, particularly for epistemic uncertainty. As shown in Table E adding the loss substan- tially improves the performance of CUPID Epistemic on the PAPILA dataset, with AUC increasing from 0.839 to 0.877 and AUPR from 0.790 to 0.854. This indicates that the differential feature loss strengthens CUPID's sensitivity to distributional shifts. similar trend is observed across the ACRIMA and CIFARIO datasets, reinforcing the importance of this component for robust epistemic uncertainty estimation in OOD scenarios.
Effect of a choice The results in Table 19 and 20 show that CUPID is generally robust to the choice of Å1- For misclassification detection, performance remains stable across all tested values. For OOD detection, λ 0.01 consistently provides the best balance between stability and accuracy, and is therefore used as the default throughout the paper.
Table 19: Performance of X1 choice on CUPID for the classification task on GLV2 dataset.
Aleatoric AURC (1)
Epistemic AURC (L)
Method
AUC (*)
Spearman (*)
AUC (1)
Spearman (f)
0.02
0.868 + 0.001
0.018 0.000
0.929 + 0.019
0.743 #: 0.044
0.045 a 0.013
0.676 1 0.021
A
0.01
0.870 + 0.002
0.018 = 0.001
0.941 + 0.004
0.769 0.015
0.034 + 0.002
0.701 + 0.051
0.001
0.868 + 0.005
0.018 0.001
0.936 + 0.010
0.754 = 0.014
0.042 + 0.005
0.674 + 0.031
Table 20: Performance of λ choice on CUPID for the OOD task.
PAPILA
ACRIMA
CIFARIO
Method and X
AUC(1)
AUPR(1)
AUC(t)
AUPR(1)
AUC(1)
AUPR())
0.02
0.457 & 0.087
0.362 # 0.043
0.727 0.124
0.651 # 0.19
0.973 0.017
0.996 0.003
Alea.
0.01
0.379 = 0.027
0.333 + 0.007
0.717 + 0.029
0.661 = 0.027
0.983 0.005
0.998 0.001
0.001
0.389 0.030
0.342
0.718 + 0.046
0.669 = 0.062
0.984 0.006
0.999 + 0.001
0.02
0.802 + 0.039
0.743 + 0.058
0.904 +: 0.058
0,921 # 0.046
0.528 + 0.293
0.923 + 0.063
Epis.
0.01
0.877 = 0.032
0.854 = 0.027
0.978 = 0.010
0.984 = 0.007
0.898 = 0.054
0.991 0.005
0.001
0.836
0.975 + 0.003
0.980 = 0.002
0.896 + 0.052
0.991 0.005
E.4 ABLATION AND HYPERPARAMETER EXPERIMENTS ON CUPID FOR
Table 21: Performance of trunk block depth variants CUPID for the super-resolution task. Best- performing results for each metric are highlighted in bold.
Set5
Set14
Trunk Num
Pearson (*)
AUSE ())
UCE (L)
Pearson (1)
AUSE (L)
UCE ())
0.525 + 0.002
0.010
0.000
0.017 0.007
0.527 0.002
0.012 0.000
0.049 0.014
4
0.528 4 0.006
0.010 0.000
0.045 0.018
0.527 0.002
0.012 0.000
0.049 0.005
Aleatoric
5
0.527 a: 0.007
0.010 0.000
0.041 +: 0.019
0.527 0.001
0.012 + 0.000
0.047 0.015
6
0.528 + 0.003
0.010 = 0.000
0,045 + 0.020
0.524 = 0.003
0.012 0.000
0,039 0.006
3
0.416 +: 0.004
0.020 +:2 0.000
0.254 + 0.024
0,428 +: 0.005
0.020 +: 0.000
0,217 +: 0.003
4
0.416 + 0.005
0.018 + 0.001
0.266 + 0.007
0.449 + 0.005
0.019 + 0.000
0.226 + 0.003
Epistemic
5
0.420 + 0.005
0.018 = 0.000
0.256 + 0.012
0.455 + 0.004
0.019 + 0.000
0.217 + 0.011
6
0.421 = 0.005
0.018 = 0.001
0.257 = 0.014
0.459 = 0.008
0.019 = 0.000
0.211 + 0.006
26

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Published as a conference paper at ICLR 2026
Effect of Trunk Block Number Table 21 reports the performance of CUPID when varying the number of Trunk Blocks in the Feature Extractor on the Set5 and Set14 datasets. Overall, both aleatoric and epistemic uncertainty estimations remain relatively stable across different block depths. For aleatoric uncertainty, increasing the trunk number from 3 to leads to a consistent improvement in Pearson correlation. While using 6 blocks achieves the best UCE on Set14 (0.039), the gains beyond blocks are marginal. For epistemic uncertainty, deeper trunk configurations (especially blocks) slightly improve the Pearson correlation, particularly on Set14 (from 0.428 0.459), and reduce UCE values modestly. These results suggest that while increasing the trunk depth can lead minor performance gains, the CUPID framework remains robust even with fewer blocks.
Effect of Perceptual Loss ESRGAN incorporates a perceptual loss derived from high-level fea- ture representations extracted by a pre-trained VGG network, encouraging the model to generate outputs that are perceptually closer to the ground truth rather than strictly minimizing pixel-wise differences. Motivated by this, we investigate the effect of incorporating a perceptual loss into the CUPID training process.
Specifically, we augment the original CUPID loss function with a perceptual term, resulting in the following objective:
LCUPID Lipis + AzC + Lipips
(38)
where pipss denotes the perceptual loss computed using the LPIPS metric.
Table 22 presents the evaluation results on Set5 and Set14. While adding the perceptual loss de- creases the Pearson correlation for both aleatoric and epistemic uncertainty (from 0.528 to 0.512 on Set5 for aleatoric), the calibration metrics AUSE and UCE remain largely unchanged or are slightly degraded. These findings suggest that although the perceptual loss enhances visual fidelity, it may introduce noise into the uncertainty estimation process, potentially due to the less pixel-aligned nature of perceptual features. As a result, we chose not to use perceptual loss.
Table 22: Performance of ê₁ choice on CUPID for the super-resolution task. Best- Best-performing results for each metric are highlighted in bold.
Set5 AUSE (L)
Set14
Loss
Pearson (f)
UCE (L)
Pearson (1)
AUSE (1)
UCE ())
Raw
0.528 + 0.006
0.010 = 0.000
0.045 0.018
0.527 0.002
0.012 =: 0.000
0.049 = 0.005
Alcatoric
Add Lipis
0.512 0.006
0.010
0.518 0.009
0.013 0.001
0.051 0.015
Epistemic
Raw
0,416 = 0.005
0.018 0.001
0.266 0.007
0,449 0.005
0.019 = 0,000
0.226 = 0.003
Add Lipis
0.389 + 0.007
0.019 = 0.000
0.253 0.035
0.399 + 0.003
0.021 + 0.000
0,230 + 0.016
Effect of A choice As shown in Table 23 for super-resolution, the Pearson, AUSE, and UCE curves are nearly identical across the full range of x values, and A 0.001 serves as a reliable default.
Table 23: Performance of A choice on CUPID for the super-resolution task.
Set5
Set14
A
Pearson (1)
AUSE (L)
UCE (L)
Pearson (t)
AUSE (L)
UCE (L)
0.001
0.527 0.002
0.010 0.000
0.034 1 0.014
0.528 0.001
0.012 = 0.000
0.064 # 0.016
Aleatoric
0.0001
0.528 = 0.006
0.010 = 0.000
0.045 : 0.018
0.527 : 0.002
0.012 = 0.000
0.049 # 0.005
0.00001
0.529 = 0.002
0.010
0.000
0.037 0.017
0.529 = 0.002
0.012 0.000
0.076 : 0.011
0.001
0.423 0.044
0.018 0.000
0.264 0.012
0.455 0.010
0.018 0.000
0.205 0.009
Epistemic
0.0001
0.416 0.005
0.018 0.001
0.266 + 0.007
0.449 0.005
0.019 = 0.000
0.226 = 0.003
0.00001
0.421 = 0.004
0.018 = 0.001
0.262 + 0.014
0.455 = 0.011
0.019 = 0.000
0.205 + 0.005
E.5 EFFECT OF ERROR MAP SELECTION
In regression- based tasks such as super-resolution, uncertainty metrics like Spearman correlation, AUSE, and UCE require comparison with error map. We conduct experiments to evaluate how the choice of error map L, or -ffeefcts the alignment between the CUPID uncertainty map and the ground-truth error.
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Published as a conference paper at ICLR 2026
Table 25
presents the results for AUSE and UCE. We observe that using the L2 enror map generally leads to lower AUSE values, indicating better aligmment with the overall uncertainty ranking. Con- versely, the L1 enror map yields lower UCE scores, suggesting improved calibration of uncertainty magnitude. This reflects the distinct sensitivities of these metrics: AUSE focuses on ranking quality, while UCE measures calibration.
In addition, Table 24 shows the Spearman and Pearson correlations between the uncertainty maps and the Li/La error maps. Across both Set5 and Set14 datasets, -based correlations consis- tently higher than those computed with L2, especially for Pearson correlation. Furthermore, Spear- man values tend exceed Pearson values, highlighting that CUPID uncertainty is more consistent with the rank ordering rather than the exact error values. These findings suggest that L1 error maps may more appropriate for evaluating uncertainty estimates in super-resolution tasks, particularly when anking-basee metrics are emphasized.
Table 24: Effect of error map selection on Pearson and Spearman metrics.
Set5
Set14
Uncertainty
Error
Spearman (1)
Pearson (t)
Spearman (f)
Pearson (t)
L1
0.643
0.002
0.560 + 0.001
0.607 = 0.002
0.569 = 0.001
Aleatoric
L
0.647 0.002
0.487 0.003
0.611 0.002
0.514 + 0.004
L1
0.229 0.007
0.258 0.005
0.312 0.006
0.327 = 0.005
Epistemic
L2
0.229 + 0.008
0.209 + 0.005
0.312 = 0.006
0.254 + 0.003
Table 25: Effect of error map selection on AUSE and UCE metrics.
Set5
Set14
Uncertainty
Error
AUSE (L)
UCE (L)
AUSE (4)
UCE (4)
L1
0.009 0.000
0.034 + 0.007
0.011 = 0.000
0.017 # 0.008
Aleatoric
L2
0.002 0.000
0.358 + 0.004
0.002 # 0.000
0.269 = 0.021
L1
0.026 0.000
0.320 # 0.004
0.023 # 0.000
0.231 # 0.005
Epistemic
L2
0.005 + 0.000
0.420 + 0.001
0.004 # 0.000
0.411 + 0.001
E.6 MORE VISUAL RESULTS
Figure presents the visual results the proposed CUPID framework on single-image super- resolution. As illustrated in the second row, the uncertainty maps produced by CUPID closely align with the pixel-wise L, and L2 error maps. This consistency suggests that CUPID accurately captures regions of high reconstruction error, reflecting its ability to localize uncertainty. Addition- ally. the difference between the original intermediate feature and the reconstructed feature generated by CUPID indicates that the Reconstruction Branch tends enhance fine-grained image details, particularly in high-frequency regions such as edges and textures.
Figure presents the visual results of the proposed CUPID framework and comparison methods.
E.7 COMPARISON OF RUNTIME ACROSS DIFFERENT UNCERTAINTY ESTIMATES.
Table and Table 27 presents the runtime for training and prediction across all models evaluated in this study. The training time reflects the total duration required to train each model on the EyePACS training dataset, while the prediction time indicates the time taken to generate predictions for the EyePACS test dataset.
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Published as a conference paper at ICLR 2026
Table 26: Runtime comparison between training and testing phases in classification model uncer-
tainty estimation.
Operation
Training Time (s)
Testing Time (s)
CUPID
4776.38
10.06
MC Dropout
49.57
Rate-in
2.85
48.46
IGRUE
5292.41
11.26
PostNet
49932.08
51.04
BNN
54974.84
217.06
DEC
1615.58
6.46
Table 27: Runtime comparison between training and testing phases in regression model uncertainty estimation.
Operation
Training Time (s)
Testing Time (s)
CUPID
26511.74
1.02
BayesCap
81 $1158.45
0.79
in-rotate
2.03
in-noise
2,86
med-dropout
2.28
med-noise
2.47
29

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Published as a conference paper at ICLR 2026
y
CUPID y
Intermediate feature my
New feature mi
Error map
Error map
Aleatoric
Epistemic uncertainty
9
CUPID SR y'
Intermediate feature m,
New feature
Error map
Error map
Aleatoric uncertainty
Epistemic uncertainty
y
CUPID SR
Intermediate fenture my
New fenture mt'
map
map
Aleatoric uncertainty
Epistemic uncertainty
Figure 8: More visual results of the uncertainty feature map of CUPID towards super-resolution model.
30

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Published as a conference paper at ICLR 2026
map
CUPID
CUPID Epis.
BayesCap
SR y
Peurson: In-rotate
0.600
0.451 Med-dropout
0.602
In-noise
Med-noise
0.4
0.0
0.560
0.396
0.231
0.316
Error map L1
CUPID
CUPID Epis.
BayesCap
Pearson:
0.637
0.648 Med-dropout
0.473 Med-noise
In-rotate
In-noise
02
0,492
0.398
0.552
0.450
Error map L.
CUPID Alea.
CUPID Epis.
BayesCap
SR y
0.6
Pearson:
0.659
0.611
0.278
In-rotate
In-noise
Med-dropout
Med-noise
044
0.584
0.429
Figure 9: More visual results of the uncertainty feature map of all the methods.
31


Reviews:
[{"review_id": "XaagFFxBXx", "reviewer": "Reviewer_Lc84", "summary": "The paper proposes a plug-in module that estimates both aleatoric and epistemic uncertainty without retraining or changing the base network. The proposed CUPID is attached to intermediate layers of an existing network and includes two branches: an uncertainty branch that learns heteroscedastic variance for aleatoric uncertainty and a reconstruction branch that generates feature perturbations to quantify epistemic uncertainty. Experiments cover misclassification detection, out-of-distribution detection, and image super-resolution regression", "questions": "Can the authors explain why CUPID Aleatoric performs poorly on the PAPILA OOD detection experiment?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "+ CUPID generally outperforms or matches baselines like MC Dropout, Rate-in, PostNet, BNN, DEC, and BayesCap, with its two branches complementing each other across tasks.\n+ A theoretical section provides a first-order Taylor expansion showing that epistemic uncertainty scales with the product of network sensitivity and feature deviation.", "weaknesses": "- The AU branch resembles standard heteroscedastic regression, and EU relies on output-preserving perturbations, which is close in spirit to prior work that estimates distributional shift via reconstruction error [RUE (Wang et al., 2023)]. It is unclear what the most significant novelty of this paper is.\n- CUPID adds a learned module plus an extra forward of the perturbed path, but there is no computational cost analysis. \n- The authors claim CUPID produces reliable uncertainty estimates, however, they do not check if high predicted uncertainty actually corresponds to higher error or misclassification rate. They only reports a calibration measure UCE in the regression (super-resolution) tasks, but not in classification or OOD detection.", "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": "RoqG0pNOhJ", "reviewer": "Reviewer_o9ug", "summary": "This paper proposes a joint estimation of aleatoric and epistemic uncertainty throuhgh a plug-and-play  auxililary module insertable at any layer of a pretrained deep model. The aleatoric uncertainty is estimated through an uncertainty branch and the epistemic uncertainty is estimated through the difference between original and perturbed outputs. The aleatoric and epistemic uncertainty loss is applied together to train the auxiliary module. The usefulness of the quantified uncertainty has been demonstrated in medical image misclassification detection, OOD detection, and image resolutions.", "questions": "The epistemic uncertainty part of the loss function seems similar with the identity mapping of the BayesCap function. Is it possible to draw a direct comparison with this paper's loss with BayesCap?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 1, "presentation": 2, "contribution": 1, "strengths": "**1. Well-written:** The paper is well-written and well-motivated with good literature review. \n\n**2. Experimental results:** The paper is enriched with numerous experimental evaluations of their methods in different tasks and demonstrated their success with AUC, AURC, spearman and Pearson coefficients. They also evaluated the calibration error. Their proposed method perform better in almost all cases.", "weaknesses": "**1.Confusing interpretation of epistemic uncertainty:** The epistemic uncertainty has been considered as the discrepancy between the original and the perturbed prediction. However, why this discrepancy captures the model's lack of knowledge is not clear from the paper. \n\n**2. Limited novelty in aleatoric uncertainty:** The estimation of aleatoric uncertainty seems similar with BayesCap [2] and therefore that part has limited novelty.  \n\n\n[1] Depeweg, S., Hernandez-Lobato, J.M., Doshi-Velez, F. and Udluft, S., 2018, July. Decomposition of uncertainty in Bayesian deep learning for efficient and risk-sensitive learning. In International conference on machine learning (pp. 1184-1193). PMLR\n\n[2] Upadhyay, U., Karthik, S., Chen, Y., Mancini, M. and Akata, Z., 2022, October. Bayescap: Bayesian identity cap for calibrated uncertainty in frozen neural networks. In European Conference on Computer Vision (pp. 299-317). Cham: Springer Nature Switzerland.", "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": "SMBOAdZwXj", "reviewer": "Reviewer_D4Tj", "summary": "The CUPID framework proposes a model-agnostic, plug-in method for estimating and decomposing aleatoric and epistemic uncertainties within deep neural networks. Instead of retraining or modifying the base model, CUPID attaches an auxiliary uncertainty branch to intermediate feature layers, allowing uncertainty estimation through a single forward pass. The aleatoric component is predicted via a regression branch that models data-dependent variance, while the epistemic component is derived from the feature reconstruction error of a decoder, both regularized under a Bayesian identity mapping to prevent mutual interference. This layer-wise design enables localized uncertainty interpretation and can be flexibly applied to pretrained networks across different tasks. Experimental results on classification, regression, and OOD detection demonstrate that CUPID achieves competitive or superior calibration and interpretability compared to existing post-hoc and evidential approaches, highlighting its practicality as a lightweight uncertainty quantification module.", "questions": "Q1.\nIn Tables 1, 2, and 3, I could not find the results of the unified model that jointly trains the aleatoric and epistemic branches. Moreover, the performance trends between the aleatoric and epistemic versions appear quite contradictory, suggesting that joint training might degrade performance. This inconsistency limits the practical benefit of the proposed method, as it becomes cumbersome to decide which type of uncertainty is more suitable for a given task.\n\nQ2.\nUsing feature reconstruction as a proxy for epistemic uncertainty is not new. What distinguishes CUPID’s reconstruction-based approach from prior autoencoder-based OOD detection or evidential methods? Furthermore, is there any theoretical justification for interpreting reconstruction error specifically as epistemic uncertainty rather than as a general measure of representation distance?\n\nQ3.\nThe aleatoric branch directly regresses variance from intermediate feature activations. How is this branch trained in classification settings where ground-truth noise levels are unavailable? Are the predicted variances calibrated, normalized, or otherwise constrained to ensure meaningful uncertainty estimates?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The proposed CUPID framework offers a simple yet flexible plug-in design that can be applied to pretrained models without retraining. This practical modularity makes the method attractive for real-world applications where uncertainty estimation must be added post hoc. The joint decomposition of aleatoric and epistemic uncertainties provides a unified view of model and data uncertainty, and the layer-wise formulation enables interpretable analysis of which network components contribute most to uncertainty. These design choices improve usability and diagnostic insight with minimal computational overhead.", "weaknesses": "**Methods**\n\nThe proposed methods for estimating aleatoric and epistemic uncertainty are conceptually straightforward and largely aligned with prior works. The aleatoric branch essentially performs feature-based variance regression, similar to heteroscedastic uncertainty modeling in Kendall & Gal (2017) and Evidential Deep Learning (Sensoy et al., 2018). The epistemic uncertainty is derived from feature reconstruction errors, an idea previously explored in autoencoder-based OOD detection (An & Cho, 2015) and variational approaches. Therefore, the underlying mechanisms are not fundamentally novel. The contribution of CUPID mainly lies in its plug-in, layer-wise formulation and the joint decomposition enabled by the Bayesian identity mapping, which improves modularity and interpretability but does not constitute a substantial algorithmic innovation.\n\nReferences\nAn, J. & Cho, S. (2015). Variational Autoencoder based Anomaly Detection using Reconstruction Probability. — Workshop on Machine Learning for Signal Processing (MLSP), IEEE.\n\n\n**Experiments**\n\nAlthough Section 3.4 describes a unified objective for jointly training the aleatoric and epistemic branches, the experiments appear to use separately trained versions (CUPID-A and CUPID-E) rather than a fully joint model. The paper would benefit from clarifying whether the joint training was actually performed, and if so, providing explicit results comparing the jointly trained version with the individual branches. Otherwise, the claimed joint decomposition may remain only conceptual rather than empirically validated.", "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": "kS43S7ROXI", "reviewer": "Reviewer_7Juj", "summary": "This paper introduces Comprehensive Uncertainty Plug-in estImation moDel (CUPID), an uncertainty estimation method that quantifies both epistemic uncertainty (EU) and aleatoric uncertainty (AU) without requiring retraining of the base model. The method introduces a CUPID module, which operates on top of intermediate network features to estimate uncertainty. This module learns to capture AU through a log-likelihood objective and EU through a loss which encourages maximal feature differences and minimal reconstruction differences. Experiments show performance of CUPID on downstream tasks like misclassification detection, out-of-distribution (OOD) detection, and regression.", "questions": "1. I'm a little unclear on the intuition behind the EU loss function in Section 3.3. Why do we want the intermediate features to be maximally different, but the outputs to be minimally different? \n2. What are the recommended strategies for choosing $\\lambda_1$ (e.g., is there a fixed value that tends to work well or does this need to be carefully tuned)?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "1. Novelty of the proposed method. The proposed objectives and CUPID module are novel ideas that seem quite robust and well-performant for downstream applications. The method also has the benefit of not requiring retraining of the base model, reducing computational overhead.\n2. Clarity. The presentation of the paper is well organized and discussion are clear / easy to follow.\n3. Experimental results. The experiments cover a good number of baselines and applications, giving a good benchmark of how the method is expected to perform in real-world scenarios.", "weaknesses": "1. Computational analysis. While the method doesn't require retraining of the base model, some metrics (e.g., runtime, memory consumption) of CUPID should be provided to give an idea of the expected computational overhead.\n2. For the misclassification experiment in Section 4.1, it would be good to find some empirical evidence that GLV2 is AU dominant and HAM1000 is EU dominant (i.e., not just by looking at other baseline results, but by examining at the data itself).", "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)