Vicinal Label Supervision for Reliable Aleatoric and Epistemic Uncertainty Estimation
Authors: Linye Li
Yufei Chen
Xiaodong Yue
Venue: NeurIPS 2025 poster ()
Keywords: evidentail deep learning; uncertainty estimation
Primary area: deep_learning

Abstract: Uncertainty estimation is crucial for ensuring the reliability of machine learning models in safety-critical applications. Evidential Deep Learning (EDL) offers a principled framework by modeling predictive uncertainty through Dirichlet distributions over class probabilities. However, existing EDL methods predominantly rely on level-0 hard labels, which supervised a uncertainty-aware model with full certainty. We argue that hard labels not only fail to capture epistemic uncertainty but also obscure the aleatoric uncertainty arising from inherent data noise and label ambiguity.
  As a result, EDL models often produce degenerate Dirichlet distributions that collapse to near-deterministic outputs. 
  To overcome these limitations, we propose a vicinal risk minimization paradigm for EDL by incorporating level-1 supervision in the form of vicinally smoothed conditional label distributions. This richer supervision exposes the model to local label uncertainty, enhancing aleatoric uncertainty quantification, while also mitigating the degeneration of the Dirichlet distribution into a Dirac delta function, thereby improving epistemic uncertainty modeling.
	Extensive experiments show that our approach consistently outperforms standard EDL baselines across synthetic datasets, covariate-shifted out-of-distribution generalization tasks, and out-of-distribution detection benchmarks, providing more reliable uncertainty estimates.

Vicinal Label Supervision for Reliable Aleatoric and Epistemic Uncertainty Estimation
Linye Li
Yufei Chen"
School of Computer Science and Technology Tongji University. Shanghai, China Linyeli@tongja @tongj edu.cn
School of Computer Science and Technology
Tongji University, Shanghai, China
yufeichen@@oongj edu. cn
Xiaodong Yue Artificial Intelligence Institute Shanghai University, Shanghai, China y yswantfly@shi edu. cn
Abstract
Uncertainty estimation is crucial for ensuring the reliability of machine learning models in safety-critical applications. Evidential Deep Learning (EDL) offers a principled framework by modeling predictive uncertainty through Dirichlet distri- butions over class probabilities. However, existing EDL methods predominantly rely on level-0 hard labels, which supervise an uncertainty-awann model with full certainty. We argue that hard labels not only fail to capture epistemic uncer- tainty but also obscure the aleatoric uncertainty arising from inherent data noise and label ambiguity. As result, EDL models often produce degenerate Dirich- let distributions that collapse to near-deterministic outputs. To overcome these limitations, we propose a vicinal risk minimization paradigm for EDL by incor- porating level-1 supervision in the form of vicinally smoothed conditional label distributions. This richer supervision exposes the model to local label uncertainty, enhancing aleatoric uncertainty quantification while mitigating the degeneration of the Dirichlet distribution into a Dirac delta function, thereby improving epistemic uncertainty modeling. Extensive experiments show that our approach consistently outperforms standard EDL baselines across synthetic datasets, covariate-shifted out-of-distribution generalization tasks, -iito--distribution detection, and selective classification benchmarks, providing more reliable uncertainty estimates.
1 Introduction
Reliable uncertainty estimation is pivotal for deploying trustworthy machine learning systems, particu- larly when models encounter distributional shifts operate in safety-critical environments. Classical approaches, such as MC Dropout [14), Deep Ensembles [26], and Bayesian Neural Networks [4] estimate uncertainty via Bayesian model averaging. While effective, these methods often ineur high computational costs due to multiple forward passes complex posterior approximations.
Evidential Deep Learning (EDL) 45
36
37
6
7
53 10 has recently emerged as promising
alternative. EDL models predictive uncertainty by representing class probabilities with a Dirichlet distribution, enabling uncertainty estimation or model ensembles. This framework enables joint quantification of two complementary types of uncertainty within single forward pass: (i) Aleatoric uncertainty, which captures inherent data noise
single forward pass without relying on sampling
"Correspondng author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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or label ambiguity, and (ii) Epistemic uncertainty, which reflects uncertainty stemming from limited data or insufficient knowledge about the gatteeennraaing process. EDL has also achieved substantial
progress in several downstream tasks, such as trusted multi-view classification [15 31
32
34
49
33
[29 and domain adaption 42
35 54
where robust and reliable uncertainty estimation essential.
However, since ground-truth Dirichlet distributions are unavailable, most EDL methods still rely on one-hot labels (i.e., level-0 supervision). similar conventional softmax classifiers. Recent studies [2] have revealed fundamental limitations of this practice: training with hard labels drives the predictive distribution to collapse into degenerate Dirac delta functions, resulting in overconfident outputs and poor uncertainty estimation. To address this issue, follow-up up work [20] introduced level-1 I supervision using crowdsourced soft labels (e.g., CIFAR- 10H 443), but such approaches require extensive human annotation (up to 500k judgments), making them costly and impractical. Moreover, existing studies mainly focus on epistemic uncertainty under level-0 supervision, while its ability to capture aleatoric uncertainty remains largely unexplored.
3,22 46
In this paper, we argue that the mismatch between fully certain level-O supervision and uncertainty- aware level-2 predictions fundamentally limits accurate estimation of epistemic and aleatoric uncer- tainty. Hard labels fail to capture class ambiguity and data noise, leading Dirichlet distributions to collapse into Dirac deltas and produce overconfident, unreliable estimates. To mitigate this issue, we propose new EDL training paradigm that narrows the supervision gap by replacing hard labels with estimated level-1 1 conditional categorical distributions. Specifically, we adopt a vicinal risk minimization ( (VRM)-based strategy that constructs level-1 supervision from local feature neighbor- hoods, inducing continuous label distributions and improving both aleatoric and epistemic uncertainty estimation without extra annotations.
Our main contributions are as follows:
We introduce a novel EDL training paradigm leveraging level-1 supervision via VRM to better capture aleatoric and epistemic uncertainty without additional annotation cost.
We provide theoretical insights into how this supervision improves generalization, supported by a risk-based analysis, and enhances aleatoric and epistemic uncertainty estimation by mitigating the Dirichlet distribution's 's collapse towards Dirac delta measure.
We empirically demonstrate the effectiveness of our method in uncertainty estimation and robustness under covariate-shine -hiffte out-of distridistiiittonn generalization, selective classification, and
detection.
2 Problem Formulation
In this section, we first discuss the key differences and connections between EDL and traditional point-estimate classifiers, such as softmax- based ones. Then, we highlight the limitations of EDL, particularly its challenges in estimating epistemic uncertainty and aleatoric uncertainty.
Basic Notations. a standard supervised K classification setting with instance space X. label space
yK ). and training dataset D = (L) ,y)"), (ax(*), ("))) C x y. Following y = (V1, ...,,,
classical settings, we also assume that the data are generated i.i.d. according to an underlying joint probability P over X x y. Correspondingly. each instance æ E is associated with a conditional distribution p(-(z)) on V, such that (()(z) is the probability to observe label y as outcome given Let IP; (D") denote the set of probability distributions over y, and ()) the set of distributions over Pi (D)).. Elements of P1 (J)) are called level-1 distributions, and those of IP2 (3²) level-2 distributions. As in 22. different levels of distributions in classification tasks can be organized as follows:
Level-0 (hard labels): A deterministic class label y {l...., KJ. which implicitly assumes that the sample belongs to a single class with absolute certainty.
Level-1 (categorical distribution): A probability vector p E A over classes, where AK1 is the (K 1) 1)-simplex. This representation captures aleatoric uncertainty by modeling ambiguity or inherent noise in class membership.
Level-2 (Dirichlet distribution): A second-order distribution Dir(a:) over categorical distributions p. This signal encodes both aleatoric and epistemic uncertainty. The expected value E E'p] = axS 5 (where S =x ag) reflects class probabilities (aleatoric), while the

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concentration S controls the dispersion around the mean. Low S indicates high epistemic uncertainty due to limited knowledge, and high S corresponds to high confidence.
In short, level-0 distribution provides no information about uncertainty, level-1 distribution can only express aleatoric uncertainty, whereas level-2 distribution can jointly represent both aleatoric and epistemic uncertainties in a principled manner.
2.1 Learning Predictive Level-1 Models
Given a K-class classification task with input space A' and label space y the model outputs a
probability vector p (P1,-... pk) E Pi ())); here, PP ()) denotes the probability simplex
the label space and IP (3)) E [0, Lj=1 Pj 1 The loss function for level-1 predictors takes the form
L A x - R.
(I)
Commonly used loss functions for level-1 predictors include the cross-entropy (CE) loss and the Brier score as
K
K
CE (p.))
Σ (==) log(p;),
I. Brier (p. v) Σ 10=)))²
(2)
j=1
j=1
Here, (1(==)) is the indicator function that equals if class index j corresponds to the true label y, and O otherwise. Let hypothesis space Hi C ()))* {h: X - P(())) to be given. In traditional supervised learning, the goal is to find a hypothesis h E Hi that minimizes the risk
R(h) S
(.(h(ar).)dP(C
(3)
Xxxy
where R(h) is the risk of hypothesis h. The hypothesis is commonly learned via Empirical Risk Minimisation (ERM), which involves minimizing the empirical risk defined as:
Remp (h; D) = Σ La (h(x (m)
Since Remp(h; of the true risk minimizer h²" They are defined as follows:
(h; D) is estimate of the true risk R(h). the learned hypothesis h is an approximation
h == arg min Remp D), h- arg min R(h). bE
(5)
Consequently, there remains an approximation gap between å and h as well as epistemic uncertainty
regarding h", as only single point estimate of the predictive distribution is obtained [14 (5].
2.2 Learning Predictive Level-2 Models
Unlike level-1 models, EDL learns hypothesis space H2 of the form H2 C P2(3/* (h: F2))))} To learn a level-2 predictor, the ideal scenario would involve access to a ground-truth distribution Q" ())), which could directly supervise the model output the target level- 2 distribution. However, such ground-truth distributions Q" are typically unavailable in practice. Consequently, existing methods adopt an alternative approach inspired by level-1 models. Specifically, a level-2 loss function is defined as
L2 P2()) --
which compares the level-2 prediction h(x) against level-0 observation y. The learning objective is minimize the empirical level-2 risk over the training data D
R(2) emp
(h)
1 N
Σ
L (2( (m)).
This paradigm, known as evidential deep learning (EDL). aims to learn reliable level-2 distribution predictor for uncertainty estimation by minimizing the L2 loss on the available data [45] 36 6]. Several previous works have proposed the minimization of an empirical loss of the form
L2(Q. v) = Ep-oLi((.1)
(8)
where the level-2 prediction Q is penalized in terms of the expected level- loss.
3

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2.3 Limitations of Aleatoric and Epistemic Uncertainty Estimation in EDL
However, recent studies have raised substantial criticisms regarding the uncertainty estimation behavior of EDL. Specifically, it has been argued that minimizing empirical risk under standard EDL frameworks with level-0 observations inevitably drives the learned evidential distribution to collapse into a Dirac measure. Consequently, epistemic uncertainty is effectively suppressed or unreported in Building upon these insights, we conduct a systematic analysis of epistemic uncertainty estimation in EDL and rigorously formalize how the distributional collapse phenomenon undermines ability to quantify uncertainty. Furthermore, we identify and theoretically characterize an additional, underexplored limitation: EDL also fails to faithfully capture aleatoric uncertainty under the same empirical risk minimization principle. Specifically, we establish the following result:
practice 23 22
Theorem 1. For any level-1 I loss function L1 IP P) x + R. that satisfies L (I(pp-Diira)))
Ep-Diaa (p..), (i.e., is convex function), such as Brier score and the log-loss in Eq. 2 the empirical risk minimizer of level-2 prediction is always a Dirac measure óp E P2(V) and the expectation of level-2 prediction is 6y E P)).. This result holds if the learner possesses a universal approximation property, allowing it to represent such degenerate distribution.
Here, 6, denotes the Dirac measure at y E Y. which is element of Pi ())) representing a prediction with no aleatoric uncertainty. While prior work 12 3 22 has established that Empirical Risk Minimization (ERM) with level-0 labels leads to collapse of epistemic uncertainty for second-order predictors like EDL, Theorem highlights that aleatoric uncertainty also vanishes under the same conditions. This observation inspired by the analysis in Theorem 3.3 of [22], which shows that the L1 loss is minimized when the prediction is a Dirac measure óy centered on the ground-truth label y. The complete proof is provided in the Appendix B An intuitive explanation is that under level-O supervision, even if the predicted distribution p accurately reflects the learner's aleatoric uncertainty, it is not the minimizer of Eq. B As result, a learner trained to minimize this loss with level-0 labels will not output such a distribution to degenerate distribution 6, that collapses the uncertainty representation into a single point mass.
Instead, the optimal prediction becomes the mode of p. leading
Proposition 1. Under the assumptions of Theorem 7 empirical risk minimization of level-2 prediction inevitably yields degenerate distributions 6p E P2()) and the expectation of the level-2 prediction is by E Pi ()²). As result, the model fails to provide any meaningful or disentangled representution of aleatoric or epistemic uncertainty.
This proposition follows directly from Theorem since the optimal prediction always collapses to a Dirac measure at the expected probability vector, the learner incentivized to output deterministic posteriors, irrespective of whether the uncertainty arises from stochastic labels (aleatoric) or from limited evidence (epistemic). Consequently, any observed uncertainty in the model's prediction cannot be separated into aleatoric and epistemic components.
As level-0 labels provide fully certain supervision, learning aleatoric uncertainty over p remains challenging. A natural solution is to leverage level-1 1 soft labels, as in (20). However, most standard datasets 24 3
only offer hard labels, and collecting accurate level-1 annotations (e.g., CIFAR 10- H (43) is impractical. To address this, we propose a VRM strategy that approximates soft labels by interpolating between neighboring samples.
3 Method
Our proposed method enhances uncertainty estimation in Dirichlet-based models by leveraging 'RM-approximated soft labels. This strategy enables learning aleatoric uncertainty from datasets restricted to hard labels. The method consists of two complementary components: one strengthens the model's capacity for alcatoric uncertainty capture, while the other preserves epistemic uncertainty by preventing the Dirichlet distribution from degenerating into Dirac delta function.
3.1 Vicinal Supervision to Enhance Aleatoric Uncertainty Estimation
The empirical risk minimization trains a model minimize loss the exact training samples (m)()) ()) but ignores the fact that the true data distribution P(æt, y) is continuous and often smooth in the instance-label space X' y. VRM (5) addresses the continuity of the instance space by introducing vicinal distribution, which augments the training set with virtual examples drawn

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from local neighborhoods of the data while keeping the labels unchanged. The Mixup method [5] extends the VRM principle by applying linear interpolations not only the input features but also to the labels. The empirical vicinal risk in this manner is defined as:
Σ / R=1
N
(h;
p( ") y (m)) dē dÿ,
N
where the vector y denotes the label in one-hot format and p(ä, y y ()) represents the joint probability density function of vicinal samples. In practice, the interpolated samples and labels are generated by linearly interpolating between pairs of training examples:
z Aa (n) + (1 A)z(m)
y Xy(
(n) + (1 A)y (m))
()
(10)
where (x)()) (m) is another randomly selected training sample, and X E [0, 1] is drawn from Beta distribution (eta () with hyperparameter 3 > O. The hyperparameter 3 controls the shape of the Beta distribution. When | < 1, the distribution is U-shaped, favoring extreme values of À = which leads to mixtures dominated by a single sample. In contrast, when 3 > the distribution concentrates around À = 0.5, promoting strongly balanced mixtures between the two samples. Inspired by the idea of Vicinal Risk Minimization (VRM), we incorporate vicinal information by generating vicinal level-1 labels ÿ. which are subsequently used as the supervision target of the L1 loss. The resulting objective formulated as:
Lvicinal
(2(()) (ynd))))("))),,.,)) ((")) ()m))))))) ( (D BewBBel Dir(a(a) (L.
(II)
In contrast to the original Mixup [51], which suggests setting 3 0.2 or 0.4, we set $ > (e.g.. 10, 20) to enforce strong mixing. Such strong mixing creates soft labels that represent high aleatoric uncertainty, encouraging the model to account for inherent label ambiguity and improve uncertainty calibration.
3.2 Noise-Augmented Vicinal Risk Minimization for Epistemic Uncertainty Estimation
In addition to simulating samples with high aleatoric uncertainty, we introduce controlled noise into the input space to generate samples that exhibit inherently high epistemic uncertainty-icce samples that are difficult to model due to insufficient, ambiguous, or incomplete information [1)] Specifically, we propose augmenting VRM with synthetic vague samples generated via Gaussian noise to account for the simple fact that the observed features æ may not contain sufficient information to fully explain the target y. We formalize this with the following noise-augmenter loss:
nois (g00))) (/(1))))))))))))))))))) (/1))> DD (.ftm)) (m(mm) ((0,0021))) E Bx (inee Pir- (e) [T. (p.
(12)
Here, (mn) is sampled from Gaussian noise, label (m)) as a uniform distribution
1 K
z (n) (m)))m)))))
Xy (m) +1 A)
(13)
K
While the added noise can inadvertently increase aleatoric uncertainty, we primarily introduce it to simulate obstacles to the model's s knowledge acquisition and encourages the model produce smoother uncertainty estimates in the vicinities of the training data. Furthermore, we observe that adding label smoothing helps control the growth of the Dirichlet strength for samples near the decision boundary, mitigating the degeneration of the Dirichlet distribution into a Dirac delta function, as formalized in Theorem 3 The Beta distribution parameters &it and 9 govern the mixing proportion. A larger 3 increases the contribution of noisy samples (i.e., smaller À on average); A larger 3 B emphasizes the clean (original) samples. We set S 2 9 9 to ensure that original samples dominate the interpolation, thereby preventing excessive degradation of predictive performance while still allowing the model to explore uncertain vicinities.
3.3 Model optimization
We follow EDL 45 to train neural network that predicts the parameters of a Dirichlet distribution.
Specifically, we set the Dirichlet prior as a non-informative prior a = [1,...,]] 1]. The neural
network's outputs (xx) are passed through a non-negative activation function o(:) e.g. ReLU
5

---

or SoftPlus, to obtain the evidence vector = (ei....,ee, ext i.e., e o (2(z)) The Dirichlet parameters are then computed as a + a. For the loss function, we adopt the entespy--o-----a based EDL loss, defined as:
K
Din
(1 (p, "pp.
/
Σ 3y log (pz)
1
II
Pj
a. -
dp
B (aa)
(14)
Σ ( (())))
where S = - Oj ((-) is the digamma function. Finally, the total vicinal loss is defined as:
L Leicinal + Lnoise-
(15)
By jointly optimizing the total loss, the model can not only fit the training data effectively but also express predictive uncertainty more reliably and generalize better to unseen or ambiguous scenarios.
4 Theoretical Analysis
We provide a theoretical analysis to elucidate how level-1 labels with strong mixing improve gen- cralization and robustness in the presence of input-dependent label noise. Then, we analyze how the hyperparameter À in Eq. 13 moderates the rapid increase of the Dirichlet concentration, thereby slowing its degeneration towards a Dirac delta function. The complete proof is provided in the Appendix B
Theorem 2. Let the ground-truth level-1 label be denoted as px)),, and let the observed level-O one-hot label (z)) be a noisy realization of (x) perturbed by input-dependet label noise p(x)
6y(2)) = (xx) + ((x)
where
p(æ) NN
(16)
Then, the test risk admits the following lower bound under mild regularity conditions
R(h; P) 2 Co²
(17)
where C' depends on the trace of the Hessian matrix of the loss function with respect to p. Then, for the level-1 label with strong mixing, the bound can be tightened
R(h; P) 2 0²
(18)
where C//C 28+1 + (VB 1/2), indicating a reduced sensitivity of the test risk to input-dependent noise.
Theorem implies that leveraging level-1 labels with strong mixing can effectively reduce the lower bound of the generalization error.
Theorem 3. Let À be the mixing hyperparameter defined in Eq.
13
Consider the optimization of the
Dirichlet parameters a Eq. 14 For samples where og S aj
(V) # kk) with lower belief assigned
to the ground-truth l class, the Jollowing properties hold
The update to the Dirichlet concentration for the ground-truth class Nork increases mono- tonically with X.
The updates to the Dirichlet concentrations for the son-ground-truth classes Dajzke de- crease monotonically with X.
The total increase in Dirichlet concentration. denoted AS, increases monotonically with X.
Theorem Sannreemminnn Bimplies that properly chosen À < 1 can effectively suppress the excessive accumulation of evidence, i.e., AS This prevents Dirichlet-hased models from collapsing into a degenerate Dirac delta distribution 6p, thereby enhancing their ability to represent epistemic uncertainty.

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Experiments
We begin by analyzing and comparing the estimated uncertainties estimated by method and baseline approaches on toy dataset. Subsequently, we conduct extensive experiments on three main tasks: OOD detection, selective classification, and OOD generalization. For the OOD detection task, we evaluate the ability of different methods to distinguish between in-distribution (ID) and out-of-distribution (OOD) samples based their estimated epistemic uncertainty. For selective classification, we assess the model's capability to differentiate correctly classified samples from misclassified ones using aleatoric uncertainty. For the OOD generalization task, we examine the classification performance of models when exposed to covariate-shifted OOD samples.
5.1 Experimental Setup
Baselines. Baseline methods include KL-PN 36 RKL-PN (37. PostNet [6]. NatPN 77. EDL 45
RED 44). I-EDL [1]. R-EDL 9 H-EDL 44
and DA-EDL [50]. For OOD detection tasks,
we further extend our experiments to include four OOD detection methods based on uncertainty
estimation: DUQ DDU
38 DUE and SNGP 30
Evaluation Metrics. We evaluate OOD detection performance using the Area Under the Receiver Operating Characteristic curve (AUROC), which measures the model's ability to distinguish between ID and OOD samples. For selective classification, we employ the Excess Area Under the Risk- Coverage Curve (E-AURC 1000; lower is better), where lower E-AURC indicates more reliable aleatoric uncertainty estimation and better selective prediction performance. OOD generalization is assessed by measuring the classification accuracy on covariate-shifter OOD test sets. For comparison, the classification accuracy on the ID test set is also reported. All results are reported as the mean + standard deviation over 10 independent runs with different random seeds.
Implementation Details. Following OpenOOD (52), we train a ResNet- 18 model [16 implemented in PyTorch [41] for 100 epochs
single NVIDIA A100 GPU. We use the SGD optimizer with a cosine annealing schedule, an initial learning rate of 0.1, and a batch size of 128. We set the hyperparameters
(Eq. and Broise
Maiss 1.0. Further implementation details for the
baselines are in Appendix
Uncertainty Measure. Existing methods adopt different strategies quantify epistemic uncertainty. KL-PN (36] and RKL-PN (37] use mutual information (Eq. 66); PostNet (6] and NatPN (7] rely on
the Dirichlet total strength ΣØ= O'j. where a smaller S indicates higher uncertainty. Methods
44 50 use vacuity (K/S) as their measure of uncertainty. We propose using conditional entropy for aleatoric uncertainty (Eq. 64 and the Dirichlet differential entropy for epistemic uncertainty (Eq. 67)
based on Dempster-Shafer Theorr and Subjective Logic 45 40
12
9
Datasets. Following prior EDL works, we conduct OOD detection using CIFAR-10 and CIFAR- 100 (24] (32 x 32 resolution). When using CIFAR-10 (or CIFAR-100) as the ID dataset, the OOD datasets include CIFAR-100 (or CIFAR-10), Tiny ImageNet 277) MNIST 28 SVHN [39]
Textures (25), and Places365 [55]. For OOD generalization, we evaluate CIFAR-10-C and CIFAR-100-C [17]. which contain 15 corruption types (c.g., snow, fog) at 5 severity levels.
Test Set Predictions
Teat Prediction
Set Precictions
9445
ndre
-134
1.26
-8-
J.12
[UZ4
-4
1000
600
5.06
(a) MSP (C (V)) x ))
(b) EDL (C (a))))) ))
(c) Ours (L P2 (.))
Figure 1: Comparison of estimated uncertainty of different methods on toy dataset.
7

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Table OOD detection (AUROC) and classification accuracy on CIFAR- 10 and CIFAR 100 dataset.
-CIIFAR-100
--Tny
-MNISS 0OD Detect
-SVHN
-TTextures
-Places365
CIFAR-10
Method
ODD Detect
OOD Detect
0OD Detect
OOD Detect
OOD Detect
Cls Acc
DUQ [CML22))
84.6011.00 1.04
86. 16.99229
9..34=1.25
91.36::1.26
86.57+
84.26:10.93
03.6010.39
DDU (CVSS25)
89.33
91.28:10.28
95.69:0.99
13.53++143
92.64=0.31
91.555+0.34
95.35
DUE 48
86,00-00,78
08.40:10.76
92.34=1.48
91.70=+1.63
1.
88.89:10.77
94.98:0.15
SNGP (NIPS20)
0
18.1
90.38:+1.00
93.18=1..00
92.16:12.40
92.88:11.37
91.211+:1200
94.63:10.18
KL-PN 1N
03.4610.88
85.2440.97
87.80=3.81
88.39422.77
1.34
84.36:10.7
90.31=1.46
RKL-PN PNN-PPN [NTPS:9
60.43-122.54
63.04:12.43
85.761:3.03
43.97:19.22
59.17+3.17
66.05:12.62
53.22:14.25
Pest (NUPS30)
3
81.46+11.00
77.78+3.88
85.19+2.19
88.27411.6
8..87+1.23
82.97+1..6
NadPN [NPS22]
78.4410.83
80.2440.41
81.0914.39
82.23+11200
82.23=5.12
80.76:10.99
86,25
EDL (NISSI]
+
86.64:0..25
90.5941.21
93,24+0.62
93.56++1.01
9..66-20.62
90.25:10.44
94.15
RED (CML22)
85.8110.34
88.07:0.27
91.6011.50
92.12:10.92
18.0811.47
88.0541.338
94.8340.18
Z-EDL (SMLL23)
88.11:00.45
90.83:10.45
9..20=
94,77-+1.62
91.29
90.38:1040
04.95:1..17
R-EDL [CCL224
86.88:10.08
89.7240.47
90.66=1.40
92.53+4.25
90,79=0.79
87.06-40.31
92.92-00.13
H- EDL (NIPS244
68.60:0.229
91.43:10.19
05.60
92.9910.67
0.397=0.34
92.07=0.322
95.0410.005
DA ED-. (ICML .4)
82.39+00665
84.0340.63
88 08.8
N6.98+00.69
82.27+0.87
K3.40+0.45
92.57
Ours
89.0910.19
91.81:1.22
97.32::0.42
96.20:1.32
0.751
916111011
0.11
-:CIFAR-I0
Tiny
-MNIST
-SVHN
TTxtures
Placce365
CIFAR-100
Method
OOD Detect
OOD Detect
OOD Detect
OOD Detect
OOD Detect
OOD Detect
Cls Acc
DUQ [CML21)
$1.20 884
23.60:12.67
13.44
614747.64
57.73=5.66
50.16:+.113
1.6610.39
DDU [VER233
15
18.1411166
1.564
79.69=6.56
76.02:13.98
83.00=1.01
74.53=:1.70
78.95:10.96
DUE 45
1.53
1..97
49.91
49.91++1.02
50.02
50.12:1.01
1.06-+0.17
SNGP INIPS20)
12.77=11.23
1..63
71.91
73.5415.33
1.85
74.53=+1.93
16.19111.19
KL.-PN 18
16
57.20:12.79
1.85
25.20:1.2.34
50.90 12.39
39.38:13.61
57.93:12..0
24.89 10.48
RKL-PN [NIPS19]
53.134:2.15
1..30
48.27422.03
15.46=10.83
44.35
54.2443.88
17. 17.6412..80
PostNet [NIPS0]]
3
54.19100.59
53.58:10.51
75.93-111.81
89.89-8.155
42.08
$3.8611
3.08:1:.23
NarPN [NPS21]
67.77+0.87
70.6940.69
65.20+4.90
75.34++2.77
66.53+1.60
69.25++0.33
59.01
EDL [NIPSI6]
56,49:1.447
1.940:11.22
28.84
13.78=14.68
49.6813.88
56.7412268
30.23++2.72
RED (CML21
78.17+0.35
81.79+0.18
78.45+2.53
80.98
77.79+0.31
78.71++0400
0..6
I-EDL [BML233
1
77.4210.31
82.39-10.29
76.22:1.83
78.9110.25
78.54=0.31
79.6510.19
77.1010.12
R EDL (CCUR24
65.45:12.01
71.28:0.53
79.44
78.50-12.49
1.335
75.35
46.70
H-EDL [NIP524]
E11
75.73:10.39
80.81+0.26
4..91
83.56:23077
75.74=0.70
79.97:0.59
77.75:0.23
DA EDL (ICML224)
13.07:11.03
56.78:10.64
6..63:3.12
48.02:12.34
49.53-2.21
54.48-2.11
16.59:10.76
Ours
81.86+00.19
83.40-
86.17 1,83
85.07++1.94
78.58+0..3
79.97-00399
78.29+0.11
5.2 Experimental Results
Uncertainty estimation toy dataset. We begin by conducting experiment toy dataset consisting of three Gaussian clusters, as shown in Fig. T To maintain consistency, the uncertainty measure for all methods is defined the vacuity of evidence, i.e., K/S. Obviously, with level-1 supervision, our method yields more precise uncertainty estimation; in particular, it assigns higher uncertainty to regions that are far from the in-distribution data and near decision boundaries.
Evaluation of epistemic uncertainty via OOD detection. As shown in Table our method achieves competitive performance in detecting OOD samples. By leveraging vicinal label information, method Iearns a smoother uncertainty landscape, leading to more reliable uncertainty estimates without imposing any regularization the output Dirichlet distribution. Besides, as demonstrated in Theorem 3 our method mitigates the tendency of the Dirichlet distribution for uncertain samples to collapse towards Dirac delta function, further enhancing the accuracy of uncertainty estimation.
Evaluation of aleatoric uncertainty via selective classification. As shown in Table 2] our method consistently achieves the lowest B-AURC across all corruption severities. This result demonstrates more reliable aleatoric uncertainty estimation and improved performance in selective classification scenarios. This suggests that many existing methods do not adequately model aleatoric uncertainty, particularly on corrupted data. In contrast, our method addresses this limitation by explicitly modeling the alcatoric uncertainty inherent in level-1 labels, amplified through strong mixup strategy, thereby achieving a more robust uncertainty characterization.
OOD generalization performance. As shown in Table our method demonstrates significantly superior OOD generalization. This addresses critical limitation of previous EDL methods, which often suffer from severe degradation in classification accuracy on OOD data, thereby limiting their practical applicability. Our approach overcomes this issue, as theoretically supported by Theorem 2 By employing a strong mixup strategy with 3 >>> 1/2, our method substantially reduces the model's sensitivity to input-dependent noise while simultaneously enhancing its generalization capabilities, leading to robust performance even out-of-distribution inputs.
8

---

Table 2: Selective classification results on CIFAR-IO-C using E-AURC at different level severity s.
Method
s=1
==2
s=3
8=1
s=5
Mean
EDL
18.12:0.31
30.16
44.54=1.95
63.61+1.90
101.6114.46
51.60+
RED
16.74=:0.0
30.15=1.40
44.80=1.66
65.26:2..11
103.87=5.73
52.16::2.36
Z-EDL
14.62:10.52
27.84:10.65
41.70=1.57
59.53+42.92
95.93-14.43
47.92+22.01
R-EDL
17.13+40.47
1.585+
45.06-10.74
63.75-+0.51
101.74+1.21
51.50--0.79
DA-EDL
20.61:2.75
35.5615.31
51.09=8.38
72.29=11.23
112.83+1453
58.4718.44
Ours
8.70
14.80:0.45
21.90=0811
35.0611.811
66.52:17.37
29.40
Table OOD generalization accuracy on CIFARIO-C and CIFARI00-C dataset.
Dataset
Method
==
8=2
= 3
8=4
8 5
Mean
EDL
87.44110.28
80.90:10.55
75.25=0.70
68.08=0.93
1..24=1.110
73.58+0071
RED
88.23++0.21
81.38:00.77
75.48=0.66
68.05=0.97
16.61
73.95=+0.70
Z-EDL
88.03++0.21
81.10+0.388
75.32+0.53
68.12-0.48
56.98=0.51
73.01=0.422
CIFARTOCC
R-EDI
85.46+0.32
80.12+0.43
74.91+0.51
67.53-0.68
56.74+0088
72.95+0.56
H-EDL
87.82++0.15
81.66+0.21
76.23++0.25
67.64+0.30
0..88+0.33
73.85++0.25
DA-EDL
85.26:10.34
80.0640.558
74.88:10.69
67.82=0.88
1.08
73.18:0.71
93.88:10.09
92.04:10.09
90.19:0.12
87.02-0.18
80.53
88.73--0.33
EDL
26.37-12.43
22.82:12.24
20.69
18.14
20.60 1.79
RED
64.69110.20
55.6510.23
50.0220.27
43.36-0.30
33.06=1.224
49.36110.25
Z-EDL
64.43-10.26
55.5240.33
49.82-10334
43.16-0.30
32.72-0.41
49.13=10.33
CIFARI00-C
R-EDL
10.42=+1.98
35.34
12.13+1.59
28.20=1.35
22.71=0097
31.76=1.53
H-EDL
64.87+00.2
55.85+0..15
50.22++0.16
43.68=0.15
33.35=00188
49.59=0.17
DA-EDL
16.76+1.55
14.94+1.32
13.81+0.99
12.53+0.99
11.10=00855
13.83+1.17
Ours 69.34.10.25
65.29110.22
63.07
08.64
50.46.10.77
61.35.1..22
Visualization of estimated uncertainty. In Figs. 2a and 2b visualize the uncertainty distributions produced by our model and a baseline method (MSP with vicinal training). These figures show that point-estimate-basee methods, despite supervision from level-1 labels, exhibit limited improvemen in OOD detection and remain overconfident on OOD samples. In contrast, our method achieves a clear separation between ID and OOD samples. To evaluate the model's 's ability to estimate aleatoric uncertainty, we further visualize its predictions on CIFAR- 10 and five CIFAR-10-C variants with increasing corruption severity (Fig. 2cp. As the corruption level increases, the estimated aleatoric uncertainty rises accordingly, indicating that our model reliably captures data uncertainty.
CIFAR-10UIDI
0.5
CIFAR-10UID)
MNIST(OOD)
MNIST(000)
CIFAR10-C/S!!
CIFARTO C/s2)
Danaky
Denaky
3,2
Donsity
CIFAR10-C(33
101
CIFAR10-C(44)
CIFAR 10 C(s5)
0.1
-1.0
-0.5
00.0
-0.4
-12
-30
-15
2.11
25
Negative Softmax Probabilty
Difforential Entropy
Alsatoric Uncortainty
MSP
(b) Epistemic Uncertainty
Alcatoric Uncertainty
Figure 2: Visualization of uncertainties.
VRM with existing techniques. We can also integrate the proposed vicinal level-1 1 labeling method into three representative models: the standard level-1 classifier (i.e., softmax classifier), level-2 evidential model (EDL) 45)) and the hyper-opinion-baste
H-EDL 44 As shown in Table our method can be seamlessly incorporated into existing techniques to enhance both OOD generalization and OOD detection. Several key observations are worth highlighting: First, vicinal level-1 labels are more compatible with level-2 distributional models than with point-estimate-basee softmax classifiers. This is likely because softmax models lack the ability to express uncertainty over multiple plausible classes, whereas distributional models can better leverage the probabilistic nature of vicinal labels.

---

Moreover, our approach is particularly effective when combined with H-EDL [44),
Table 4: Level-1 and level-2 models with VRM.
Model generalization
OOD detection
which explicitly captures the possibility of an instance be- longing to multiple classes through feature-based hyper- opinions. The synergy be- tween vicinal labels and hyper-opinion enables more accurate and continuous mod- eling of probability distribu- tions.
Method
ID-Acc
OOD-Acc
AUROC
MSP 18
95.06
74.75
89.83
EDL 45
95.17
74.51
90.67
H-EDL [44]
95.04
73.84
92.27
MSP w. Vic
96.33+1.27
87.92 87.92+13.17
89.59.u.24
EDL w. Vie
96. 18.0.
88.73+14.22
93.08+2.39
H-EDL w. Vic
96.43+1.39
88. 88.63+14.79
93.89+1.72
Ablation study. As shown in
Fig. 3 we investigate the im-
Accuracy (%g
68.75
Oo0 AUROD (%)
OCD Accuracy (%)
000 AUROC (%)
pact of two hyperparameters: 3 and 8 We begin by analyzing As B increases, the corresponding Beta distri- bution becomes more peaked around 0.5, with narrower tails on both sides. This sharper concentration near 0.5 facilitates improved OOD generalization, as supported by Theorem 2 However, an overly concen- trated Å distribution around 0.5 can hinder smooth sample mixing, which in turn may degrade OOD detection performance. Empirically, we find that setting 33 to around 10 provides the best trade-off. For the analysis of 8 we first consider the case where 3 B Buoise 1.0. In this scenario, the resulting values follow a uniform distribution over the interval [0, 1]. In this case, OOD detection performance reaches its peak, smaller À values (according to Theorem help suppress the rapid growth of Dirichlet concentration, thereby enhancing epistemic uncertainty estimation. However, as 3 increases, the sampled values become increasingly concentrated near 1, which accelerates the Dirichlet concentration growth and compromises the model's ability to estimate epistemic uncertainty accurately. While primary analysis focuses their joint effect, we further isolate and analyze the contribution of each component through detailed ablation experiments in Appendix D.2
68.50
68.25
68.5
68.00
Accuracy AUROC
Accuracy
88.0
10
Misee
Figure 3: Ablation study on the hyperparameters.
Conclusion
This work addresses a key limitation of current EDL methods. namely their reliance on hard labels that ignore inherent label uncertainty. We propose vicinal risk minimization framework that employs level-1 supervision through smoothed conditional label distributions. This approach enhances aleatoric uncertainty modeling and mitigates Dirichlet degeneration, also resulting in improved epistemic uncertainty estimation. Extensive experiments demonstrate consistent improvements across both dustof--isttibuttin and selective classification benchmarks.
Limitations. While our method effectively alleviates the degeneration problem in evidential uncer- tainty estimation, it introduces two hyperparameters that control the distribution the generated level-1 vicinal labels. The choice of these hyperparameters alcatoric and epistemic uncertainty, similar to how the regularization strength in previous works based on entropy or Fisher information affects standard EDL methods. Although we empirically found our approach to be robust within reasonable range of parameter values, developing adaptive or data-driven strategies automatically calibrate the vicinal smoothing strength remains an important direction for future work.
influence the balance between
Acknowledgments and Disclosure of Funding
This work was supported by the National Natural Science Foundation of China (No. 62472315, 62476165). We thank Wei Liu and Xujing Zhou for their constructive discussions on this work.
10

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A List of Symbols
list of symbols used in the main paper as well as in the following supplementary material, most of
symbols keep same 2 3
Table 5: Notation summary for the general, level-1, and level-2 learning settings.
General Symbols
K
number of classes instance space
X
label space with hard labels (yi....; 4K
D
training data { ((x("), ( C X'
data generating prohability
p(- a))
conditional distribution on V, i.e., p(v x.), represents the probability of observing given æ the set of probability distributions
(O))
Ax
the K -simplex, ie., Ax := (0 (O...... 0x) E [0.1]* ||0|| = 1}
(01,... "0)
prohahility vector with singletons
Level-1 Learning Setting
H1
(level-1) hypothesis space consisting of hypothesis h: X Ax
La
loss function for level-1 hypothesis, i.c., L1 P: P.()) 3 I
R(-)
risk or expected loss a level-1 hypothesis (Eq) 3
Remp (+)
empirical loss of level-1 I hypothesis (Eq.
empirical risk minimiser, i.e., true risk minimiser or Bayes predictor, i.e., h² arg milNEH R(h)
min.ew Remp (h)
h
Level-2 Learning Setting
A
the set distributions on simplex A the set of distributions on Pr (the set of level-2 distributions)
P2())
H2
(level-2) hypothesis, ie., a mapping h X' + A (2)
Q
probability distribution on Pi (3)), ie.., element of (2
La
loss function for level-2 hypothesis, e.g., F2()) x (.) R,
R (-)
empirical (level-2) loss of level-2 hypothesis
Re( (-)
(level-2) risk or expected loss of a level-2 hypothesis
Distributions
of,
Gaussian distribuiton with location parameter and scale parameter a >> 0 Dirichlet distribution with parameter E R
Dir(a)
By
Dirac measure y E y (i.c. 6y is an element Pi ()())
6p
Dirac measure at p E Pr (J) (i.e., 8p is an element of (2()))
Entropy and Divergence Shannon Entropy of categorical distribution Kullback-Leibler divergence on P2()) x P2()))
H(-)
kL()))
B Proof of Theorem
Theorem 1. For any level-1 loss function L, (P(()) x - R that satisfies LA (By-Diiraa)) P p. Ep-Di Lx(p.)), -), (i.e., is convex function), such Brier score and the log-loss in Eq. 2 the empirical risk minimizer of level-2 prediction is always Dirac measure 8n E P2()) and the expectation of level-2 prediction is 6g E (Y). This result holds if the learner possesses a universal approximation property, allowing represent such degenerate distribution.
15

---

Proof. Let the empirical risk of level-2 prediction Q E P2()))
[22)
(Q)
Σ L2 (0. (m))
N
(19)
E E (wu)
By assumption on the level-1 loss L (i.e. convexity), it holds that
1
Q
Z Σ L1 (Ep-o(p).y
(20)
Let ((N) be the minimizer over all Q of the right- -aandaand side, then ((,) B p (p) an element Ág Define ((N) 8pCN) and note that pp- [p] (N) Then,
[
(Q(N))
1
E
pupeen Li (p. (n)
N
N
1
Σ
Li
(p'N),y(0) yn())
(21)
N
n=1
N
1
Σ
La
B ( [p]. s(u))
N
n=1
This proves that the empirical level-2 risk is minimized by Dirac distribution over a single level-1 prediction, i.c., Q(N) = dIN,,, implying vanishing epistemic uncertainty. We now show that the corresponding level-1 1 prediction also collapses Dirac measure, indicating vanishing alcatoric uncertainty. Consider the empirical level-1 risk:
R emp (p)
Σ ((, 1=1
(22)
For any strictly proper loss function L1 (e.g., Brier score, loss) it is uniquely minimized when p 5y(,, i.e., the one-hot encoding of the ground-truth label. That is,
arg min L1 y (*) = 5ytm,, with L1 (6gt)), v()) 0. PEAK
(23)
Hence, the optimal level-1 predictor
(N) that minimizes the empirical risk is
p (N) 5y0s) for all n.
(24)
follows that the expected level-1 prediction under the optimal level-2 distribution is
F P- 6ybs,
(25)
i..., one-hot distribution that assigns all probability mass the ground-truth class. This indicates that aleatoric uncertainty also vanishes.
Therefore, the empirical level-2 risk is minimized by a Dirae measure over level-] Dirac prediction
Q(N) 68 (m)'
(26)
This implies that:
Epistemic uncertainty vanishes, since Q is a Dirac measure.
Aleatoric uncertainty vanishes, since the expected level-1 prediction under Q a one-hot vector.
16

---

This highlights a critical degeneracy of empirical risk minimization with strictly proper convex losses in the level-2 setting: it collapses all predictive uncertainty, providing no representation of uncertainty despite operating in distribution-over pirrruutioo-ove--ddsrtttttiooo framework.
Proposition 1. Under the assumptions of Theorem 1 empirical risk minimization of level-2 prediction inevitably yields degenerate distributions 6p P2()) and the expectation the level-2 prediction is By E 1 ())). As a result, the model fails to provide any meaningful or disentangled representation of aleatoric epistemic uncertainty.
Proof. Assume that the optimal strategy under ERM is to collapse the Dirichlet distribution to a delta distribution centered on the one-hot vector oy, i.e., Dir(()) 6x, in Theorem | This degeneracy has consequences for uncertainty estimation. Consider the standard decomposition of predictive uncertainty in Dirichlet-based models as in Theorem D we have
Total Uncertainty (TU) = H [irpDirt(o) p(U p)]]
(27)
Aleatoric Uncertainty (AU) Ep H [p(y p)]l, Epistemic Uncertainty (EU) TU AU.
(28)
(29)
When the Dirichlet degenerates to 65,, both the expected predictive distribution and the samples from Dir(ax) are deterministic, yielding
TU + O, AU +
EU + 0.
(30)
Thus, the model expresses neither AU nor FU, regardless of the true nature of the data distribution. Consequently, the level-2 model fails to provide any meaningful or disentangled representation of aleatoric or epistemic uncertainty.
Theorem 2. Let the ground-truth level-1 label be denoted as (x)) and let the observed level-O one-hot label (z) be noisy realization of p" (xx) perturbed by dependee label noise μ(x)
(x)x) = (x())) + u(x) where p(x) ~ 00,
(31)
Then, the test risk admits the following lower bound under mild regularity conditions
R(h; P) 2 Co²,
(32)
where C' depends on the trace of the Hessian matrix of the loss function with respect to p. Then, for the level-1 I label with strong mixing, the bound can be tightened
R(h; P) 2 C'
(33)
< 1 (VB > 1/2), indicating a reduced sensitivity of the test risk to where C²/C ~ 2 input-dependent noise.
1 +
Proof. We suppose the label noise μ follows isotropic Gaussian distribution I-BDL 12
J ~ ~~10000²
(34)
Then, even if the optimization loss R(h; D) is minimized (or approaches zero), the population loss R(h; P) will have an irreducible component that is at least on the order of o², As we assume that the training labels y are generated from the true labels pd with added noise:
(x)) p" (xx) + (x(x))
(35)
where (xx))= ~ N(0,0²1). The expected test loss can be expressed as
R(h; P) = Eig,y>> La [a(x)))))))
(36)
Since the label itself is affected by noise, we can decompose the expectation
E [2(())) o.)] = [h(z)), + +))
(37)
Using second-order Taylor expansion to approximate the loss function:
(((xx).p' p" + +1) (2((x))pp) (V 1) + 2 Hμ.
(38)
17

---

where H represents the Hessian matrix of the loss function L2(h(x)). w.r.t p", defined as
H = v (x(((x))
(39)
and (VL2, (4) is the inner product between the gradient of the loss function and the noise vector μ.:
K
0L2
LV...))
Σ
Pk.
(40)
k
Since the noise j follows a zero-mean Gaussian distribution, the expectation of the first-order term vanishes:
L2 [J] O,
(41)
while the expectation of the second-order term is given by the noise covariance:
Eju H TH.))
(42)
Thus, the lower bound of the test loss can be approximated as
R(h; P) 2 Co?,
(43)
where C depends on the trace of the Hessian matrix. We then show that incorporating VRM leads to lower test risk. Let the original label noise (m) ((()) N(0,o²1) be iid. After vicinal interpolation, the noise in vicinal labels becomes
)pe + (1 1))pp(m))
(44)
with variance
E = + +(1 (1-1)002 =
0² [A² + +11 (1-
(45)
When À ~ Beta(B, B), the expected variance is
EA 1X + (1 - )) = 2E[ - 2E[A] +
(46)
Using properties of Beta distribution E[A] and Var(A) 1(2++JJ we obtain
E[A²] = Var(A) + (E[A))² 4(28 11 4'
(47)
Substituting yields
Ex [X + (1- -19] 233 +1 + <1 (V3 > 1/2).
(48)
Thus, the effective noise variance after Mixup is ko² where k = 2 than the original a Substituting into the theorem's lower bound gives
+ 1/2 < 1, significantly lower
R(h; P) 2 C ko² Co?.
(49)
Although distribution of the noise μs is unknown, and assumptions about it are modeling questions, most statistical methods rely on certain mathematical conditions, known regularity assumptions, to ensure their validity. In our proof, i.e., we assume that μ follows an additive Gaussian noise.
Theorem 3. Let X be the mixing hyperparameter defined in Eq. 13
Consider the optimization of the Dirichlet parameters a in Eq, 14 For samples where on S ozj (V) + k) with lower belief assigned the ground-truth k class, the Jollowing properties hold
The update to the Dirichlet concentration for the ground-truth class Doy increases mono- tonically with X.
The updates to the Dirichlet concentrations for the non- non-ground-truth classes Aajzko de- crease monotonically with X.
The total increase in Dirichlet concentration, denoted AS, increases monotonically with X.
18

---

Proof. Lets take the L1 loss as cross-entropy loss for example, which calculate loss between the sampled p from Dir(a) with ÿ. Then, we derive the following analytical form of Cell as
Leala, 73) /
Σ By log (pj)
1
9. Pj j=1
K
dp
(
(50)
Σ
(bb
1/2 (aa))
where S Σj= oz. Then, with gradient descent, the update of aj, denote -1j
aLin where aj
the learning rate. Let j denote the index of class, we have
aLen(ax 5)
K
K Σ
(S)
5 = (ay)
as
(51)
where 1 is the trigamma function, which is a positive, monotonic decreasing function. Then, we have the updates of a, as Eq. 52 with the negative gradient descent update
Aaj = 14/1 (S) 0y41(as) (ay))
(52)
As the vicinal label is obtained by ÿ = Xy(()) smoothed target labels explicitly as
(n) + (1 - X)
R. we can also express the
X K
1-
Uk X
(53)
K
where & denotes the index of ground-truth class. By substituting Eq. 53 into Eq. 52
we have
Aan -1 (1)())) (*+
1-1 K
tba (axk)
(54)
Aaj -n (/())) 1- X
kn(as))
(55)
K
and
K AS
Σ Daj = -1 K (1) - (x+ 1-11 1
1-
1- M Eml(as))
th (as)))
(56)
K
K
j=1
jtk
To analyze how affects Aax
Daj, and AS, consider the derivatives as follows.
AΔak
17 1
1
v (ax) > 0
(57)
K
and
dDoj
-1 K
24 (ar)) <0,
jtk
(58)
0X
and
dAs
K Σ
dDaj
0x
ax
1
1
w vi((a))
1
Σ
vilass
K
K
(59)
T
K
#1
Σ wh (az)
K
j=1
K
(v' - - (()())))
K
j=l
19

---

As the label smooth process takes the following
Aa (m)) (1
A)ar (m))
カ= Xyy + (1 - X)
(60)
K
This analysis reveals how label smoothing influences the accumulation of Dirichlet strength. When the model is not yet confident the class its corresponding Dirichlet strength og is relatively small. Given that the trigamma function 1/1 (ar) is monotonically decreasing, smaller on results in th (ax) being larger than the average trigamma value across all classes (i.e., 1h1 (ax) > k Σド (ay)).
Consequently, the derivative OAS becomes positive. This positive derivative indicates that a decrease aa
in Å (which corresponds an increased degree of label smoothing) will lead to a smaller increment AS, thus slowing the growth of the total Dirichlet strength S.
C Uncertainty Measures
C.1 Uncertainty Decomposition in Birichlet-Basee Models
fundamental identity in information theory is that the Shannon entropy of a random variable X be additively decomposed into the mutual information between X and Y, and the conditional entropy of X given Y [I:
((X) = -1;;)) [[X IY)
(61)
Follow this idea, Prior Networks (36) propose method explicitly model and decompose predictive uncertainty into two components: aleatoric uncertainty and epistemic uncertainly. This is achieved by treating the output of the classifier the parameters of a Dirichlet distribution over categorical class distributions. Given Dirichlet distribution parameterized by a (a1...., ,ak over the probability simplex Ak. the expected predictive distribution over class labels
K
ply j x)) = Ep-
Oj S
where S
Σ
aj
(62)
The total uncertainty in the prediction is measured by the Shannon entropy of the expected categorical distribution conditioned:
Hana [p(ylp)] = Ep- pll p)] =
jj
Oj log
(63)
S
C.2 Conditional Entropy
Alcatoric uncertainty corresponds to the expected entropy of the categorical distributions sampled from the Dirichlet prior, commonly referred to the conditional entropy
E Ep-Dir()) [M[p( p)]l = Ex
Pj log
j=1
k
= Σ
(wba +1) - +0(S+ 1))
(64)
S
j=1
k
+ 1)
Σ
a
_ s
v +1)
C.3 Mutual Information
Epistemic uncertainty can be measured by the mutual information between predictions and the Dirichlet parameters, capturing uncertainty about the model itself:
Mi(y. p) Hottll [Ø(y(p)] Ep [p(y p)]]
(65)
20

---

This mutual information quantifies how much of the total uncertainty arises from uncertainty in the model parameters (i.c., distribution over categorical distributions), and thus reflects epistemic uncertainty.
y P
2 H Dirlaa (p(P
E Dirla [ [p((lp)]
Epistemic Uncertainty
Total Uncertainty
Alcatoric Uncertainty
aj
(v(a, + +1) (5 1))
(66)
8
In
vaa + 1) + 6S
g
C.4 Differential Entropy
The differential entropy is defined as
S Dir(p a) log Dir(p a) dp,
ENT((iir(p) a)) = -
(67)
Ax
where Ag denotes the probability simplex. The closed-form expression is given by
ENT(Dir(p) x)) log B(x) (S K)((S) - E(a 1))())))
(68)
Differential entropy is also a prevalent measure of epistemic uncertainty, where lower value indicates that the model yields sharper distribution, and higher value means more uniform Dirichlet distribution.
C.5
Vacuity of Evidence
For EDL RED 40 I-EDL 12
R-EDL 99, H-EDL
44
which grounded in Subjective
Logic and DS- Theory M. Subjective Logic provides principled framework for modeling predictive uncertainty by interpreting the output of neural network as an opinion-a structured representation uncertainty over discrete set of classes. Unlike conventional classifiers that output categorical probabilities, EDL models produce non-negative evidence values e = (2, for each of the K classes. These evidence values parameterize a Dirichlet distribution Dir(a.) where aj ej + 1. In Subjective Logic, an opinion over finite domain is characterized by three components: the belief mass bj. the base rate aj, and the uncertainty mass u, satisfying:
62,- ......... ex]
K
by a, Elp,l. and
b +u=1
(69)
where py denotes the probability assigned to class j. These quantities relate to the Dirichlet parameters as follows: The belief mass by is proportional to the evidence for class k:
K
K Σ aj j=1
by
es S
where Σ (e, +1)
(70)
The base rate ag is typically assumed to be uniform, i.c., as 1/K. The uncertainty mass is defined as:
w K K Σ5=1 03
(71)
S
This uncertainty mass u is referred to as vacuity in EDL literature, and it quantifies the degree of epistemic uncertainty due to a lack of evidence. When the total evidence is low (e.g.. under out-of-distne difsrriibbtiinn or ambiguous inputs), S becomes small and vacuity u approaches 1, indicating that the model abstains from committing belief any specific class. Conversely, high total evidence yields a low vacuity, reflecting confident predictions based on strong feature-based support. This opinion-based interpretation highlights the epistemic nature of uncertainty in EDI. and differentiates it from alcatoric uncertainty captured by distributional spread in conventional probabilistic models.
21

---

D Additional Experimental Details
D.1 Implementation Details
Since different baseline methods involve distinct activation functions and regularization terms, we provide detailed implementation settings below.
EDLs based on Subjective Logic. For EDL [45], we adopt the mean squared error (MSE) loss, also known as the barrier score. For Z-EDL (12.. we follow their original paper and use the Fisher- MSE loss, setting the Fisher information regularization weight to 0.05. The activation function is Softplus, as specified in their implementation. For R-EDL [9] we follow the settings in the original paper and set the prior strength to 0.8 for the CIFAR datasets and using the MSE loss variant without the variance minimization term. For all three methods (EDL, I-EDL, and R-EDL), the KL divergence term which aims to remove mislcading evidence with an annealing weight schedule of A min(epoch_ Bancch_ixx111 idx/10, The KI. divergence term which is used to regularize the predicted Dirichlet distribution by encouraging it to stay close to a non-informative prior for incorrect classes, typically Dir(p 1). where each class has concentration parameter of 1. The KL divergence between the predicted Dirichlet distribution Dir(p α) and the uniform Dirichlet prior
LKL. = (L]Dir(P) @) Dir(p)
1)]
T [E 83)) K 11 M(a))
K
(72)
log
Σ
(aj 1)
(az) +
Oj
we set the target class Dirichlet concentration parameter Or to 200. Since both methods require out-of-distribution (OOD) samples during training to constrain their predicted Dirichlet distributions, we follow the setup in 12] and use random noise as the OOD dataset to ensure a fair comparison.
PriorNets. For KL-PN (36] and RKL-PN 37
Our method. For our method, the non non-negative activation function a is set to Softplus for CIFAR- 10. For CIFAR-100, due to the large zero-evidence regions observed in prior work 440), we warm up the model using an Exponential activation for the first 10 epochs to help the model avoid these regions, and then switch Softplus activation for the remainder of training.
D.2 Further Ablation Study Vicinal Supervision and Noise Augmentation
To clarify the individual contributions of the two components the total loss (5). we conduct an ablation study isolating the effects of vicinal supervision (Evicina) and noise augmentation (Cuoise).
Effect of Vicinal Supervision. tion term Croise- Jublea/ssmmarize the results as 3 varies.
We first isolate the influence of 3 by removing the noise augmenta-
Table 6: Ablation on vicinal supervision by varying 3 while removing Luoise-
B
8
E-AURC +
OOD AUROC
OOD Acc t
ID Acc t
56.54-13.98
90.6710.35
74.51
95.17=0.18
0.2
58.10-+6.04
89.77+-0.63
0..5
95.31++0.08
0.4
49.44
90.28+0.05
77.82+0.61
95.61+0.11
1.0
45.18+2.11
91.11+0.40
7..1
96.01++0022
5.0
42.94.14.37
91.68+1048
79.75++1002
95.89++.21
10.0
39.15114.3
91.7311..1
80.41+1.233
95.86
We observe three consistent trends: (1) Improved aleatoric uncertainty estimation. Increasing yields monotonic decrease in E-AURC, from 56.54 to 39.15, indicating better calibration for selective classification. (2) Enhanced OOD detection. Despite being designed for aleatoric calibration, vicinal supervision also improves OOD AUROC, suggesting stronger discrimination between ID and OOD samples due to its regularizing effect on the data manifold. (3) Improved generalization. Both OOD and ID accuracies increase with 3, supporting Theorem that stronger mixup enhances generalization across both seen and unseen distributions.
22

---

Effect of Noise Augmentation. Next, we isolate Cnosse by setting L = Lnoise and varying 3 while fixing Posise 1.0. The results are shown in Table
Table 7: Ablation noise augmentation by varying 3
while removing Lvicinal.
B
OOD AUROC
E-AURC +
OOD Acc f
ID Acc f
hoise
90.67110.3
56.5411.998
74.51=0.49
95.17:0.118
0.2
90.95:10.32
56.11
74.32.10:33
95.00:10.02
0.4
91.1810.42
62.81=10.15
73.76:10.67
95.03:10.21
1.0
91.85++0.24
57.22:16.14
73.96+10.43
95.17:+0.09
5.0
90.74+0.20
55.95=6.40
74.67=0.75
95.02++0.20
10.0
90.37=0.55
60.17+2.85
73.42::0.53
94.86++0441
10.0
1.0
93.08+0.33
29.40-2.16
88.73-10.13
96.1811.113
From Table 7 we draw three conclusions: (1) Limited effeet of isolated 5 Varying S alone causes only minor fluctuations in E-AURC and accuracy, suggesting that noise augmentation without vicinal supervision is insufficient for consistent gains. (2) Importance of balanced noise intensity. Too small Poois leads to excessive perturbations that harm learning, while too large values make the sampled À concentrate near 1, effectively disabling noise augmentation. 3 1.0 yields the best balance. (3) Synergistic effect. The best overall performance is achieved when both components are combined (3 10.0, ( 1.0), achieving the highest OOD AUROC (93.08%). lowest E-AURC (29.40), and best ID/OOD accuracies, demonstrating the complementary benefits of vicinal supervision and noise augmentation.
Discussions
E.1
Why do some baseline methods perform poorly CIFAR-100?
that require Dirichlet concentrations of incorrect classes approach zero, observe that they struggle converge when the number of classes is large (e.g., K = 100). Since the original papers do not provide CIFAR- experimental settings, we adopt the same configurations as used for CIFAR-10, which may limit their performance.
For models like EDLs 45 and PriorNets 36 37
E.2
Social Impact
Our work addresses the challenges of uncertainty estimation, out-of-distribution (OOD) detection, and OOD generalization, which are critical for ensuring the safety, reliability, and fairness of machine learning systems in real-world applications. By improving models' ability to recognize and appropriately respond unfamiliar ambiguous inputs, methods help reduce the risk of overconfident mispredictions in high-stakes domains such as healthcare, autonomous driving, and finance. These advances have the potential to increase trust in Al systems and support more responsible deployment practices. Moreover, enhanced OOD generalization may help mitigate performance disparities when models applied across diverse populations and settings.
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Reviews:
[{"review_id": "IfVpEkM6bU", "reviewer": "Reviewer_Vmcj", "summary": "The paper proposes to further improve the epistemic and aleatoric uncertainty estimation using vicinal risk minimization for evidential deep learning. The paper provides a three-level view: level-0 being hard labels, level-1 being categorical distributions, level-2 being Dirichlet distributions. The paper analyzed the limitations of EDL (level-2 learning against level-0 labels) and proposes to use vicinal supervision to enhance the aleatoric uncertainty estimation and noise-augmented vicinal risk minimization for epistemic uncertainty estimation. The proposed method is used for OOD detection and generalization on several common image datasets.", "questions": "1. How realistic are the assumptions in Theorem 1?\n\n2. Can the analysis in Theorem 3 be generalized to other types of evidential losses?\n\n3. Is there more analysis on the ablation study in Figure 3?", "limitations": "The limitation of datasets and tasks is briefly mentioned, but could be better discussed.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The paper is well written and most information is clearly presented.\n\n2. Without checking all mathematical details, judging from the high level propositions and conclusions of the theoretical analysis, the theorems are of significance for the uncertainty estimation problem. The theoretical analysis is also essential to the proposed method, showing good consistency. \n\nThe main novelty with the theoretical framework is the detailed analysis on learning from level-0/level-1 supervision with level-2 models.  Although level-2 models have already been studied by many (e.g. [3],[2] from the references), Theorem 1 and Proposition 1 formally identifies the insufficiency of traditional risk minimization of level-2 models that also includes the aleatoric uncertainty perspective (rather than epstemic uncertainty alone). To my knowledge, this is a novel contribution and reveals a meaningful characteristic of evidential learning.  \n\nThe proposed method is not limited to a specific learning task, and the difference comes from the supervision side of things (X and Y). This makes the solution more applicable to various problems, and aligns with the trend of introducing noise into machine learning. This also shows the advantage of VRM in the uncertainty quantification aspect, in addition to generalization. \n\n3. The experimental results show good OOD detection and generalization performances. A sufficient number of recent EDL baselines are included in the comparison.\n\nWeaknesses:\n\n1. As mentioned in the paper, the evaluation is limited to CIFAR scale datasets and the OOD behavior on more realistic tasks is not clear.\n\n2. In the hyperparameter ablation study, the accuracy trend is not clear and the analysis is limited.\n\n3. The notation of $\\mathbb{P}_1$ and $\\mathbb{P}$ from lines 75-77 can be confusing.", "was_revised": true, "final_justification": "I think the paper provides a novel improvement to EDL with some theoretical justification, which should be considered as a useful contribution to the uncertainty quantification and OOD detection/generalization tasks in ML. I agree with Reviewer 9dhq that the proposed method somewhat depends on the hyperparameter and hope that the revised version can better analyze the ablation study part and provide more comprehensive results (specific datasets, detailed guidelines from rebuttal). Overall, I maintain a positive opinion about the paper.", "rebuttal": "# Response to Weakness\n\n##  Response to Comment 1\nWe sincerely appreciate the reviewer’s suggestion regarding the need for evaluation on more realistic large-scale datasets. In response, we conducted additional experiments on the ImageNet-200 dataset, which consists of 200 categories sampled from the full ImageNet-1K. The images are of size 224×224, consistent with the standard ImageNet-1K setting.\nTo evaluate the reliability of epistemic estimation in large-scale settings, we first focused on OOD detection tasks where ImageNet-200 serves as the in-distribution (ID) dataset, and SSB-Hard, NINCO, iNaturalist, Textures, and OpenImage-O are used as OOD datasets. We report both in-distribution classification accuracy and OOD AUROC as evaluation metrics.\nThe results show that our approach significantly outperforms prior EDL-based methods across all metrics. In particular, models employing KL-PN, RKL-PN, or Brier score-based losses—such as EDL, R-EDL, DA-EDL, and I-EDL—suffered from poor convergence and suboptimal performance. This phenomenon, previously observed on CIFAR-100, becomes even more pronounced on ImageNet-200. \nAs noted in the previous work RED, Brier score-based losses tend to struggle with maintaining high classification accuracy when the number of categories is large. RED addresses this issue by introducing evidence regularization to improve class discrimination.\nOur method, by contrast, adopts the expected cross-entropy loss, which avoids the accuracy degradation seen in Brier-based methods. Furthermore, by leveraging vicinal label supervision, our model learns more reliable uncertainty estimates while preserving high classification performance.\n\nIn addition, we also include a misclassification detection experiment to evaluate the reliability of aleatoric uncertainty estimation. Specifically, samples that are misclassified should exhibit higher aleatoric uncertainty, while correctly classified samples should have lower uncertainty. \nWe use the Excess Area Under the Risk-Coverage Curve (E-AURC × 1000, lower is better), a standard metric for misclassification detection task. \nWe find that our method achieves not only higher classification accuracy but also a lower E-AURC, indicating better aleatoric uncertainty calibration.\n\n    Experimental settings:\n    Network: ResNet-18\n    Optimizer: SGD\n    learning rate:0.1 \n    lr_scheduler: cosine annealing schedule\n    epochs: 100\n    KL regularization weight for EDL RED I-EDL R-EDL: 0.001 (as suggested in RED for large-scale dataset)\n    Fisher information regularization weight for I-EDL: 0.05\n    Hyperparamter for our method:  $ \\beta=10 $, $\\beta_noise^+=2$\n\n\n\n| Method     | →SSB-Hard   (OOD AUROC)     | →NINCO   (OOD AUROC)           | →iNaturalist   (OOD AUROC)     | →Textures    (OOD AUROC)      | →OpenImage-O    (OOD AUROC)    | Cls Acc         | MisD E-AURC|\n|------------|------------------|------------------|------------------|------------------|-------------------|------------------| ------------------|\n| KL-PN      | 50.00        | 50.00        | 50.00       | 50.00      | 50.00        | 0.50         | 48.75 |\n| RKL-PN     | 50.00      | 50.00         | 50.00      | 50.00       | 50.00   | 0.50        | 50.25  |\n| EDL        |45.55\t| 46.34| 40.91| 36.08 | 52.42\t |2.30|  49.91 |\n| RED        |77.81| 82.25 | 85.42| 80.32 | 83.72 |86.19| 22.62 |\n| I-EDL     |48.50| 52.49\t| 52.57 | 54.24 | 49.58 |1.09| 55.98 |\n| R-EDL      |50.88| 53.99 | 41.10\t| 53.99 | 55.28 |2.67| 53.60 |\n| Our | **79.80**|  **83.58** | **93.93**| **90.71** | **89.84** |**87.33**| **14.65**  |\n\n\n\n\n## Response to Comment 2\nThank you for the insightful comment. We agree that the classification accuracy trend in the original hyperparameter ablation study was not very clear.\nWith further ablation study, we found that the classification accuracy is relatively insensitive to variations in these hyperparameters when the other one is held fixed. \nTo provide a more thorough analysis, we have added additional experiments where we isolate each loss component and vary the corresponding hyperparameter. \nPlease kindly refer to our detailed discussion in the response to Comment 2 from Reviewer 9dhq for further ablation study on hyperparameters.\n\n\n\n\n\n## Response: Comment 3\nThank you for the suggestion. We acknowledge that the notation of $\\mathbb{P}_1$ and $\\mathbb{P}$ could be confusing. We will revise the corresponding expressions in the paper to make the distinction clearer.\n\n\n\n\n\n# Response to Questions\n\n## Respoonse to Question 1 \nThe assumptions in Theorem 1 are partially idealized, but aligned with standard practice (level-0 labels + convex level-1 losses). The Dirac assumption enables a clear theoretical analysis of uncertainty collapse. We empirically validate this behavior: the variance of the sampled level-1 distributions $ p \\sim \\text{Dir}(\\alpha) $ is typically less than $10^{-5}$, indicating that level-2 predictions degenerate into Dirac measures $ \\delta_p $. \nFurthermore, we observe that the expected predictive distribution over i.i.d. test samples exhibits a sharp peak near 1.0 with a thin tail toward 0, implying that the model assigns nearly all probability mass to a single class. This behavior is consistent with a level-1 Dirac distribution $ \\delta_y $, suggesting that the learned uncertainty representation collapses into a single-point prediction, as our theorem characterizes.\n\n\n## Response to Question 2\nYes, the analysis in Theorem 3 can be generalized to other types of evidential losses.\nFor example, the negative log-likelihood (NLL) loss takes the form\nFor log likelihood  takes the form \n$$\nL_{nll}=\\sum_{j=1}^K y_j \\left( \\log (S) - \\log (\\alpha_j) \\right)\n$$\nwhich can be easily shown to satisfy the conditions in Theorem 3. This is because the derivative of the logarithm function, \n$1/x$, shares similar mathematical properties with the trigamma function $\\psi_1(\\cdot)$—both are strictly decreasing and strictly convex—making the theoretical analysis straightforward to extend.\nOn the other hand, the Brier score loss is given by\n $$  L_{Brier} = L_{ j}^{ {bias }} + L_{ j}^{ {var }} = \\sum_{j=1}^K  \\left(y_{ j}-\\alpha_{ j} / S\\right)^2+ \\frac{\\alpha_{j}\\left(S-\\alpha_{j}\\right)}{S^2\\left(S+1\\right)}  $$\nwhere the variance term \n$$L_{ j}^{ {var }}=\\frac{\\alpha_{j}\\left(S-\\alpha_{j}\\right)}{S^2\\left(S+1\\right)} $$ \nis independent of the label and reaches its minimum only as \n𝑆→∞, thereby forcing the Dirichlet distribution toward a degenerate (Dirac) distribution. This behavior is problematic because the minimization pressure is label-agnostic—i.e., even if we use softened labels (e.g., from noise augmentation supervision), this term still encourages overconfident (sharp Dirichlet) predictions regardless of the actual supervision signal.\nAdditionally, we include experiments on two types of evidential losses to support our theoretical insights. Specifically, we evaluate our method under both Brier Score and negative log-likelihood losses, comparing models trained with and without vicinal label supervision. To ensure a fair comparison, we exclude the KL regularization term in all settings.\nExperimental results show that, while Brier Score does benefit from vicinal label supervision—particularly in tasks that rely on epistemic uncertainty estimation, such as OOD detection—the improvement in AUROC is noticeably smaller compared to that achieved with log-likelihood loss. This observation aligns well with our earlier analysis: due to the label-agnostic nature of the variance term in Brier Score, its effectiveness in leveraging uncertainty signals is inherently limited.\n\n\n| Loss Type            | Method Used     | ID Acc ↑   | OOD AUROC ↑ | \n|--------------|-----------------|---------|-------------| \n| Brier Score          | w/o Vic   |   94.83 $\\pm$ 0.38      |   89.95 $\\pm$ 0.29          | \n|  Brier Score                    | w/ Vic   |    95.89 $\\pm$ 0.29      |        91.70 $\\pm$ 0.26     |       \n| Log Likelihood       | w/o Vic   |     94.97 $\\pm$ 0.03    |       90.86 $\\pm$ 0.18      |     \n|     Log Likelihood                 | w/ Vic  |  96.05 $\\pm$ 0.13       |    92.8 $\\pm$ 0.28         |     \n\n## Response to Question 3 \nThank you very much for your question. Due to space limitations, we were unable to include a more comprehensive analysis in the main paper. Please kindly refer to our detailed discussion in the response to Comment 2 from Reviewer 9dhq for further ablation study on hyperparameters."}, {"review_id": "y3Y6NuDPT3", "reviewer": "Reviewer_9dhq", "summary": "Evidential Deep Learning aims to estimate both aleatoric and epistemic uncertainty using Dirichlet distributions, but existing approaches relying on hard labels (level-0 supervision) often produce degenerate outputs with unreliable uncertainty estimates, including both aleatoric and epistemic uncertainty. This paper proposes a vicinal risk minimization framework that constructs level-1 supervisory signals through local label interpolation, without requiring additional annotations. The authors provide a theoretical analysis showing improved generalization by reducing the lower bound of generalization error, and empirical results demonstrate consistent gains over baselines in out-of-distribution detection and generalization tasks.", "questions": "1. The proposed method builds on mixup and label smoothing strategies, and Table 3 shows it improves ID accuracy even for MSP. How can we tell whether the performance gains in Tables 1 and 2 come from improved representation learning versus improved uncertainty modeling? For example, can we analyze latent representations (e.g., in PostNet) before and after applying the proposed method?\n\n2. The paper introduces two new uncertainty quantification measures—Dirichlet differential entropy and conditional entropy. What is the motivation for using these specific measures, and is there any experimental evidence showing their advantages over existing metrics?\n\n3. How does Theorem 2 theoretically support improved generalization and robustness? Does it also suggest an improvement in the quality of the uncertainty estimates?", "limitations": "yes", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n1. The paper is well-written and easy to follow; the motivation is clearly articulated, the methodology is well structured, and the appendix provides detailed derivations and theoretical proofs.\n2. The experimental evaluation includes a broad set of EDL-related baselines across both out-of-distribution (OOD) detection and OOD generalization tasks, showing consistent improvements of the proposed method.\n3. The proposed approach is simple to implement and can be readily integrated into other uncertainty-aware models and tasks without requiring additional annotation or architectural changes.\n\n**Weaknesses**\n1. The paper claims that standard EDL methods fail to capture reliable aleatoric uncertainty and that the proposed method improves this; however, the evaluation would benefit from more targeted experiments measuring aleatoric uncertainty quality, such as misclassification detection or selective prediction.\n\n2. The total loss consists of two components (Equation 14), i.e., vicinal supervision and noise augmentation. However, the paper lacks an ablation study isolating their individual contributions, which would help clarify the effectiveness of each.\n\n3. Since the proposed method involves mixup and additional data generation, a brief analysis or discussion of computational complexity and training overhead would strengthen the practical understanding of the method's scalability.", "was_revised": true, "final_justification": "EDL methods offer valuable explainability through multi-dimensional uncertainty estimates, but are known to produce unfaithful uncertainty estimation. Learning a faithful second-order distribution remains a challenging and active research topic. While this paper can not fully resolve the collapsing problem in EDL, it may provide useful ideas for further exploration of EDL methods. Therefore, I maintain my score.", "rebuttal": "# Response to Weakness\n## Response to Comment 1\nThank you for highlighting the need for more targeted evaluations of aleatoric uncertainty. In response, we conduct selective prediction experiments on CIFAR-10-C using the Excess Area Under the Risk-Coverage Curve (E-AURC × 1000, lower is better), a standard metric for misclassification detection task. We adopt conditional entropy to measure aleatoric uncertainty.\nOur method consistently achieves the lowest E-AURC across all severities, demonstrating more reliable aleatoric uncertainty estimation and improved performance in selective prediction scenarios.\nWe will include these results in the revised manuscript.\n\n| Method       | s=1   | s=2  | s=3   | s=4   | s=5   | mean  |\n| ------------ | --- | --- | --- | --- | --- |  --- |  \n| EDL          |  18.12 $\\pm$ 0.31 | 30.16 $\\pm$ 0.96  |   44.54 $\\pm$ 1.95 |  63.61 $\\pm$  1.90  |  101.61 $\\pm$ 4.46 |      51.60 $\\pm$ 1.91\n| RED          |   16.74 $\\pm$ 0.40  |   30.15 $\\pm$ 1.40   |  44.80 $\\pm$ 1.66   |    65.26 $\\pm$ 2.61  | 103.87 $\\pm$ 5.73    |    52.16 $\\pm$ 2.36 |   \n| I-EDL        |   14.62 $\\pm$ 0.52 |   27.84 $\\pm$ 0.65   |  41.70 $\\pm$ 1.57   |   59.53 $\\pm$ 2.92  |  95.93 $\\pm$ 4.43   |      47.92 $\\pm$ 2.01 | \n| R-EDL        |  17.13 $\\pm$ 0.47  |  29.85 $\\pm$ 1.05  |  45.06 $\\pm$ 0.74   |   63.75 $\\pm$ 0.51  | 101.74 $\\pm$ 1.21\t    |      51.50 $\\pm$ 0.79 |\n| DA-EDL       |   20.61 $\\pm$ 2.75  |  35.56 $\\pm$ 5.31   |  51.09 $\\pm$ 8.38  |   72.29 $\\pm$ 11.23  |  112.83 $\\pm$ 14.53   |      58.47 $\\pm$ 8.44 |\n| **Ours**     | **8.70 $\\pm$ 0.35** | **14.80 $\\pm$ 0.45** | **21.90 $\\pm$ 0.81**|**35.06 $\\pm$ 1.81** | **66.52 $\\pm$ 7.37** | **29.40 $\\pm$ 2.16**|      \n\n## Response to Comment 2\nThank you for the valuable suggestion. We agree that isolating the contributions of the vicinal supervision and noise augmentation losses would provide a clearer understanding of their individual effects.\nTo address this, we have conducted an ablation study to individually analyze the contributions of the two components in the total loss.\nFirst, to isolate the effect of $ \\beta $, we first remove the loss term  $ L_{noise} $  and focus solely on its influence.\n\n| $\\beta$ | $\\beta_{\\text{noise}}^+$| MisD E-AURC ↓ | OOD AUROC ↑ | OOD Acc ↑ | ID Acc |\n|--------:|--------:|-----------------------------:|------------:|----------:|---------:|\n|-| -            |        56.54   $\\pm$ 3.98   |           90.67 $\\pm$ 0.35    |      74.51   $\\pm$ 0.49       | 95.17  $\\pm$ 0.18 |\n| 0.2     |   -  |          58.10   $\\pm$ 6.04             |     89.77 $\\pm$ 0.63        |     76.39 $\\pm$ 0.55      | 95.31 $\\pm$ 0.08|\n| 0.4     |  -    |        49.44 $\\pm$ 4.40                |    90.28 $\\pm$ 0.05          |   77.82 $\\pm$ 0.61        | 95.61 $\\pm$ 0.16|\n| 1.0     |   -     |         45.18 $\\pm$ 2.19          |   91.11 $\\pm$ 0.40              |     79.13 $\\pm$ 0.32      | 96.01 $\\pm$ 0.02|\n| 5.0     |   -      |      42.94 $\\pm$ 4.37               |       91.68 $\\pm$ 0.48      |    79.75 $\\pm$ 1.02       | 95.89 $\\pm$ 0.21 |\n| 10.0    |   -       |    39.15 $\\pm$ 4.33                |   91.73 $\\pm$ 0.14          |   80.41 $\\pm$ 1.23        | 95.86 $\\pm$ 0.16 |\n\nWe highlight several key observations from our ablation study:\n1. Improved Aleatoric Uncertainty Estimation for Misclassification Detection:\nIncreasing $\\beta$ leads to a monotonic decrease in E-AURC, indicating better calibration of aleatoric uncertainty for misclassification detection. E-AURC drops significantly from 56.54 (baseline) to 39.15 at $\\beta = 10.0$.\n\n2. Enhanced OOD Detection Performance:\nAlthough our primary goal is to improve aleatoric uncertainty estimation with $ L_{vicinal} $ , we observe consistent gains in both OOD AUROC as $\\beta$ increases. This suggests that vicinal supervision not only calibrates aleatoric uncertainty but also enhances the model’s discriminative ability between ID and OOD samples. We attribute this to the regularizing effect of vicinal labels, which expose the model to local variations around the data manifold and thus improve generalization beyond the training distribution—even in the absence of $\\mathcal{L}_{\\text{noise}}$. \n\n3. Enhanced Generalization Performance:\nWe also observe simultaneous improvements in both OOD classification accuracy and ID accuracy as $\\beta$ increases. Specifically, OOD accuracy rises from 74.51% to 80.41%, while ID accuracy improves from 95.17% to 95.86%. These gains follows Theorem 2 that strong mixup strengthens the model’s overall generalization across both known and unseen data distributions.\n\nWe further analyze the impact of $\\beta_{noise}^+$ by isolating the loss to $L = L_{noise}$. Following the setup in our main paper, we fix $\\beta_{noise}^- = 1.0$ and vary $\\beta_{noise}^+$.\n\n|$\\beta$| $\\beta_{\\text{noise}}^+$ | OOD AUROC (↑) | MisD E-AURC   (↓) | OOD Acc  (↑) | ID Acc  (↑)|\n|------|----------------------------|-----------|-------------|--------------|--------------|\n|-| -            |     90.67  $\\pm$ 0.35      |     56.54   $\\pm$ 3.98       |      74.51   $\\pm$ 0.49     | 95.17 $\\pm$ 0.18|\n|-| 0.2                        |   90.95 $\\pm$ 0.32       |   56.11 $\\pm$ 4.92        |  74.32 $\\pm$  0.43            | 95.00 $\\pm$ 0.02  |\n|-| 0.4                        |   91.18 $\\pm$ 0.42      |      62.81 $\\pm$ 10.15       |       73.76 $\\pm$ 0.67     | 95.03 $\\pm$ 0.21 |\n|-| 1.0                        |   91.85 $\\pm$ 0.24        |     57.22 $\\pm$ 6.14        |     73.96 $\\pm$ 0.43         | 95.17 $\\pm$ 0.09  |\n|-| 5.0                        |    90.74 $\\pm$ 0.20       |     55.95 $\\pm$ 6.40        |     74.67 $\\pm$ 0.75         | 95.02 $\\pm$ 0.20 |\n|-| 10.0                       |        90.37 $\\pm$ 0.55    |    60.17 $\\pm$ 2.85       |      73.42 $\\pm$ 0.53        | 94.86 $\\pm$ 0.41 |\n| 10.0  |          1.0         |    **93.08 $\\pm$ 0.33**       |   **29.40 $\\pm$ 2.16**             |      **88.73 $\\pm$ 0.13**        | **96.18 $\\pm$ 0.13** |\n\nWe highlight several key observations:\n1. Isolated Variation of $\\beta_{\\text{noise}}^+$ Alone Has Limited Impact on MisD and Cls Acc:\nWhen $L_{vicinal}$ is removed and only $\\beta_{\\text{noise}}^+$ varies (rows with “-” under $\\beta$), E-AURC and ID Acc fluctuate slightly but do not show consistent improvement,  indicating that tuning $\\beta_{\\text{noise}}^+$ alone without adjusting $\\beta$ with vicinal information may be insufficient for substantial gains.\n\n2. Balancing Noise Intensity Is Crucial: \nSetting $\\beta_{\\text{noise}}^+$ too small can impair classification accuracy, as overly aggressive noise perturbations dominate the input and hinder effective learning. Conversely, a very large $\\beta_{\\text{noise}}^+$ causes the sampled mixing coefficient $\\lambda$ to concentrate near 1, effectively disabling noise augmentation and reverting the method back to standard EDL behavior. In contrast, $\\beta_{\\text{noise}}^+ = 1.0$ achieves the best trade-off for epistemic uncertainty estimation and OOD detection.\n\n3. **Best Performance Achieved with Combined $\\beta = 10.0$ and $\\beta_{\\text{noise}}^+ = 1.0$:**\n The model attains the highest **OOD AUROC (93.08%)**, lowest **MisD E-AURC (29.40)**, best **OOD accuracy (88.73%)**, and highest **ID accuracy (96.18%)** when both parameters are optimized together. This highlights the **synergistic effect** of these two components.\n\n\n## Response to Comment 3\nOur method introduces minimal overhead from mixup, as it only involves simple linear interpolation. The main cost comes from extra forward and backward passes, roughly doubling training time compared to standard EDL. Importantly, our method adds no overhead at inference, keeping test-time cost unchanged.\n\n# Response to Questions\n### Response to Question 1\nTo further investigate whether the improvements in classification accuracy and uncertainty estimation stem from enhanced feature representations, we conducted a feature visualization analysis using t-SNE. We exactly observed that our method with vicinal labels learns more distinct boundaries between ID and semantic-shift OOD features, leading to more reliable epistemic uncertainty estimates. Additionally, it yields more robust feature representations for both covariate-shift OOD samples and ID samples, contributing to improved classification performance on both ID and OOD data. We will include these feature visualizations in a future revision of the paper.\n### Response to Question 2\nDirichlet differential entropy reflects both the concentration and dispersion of the distribution, while conditional entropy more directly corresponds to aleatoric uncertainty, as it measures the expected entropy of the predictive categorical distribution [1].\nEmpirically, we find that entropy-based metrics are particularly effective for models trained with level-1 supervision. We will include a comparison against other uncertainty measures such as vacuity of evidence and mutual information in the updated appendix.\n\n### Response to Question 3\n Theorem 2 rigorously proves that strong MixUp effectively reduces sample-dependent label noise introduced during training. Specifically, the conventional level-0 hard labels act as supervision signals that inject overconfident noise, which lowers the signal-to-noise ratio between the true label information and noise inherent in each sample. This often causes the model to overfit near class boundaries, negatively impacting generalization and robustness.\nBy constructing vicinal samples through strong MixUp, the strength of this overconfident noise is effectively diluted, enhancing the purity and consistency of label signals. As a result, the model achieves improved uncertainty estimation capability. Therefore, Theorem 2 not only theoretically supports improved generalization and robustness, but also suggests an enhancement in the quality of uncertainty quantification.\n\n[1] Malinin A, Gales M. Predictive uncertainty estimation via prior networks[J]. Advances in neural information processing systems, 2018, 31."}, {"review_id": "xq2uXeKjF0", "reviewer": "Reviewer_LVLn", "summary": "Evidential Deep Learning (EDL) is a widely used approach for uncertainty quantification (UQ) that models predictive uncertainty via a second-order distribution---specifically, a Dirichlet distribution over the probability simplex in classification tasks. Recent studies have pointed out that most existing EDL methods rely on level-0 supervision (i.e., hard labels), which can lead to unfaithful uncertainty representations. In particular, the learned Dirichlet distribution may become degenerate, collapsing to a Dirac delta function.\n\nThis work aims to address this issue by incorporating level-1 supervision, i.e., categorical label distributions. Since such ground-truth level-1 labels are typically expensive to obtain, the authors propose generating them using techniques inspired by MixUp. Two types of level-1 labels are constructed: one to model aleatoric uncertainty and another for epistemic uncertainty. The UQ model is then trained using a vicinal risk minimization (VRM) objective based on the constructed labels. The authors argue that this approach mitigates the degeneracy issue of Dirichlet distributions and enables more faithful uncertainty modeling. Empirical results demonstrate that the proposed method yields improved performance on downstream tasks such as out-of-distribution (OOD) detection.", "questions": "1.   Can you elaborate on the sensitivity to hyperparameters?  More generally, it is helpful to understand in what scenarios the method will fail or otherwise not be useful, and how it will fail when it does.\n\n2.  How can the approach of Section 3.2 be justified in light of the comments above?\n\n3.  How would the authors address the comment regarding one of the claims, as mentioned above?", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\n- This work tackles an important research problem: overcoming the limitations of existing EDL methods that struggle to learn faithful representations of uncertainty.\n\n- The motivation and overall direction are intuitive. Specifically, the paper clearly identifies the key issue, EDL methods learning second-order distributions from level-0 (hard) labels, and proposes constructing level-1 (soft) labels as a principled solution.\n\n- The proposed method demonstrates strong empirical performance on downstream UQ tasks.\n\nWeaknesses:\n\n- The contribution of the paper is somewhat limited, as several of the key arguments have already been well-established in the existing uncertainty quantification (UQ) literature. For example, prior works [2, 3] have pointed out that a core issue with EDL lies in its reliance on level-0 hard labels, which can cause the learned Dirichlet distribution to collapse into a Dirac measure. Additionally, [39] has shown that existing EDL methods fail to faithfully capture both aleatoric and epistemic uncertainty. Furthermore, both [22] and [39] emphasize the importance of incorporating level-1 supervision into the EDL framework, and [39] advocates for distillation-based approaches that construct level-1 labels through ensemble or bootstrap-based techniques (see Section 6 of [39]).\n\n- The proposed method is heuristic and appears to be sensitive to hyperparameters. While level-0 hard labels are sampled from an underlying true distribution, the level-1 labels in this work are constructed heuristically via MixUp. As a result, the distribution of these crafted labels, and by extension, the behavior of the learned UQ model, depends on the specific MixUp parameters. The ablation studies in Section 5 further confirm that the proposed method performs well only under well-tuned hyperparameter settings.\n\n- Some of the paper’s claims are not entirely accurate. Specifically, the concern regarding the Dirac delta collapse applies only to EDL objectives without regularization. In practice, many EDL methods employ regularizers, which prevent the learned Dirichlet from collapsing into a delta function. As shown in [39], with regularization, the learned second-order distribution converges to a fixed-form Dirichlet (see Example 5.2), determined by the regularization hyperparameter. Thus, while the distribution may not collapse, it does not necessarily mean the model has learned faithful or meaningful uncertainty.\n\n- The justification for the proposed approach in Section 3.2 is unconvincing. The authors suggest that mixing in-distribution samples with random noise helps epistemic uncertainty estimation and argue that perturbed samples should exhibit greater epistemic uncertainty than the clean samples. However, this assumption is questionable. Epistemic uncertainty reflects the model's lack of knowledge and should not depend on the input distribution. A simple counterexample illustrates this: consider a human expert acting as a highly confident classifier with negligible epistemic uncertainty. If a clean image is perturbed with sufficient random noise, even a human may become uncertain, not due to lack of knowledge of a human expert (epistemic uncertainty), but because the image has become ambiguous or unrecognizable (aleatoric uncertainty). This suggests that adding noise primarily increases aleatoric uncertainty, not epistemic uncertainty, as the model would struggle to make reliable predictions regardless of additional training data.", "was_revised": true, "final_justification": "While justifying analysis is limited, empirical benefits have been demonstrated.   Having gone through the responses to my review as well as the other reviews and responses, I am willing to raise my score from a 3 to a 4.", "rebuttal": "# Response to Weakness\n\n## Response to Comment 1\nWe sincerely appreciate the reviewer’s careful reading and valuable comments. We fully acknowledge that the core issue of Dirichlet degeneration under level-0 supervision has been identified in prior works such as [2, 3, 22, 39]. However, we would like to clarify several key differences and contributions of our method compared to these existing works:\n1. Consideration of aleatoric uncertainty:\nWhile [2, 3, 22] primarily focus on epistemic uncertainty, their methods largely overlook the modeling and improvement of aleatoric uncertainty. Our work explicitly addresses this gap by designing strong MixUp-based vicinal level-1 supervision to provide richer and more effective signals for aleatoric uncertainty estimation, which is often under-explored in existing EDL literature.\n\n2. Limitations of distillation labels for aleatoric uncertainty estimation:\nWhile [39] proposes a distillation-based approach to construct level-1 soft labels without requiring ground-truth distributions, we argue that such soft labels are inherently limited in representing aleatoric uncertainty. Specifically, distilled labels are typically:\n\n    - derived from ensemble predictions that tend to average out input ambiguity, and\n    - often remain close to hard labels, failing to reflect the true uncertainty present in ambiguous or borderline samples.\n    \n    As a result, we believe distillation is not well-suited for modeling aleatoric uncertainty.\n    In contrast, our method generates level-1 supervision using strong MixUp between semantically distinct examples, explicitly injecting ambiguity and producing soft labels that more faithfully capture aleatoric uncertainty.\n    For instance, although [39] lacks public code for direct comparison, Figure 4 in [39] shows their method underperforms I-EDL on misclassification detection. In our experiments, we surpass I-EDL on this task (see Response to Reviewer XHmk, Comment 2), highlighting the strength of our approach for aleatoric uncertainty estimation.\n\n\n3. Efficiency and scalability:\nEnsemble-based distillation approaches like [39] require training multiple models, which can be computationally expensive and less practical in large-scale or resource-limited settings. Our method requires training only a single model, making it significantly more efficient and scalable.\n\n\n\n## Response to Comment 2\nWe appreciate the reviewer’s concern regarding the heuristic nature and hyperparameter sensitivity of our proposed method.\nWhile it is true that our method involves hyperparameters such as $\\beta$ and $\\beta_{\\text{noise}}$, we would like to emphasize two key points:\n\n1. Theoretical guidance for hyperparameter selection: Unlike existing methods that rely purely on empirical tuning (e.g., KL divergence regularization or entropy constraints often require manually chosen weight coefficients without theoretical justification), our approach is supported by theoretical analysis. Specifically:\n\n\n    - Theorem 2 suggests that using a larger $\\beta$ (i.e., $\\beta \\gg 1/2$) encourages stronger interpolations, which in turn enhance model generalization by mitigating input-dependent label noise introduced by level-0 hard labels. This insight departs from standard MixUp settings (e.g., $\\beta = 0.2$) and offers a principled approach to narrowing the hyperparameter search range.\n\n    - Theorem 3 further links the choice of $\\beta_{\\text{noise}}$ to the statistical behavior of sampled $\\lambda$ values in the noise-based mixup. It highlights that overly large $\\beta_{\\text{noise}}^+$ leads to $\\lambda \\approx 1$, which may lead to an excessively rapid increase in Dirichlet strength.\n    Thus, our theory also guides the selection of $\\beta_{\\text{noise}}^+$ to avoid such degenerate cases.Together, these theoretical results significantly reduce the burden of hyperparameter tuning by informing practitioners what regimes to avoid and where optimal values are likely to lie.\n\n\n2. We also point out that existing EDL methods are not free from hyperparameter sensitivity. For example, methods that incorporate KL divergence or entropy constraints often require selecting an appropriate regularization weight (e.g., $\\lambda_{\\text{KL}}$), yet such choices are entirely empirical and lack theoretical grounding. In contrast, our method provides both practical effectiveness and theoretical explainability.\n\n\n## Response to Comment 3\nWe thank the reviewer for pointing out that Dirac delta collapse primarily arises in EDL without regularization. \nWe completely agree with this point, and indeed, our Theorem 1 is based on the same observation. \nAt the same time, we would like to clarify that our paper does not claim that non-degenerate Dirichlet distributions (e.g., those obtained via regularization) necessarily lead to reliable uncertainty estimation. In this regard, we fully share the reviewer’s perspective.\nHowever, our work focuses on a related but distinct issue: models trained solely with level-0 labels often fail to learn faithful or meaningful uncertainty estimates. \nIn other words, avoiding collapse is necessary but not sufficient for achieving well-calibrated uncertainty. This insight underpins our motivation to incorporate level-1 supervision.\nWhile our work does not directly investigate regularization-based EDL objectives, we believe we share the reviewer’s perspective on their limitations. In particular, many commonly used regularizers (e.g., entropy-based penalties) primarily control the overall spread or shape of the Dirichlet distribution, without necessarily aligning the uncertainty estimates with the underlying data ambiguity or model ignorance.\nAs a result, such regularization may help prevent degenerate solutions but still fail to produce reliable or calibrated uncertainty estimates, especially in regions of input space that are ambiguous or underrepresented. This further motivates our use of level-1 supervision, which provides more targeted guidance for modeling both aleatoric and epistemic uncertainty.\nWe appreciate the reviewer’s clarification and will revise the manuscript to better position our work relative to regularization-based methods and to make it explicit that we do not claim regularization alone ensures reliable uncertainty quantification.\n\n\n## Response to Comment 4\nWe thank the reviewer for the insightful comment on Section 3.2. We would like to clarify that aleatoric uncertainty arises from inherent  semantic ambiguity—such as when multiple annotators assign different labels to a perceptually ambiguous image. In contrast, our method does not introduce label randomness. Even when the input is perturbed with noise, we assign a consistent ground-truth label to each generated sample, and thus do not increase label uncertainty.\nMoreover, the Gaussian noise introduced in Section 3.2 is milder compared to the strong MixUp strategy discussed in Section 3.1, and therefore does not lead to a higher level of aleatoric uncertainty.\nThe objective of Section 3.2 is to simulate scenarios where the input becomes ambiguous or unfamiliar due to perturbations. By applying smoothed level-0 labels, we avoid excessive Dirichlet concentration, following the principle that epistemic uncertainty is closely tied to the dispersion of the Dirichlet distribution.\nIn summary, our method does not conflate aleatoric and epistemic uncertainty. Instead, it leverages fixed supervision to guide the model toward expressing lower confidence (i.e., higher epistemic uncertainty) when encountering inputs that lie off the data manifold. We will revise Section 3.2 to make this distinction clearer in the final version.\n\n\n\n# Response to Questions\n\n## Response to Question 1\nThank you for raising this important point regarding the sensitivity to hyperparameters and potential failure cases of our method.\nTo provide a more thorough understanding of the robustness of our approach, we kindly invite the reviewer to refer to our response to Comment 2 from Reviewer 9dhq, where we independently activated each loss component to assess its individual contribution. This analysis indicates that, although the method does benefit from appropriate tuning, the effective range of hyperparameters aligns well with the theoretical guidance provided by Theorems 2 and 3.\n\n## Response to Question 2\n \nWe thank the reviewer for raising this important question regarding the justification of our approach in Section 3.2. The key intuition behind our method is to approximate epistemic uncertainty via a smooth supervision signal, inspired by label smoothing and vicinal risk minimization principles.\nSpecifically, we perturb in-distribution samples with increasing levels of noise, and assign increasingly smooth target distributions (i.e., closer to uniform) as supervision. This construction is based on the observation that such perturbed inputs are more likely to lie away from the training data manifold, and hence the model should express lower confidence (higher epistemic uncertainty) in its predictions.\nImportantly, we emphasize that our method does not inject label noise—the supervision remains controlled and structured. Rather than introducing randomness, we deliberately reduce the confidence on the original label as the input becomes less familiar to the model. This aligns with the core idea of epistemic uncertainty: uncertainty due to limited knowledge or lack of training coverage.\nWhile our approach is heuristic in nature, we find it empirically effective for improving model behavior on OOD and ambiguous inputs. We will revise Section 3.2 to clarify this motivation and better distinguish between label smoothing for epistemic modeling versus aleatoric uncertainty due to intrinsic ambiguity.\n\n\n## Response to Question 3\n Thank you for your question. Please kindly refer to our previous comments."}, {"review_id": "6SIx4Ian4z", "reviewer": "Reviewer_XHmk", "summary": "This paper adresses a limitation in a class of deep learning uncertainty quantification (UQ) methods referred to as Evidential Deep Learning (EDL) methods, which rely on level-0 hard labels for supervision. The authors argue that hard labels do not reflect inherent data noise (aleatoric uncertainty). In practice, they observe that this mismatch encourages EDL models to produce degenerate Dirichlet distributions that collapse into near-deterministic outputs, leading to poor uncertainty quantification.", "questions": "A concern with this paper is that the experiments are not sufficiently convincing, as the authors seem eager to claim that their method establishes new standards in OOD input detection. Generally speaking, UQ methods are not explicitly designed for outlier/OOD input detection but may develop this capability as a byproduct. Specifically, the benchmark should include at least one UQ method known to exhibit strong outlier-detection performance, such as Liu et al. (2020). However, the experiments do validate the added value of their method compared to EDL for this use case.\n\n(Liu et al. 2020) Liu, J., Lin, Z., Padhy, S., Tran, D., Bedrax Weiss, T., & Lakshminarayanan, B. (2020). Simple and principled uncertainty estimation with deterministic deep learning via distance awareness. Advances in neural information processing systems, 33, 7498-7512.\n\nAnother important aspect to be addressed in the experiments is evaluating other use cases for quantified uncertainties, particularly whether those uncertainties align with classification errors. Indeed, one generally expect higher levels of uncertainties in the regions of the input space where the model fails to predict the correct class labels. \n\nBesides, many competing methods in the benchmark exhibit terrible performances. Could the authors comment on why they fail to so dramatically ?\n\nThe main contribution of the proposed modification to EDL is its ability to capture the input distribution (as Fig. 1 suggests) and leverage it to assign high uncertainty in low-probability regions of the input space. Yet this emerging capability is neither fully explored nor theoretically explained. Based on the authors’ comments that they use a large value for β, I suspect that most mixed samples lie near class boundaries; thus, it is unclear why the model would also learn to assign high uncertainty to inputs far from the training samples.\n\nMinor: In Figure 1, unless I’m mistaken, the displayed uncertainty-scoring function is not specified.", "limitations": "The limitations are addressed very briefly (one sentence - 2 lines) in the conclusion. Usually, one would expect at least a whole paragraph.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Pros : \n- Avoiding EDL degenerate solutions seems to be achieved by the proposed method\n- The method is simple in its construction and thus does not add significant complexity on top of EDL algorithms.\n- The efficiency of the method is theoretically grounded\n\nCons : \n- The empirical comparisons in the experimental section omit important state-of-the-art works leveraging UQ methods to detect outliers/OOD inputs. \n- The experiments focus on OOD detection uses cases and do not examine whether quantified uncertainties align with incorrect predictions, as is commonly done\n- [Minor] The experiments involve training runs on two vision datasets and one toy dataset. The paper would gain in impact with a more solid evaluation involving more training datasets and other data modalities.", "was_revised": true, "final_justification": "I thank the authors for addressing the raised concerns. In light of the additional empirical and critical evidence provided by the authors, I raised my score by one unit.", "rebuttal": "# Response to Weakness\n## Reponse to Comment 1\nThank you for your valuable comment. We extend our experiments to include four SOTA UQ-based OOD detection methods—DUQ [1], DDU [2], DUE [3], and SNGP [4]. Our method achieves competitive performance, especially on CIFAR-100, without relying on additional techniques like spectral normalization adopted by competitive methods DDU and SNGP. This highlights the effectiveness of using level-1 vicinal labels to improve uncertainty estimation based OOD detection.\n\n#### → ID: CIFAR-10\n\n| Method | → CIFAR-100 | → Tiny | → MNIST | → SVHN | → Textures | → Places365 | CIFAR-10 Acc |\n|--------|-------------|--------|---------|--------|------------|-------------|--------------|\n| DUQ [1]  | 84.60 $\\pm$   1.04 | 86.16 $\\pm$  0.92 | 92.34 $\\pm$  1.25 | 91.36 $\\pm$ 1.26 | 86.57 $\\pm$ 1.27 | 84.26 $\\pm$ 0.93 | 93.60 $\\pm$ 0.39 |\n| DDU [2]  | **89.23 $\\pm$ 0.33** | 91.28 $\\pm$ 0.28 | 95.69 $\\pm$ 0.99 | 93.53 $\\pm$ 1.43 | 92.64 $\\pm$ 0.31 | 91.55 $\\pm$ 0.34 | 95.35 $\\pm$ 0.05 |\n| DUE [3]  | 86.00 $\\pm$ 0.78 | 88.40 $\\pm$ 0.76 | 92.34 $\\pm$ 1.48 | 91.70 $\\pm$ 1.63 | 89.25 $\\pm$ 1.16 | 88.89 $\\pm$ 0.77 | 94.98 $\\pm$ 0.15 |\n| SNGP [4] | 88.14 $\\pm$ 1.18 | 90.38 $\\pm$ 1.00 | 93.18 $\\pm$ 1.20 | 92.16 $\\pm$ 2.40 | **92.88 $\\pm$ 1.37** | 91.21 $\\pm$ 1.20 | 94.63 $\\pm$ 0.18 |\n| **Ours** | 89.09 $\\pm$ 0.19 | **91.81 $\\pm$ 0.22** | **97.32 $\\pm$ 0.42** | **96.20 $\\pm$ 0.32** | 92.51 $\\pm$ 0.73 | **91.61 $\\pm$ 0.15** | **96.18 $\\pm$ 0.13** |\n\n#### → ID: CIFAR-100\n\n| Method | → CIFAR-10 | → Tiny | → MNIST | → SVHN | → Textures | → Places365 | CIFAR-100 Acc |\n|--------|------------|--------|---------|--------|------------|-------------|----------------|\n| DUQ [1]  | 51.20 $\\pm$ 1.84 | 53.60 $\\pm$ 2.67 | 39.44 $\\pm$ 13.20 | 61.47 $\\pm$ 7.64 | 57.73 $\\pm$ 5.66 | 50.16 $\\pm$ 3.13 | 1.66 $\\pm$ 0.39 |\n| DDU [2]  | 68.14 $\\pm$ 1.63 | 78.64 $\\pm$ 1.57 | 79.69 $\\pm$ 6.56 | 76.02 $\\pm$ 3.98 | **83.00 $\\pm$ 1.01** | 74.53 $\\pm$ 1.70 | 78.05 $\\pm$ 0.96 |\n| DUE [3]  | 50.30 $\\pm$ 1.54 | 49.97 $\\pm$ 1.39 | 49.91 $\\pm$ 1.43 | 49.91 $\\pm$ 1.02 | 50.02 $\\pm$ 1.65 | 50.12 $\\pm$ 1.01 | 1.06 $\\pm$ 0.17 |\n| SNGP [4] | 72.77 $\\pm$ 1.23 | 76.63 $\\pm$ 1.51 | 71.91 $\\pm$ 6.19 | 73.54 $\\pm$ 5.36 | 73.91 $\\pm$ 1.85 | 74.53 $\\pm$ 1.93 | 76.19 $\\pm$ 1.19 |\n| **Ours** | **81.86 $\\pm$ 0.19** | **83.40 $\\pm$ 0.09** | **86.17 $\\pm$ 1.83** | **85.07 $\\pm$ 1.94** | 78.58 $\\pm$ 0.33 | **79.97 $\\pm$ 0.39** | **78.29 $\\pm$ 0.11** |\n\nWe found that DUQ and DUE fail to converge on CIFAR-100 when using the same settings that work on CIFAR-10. As their original papers do not report CIFAR-100 results, there is little guidance for adapting these methods to more complex datasets.\n\n\n## Response to Comment 2\nThank you for your constructive comment. We have added misclassification detection experiments using Excess Area Under the Risk-Coverage curve (E-AURC *1000) as the evaluation metric, which is widely adopted for misclassification detection task. We use conditional entropy as the aleatoric uncertainty measure.\n\nOur method consistently achieves the lowest E-AURC scores across different corruption levels on CIFAR10-C, indicating its strong ability to provide well-calibrated uncertainty that correlates with misclassification likelihood. This further validates the practical utility of our approach beyond OOD detection. \n\n| Method       | s=1 | s=2 | s=3  | s=4  | s=5   | mean |\n| ------------ | --- | --- | --- | --- | --- |  --- |  \n| EDL          |  18.12 $\\pm$ 0.31 | 30.16 $\\pm$ 0.96  |   44.54 $\\pm$ 1.95 |  63.61 $\\pm$  1.90  |  101.61 $\\pm$ 4.46 |      51.60 $\\pm$ 1.91\n| RED          |   16.74 $\\pm$ 0.40  |   30.15 $\\pm$ 1.40   |  44.80 $\\pm$ 1.66   |    65.26 $\\pm$ 2.61  | 103.87 $\\pm$ 5.73    |    52.16 $\\pm$ 2.36 |   \n| I-EDL        |   14.62 $\\pm$ 0.52 |   27.84 $\\pm$ 0.65   |  41.70 $\\pm$ 1.57   |   59.53 $\\pm$ 2.92  |  95.93 $\\pm$ 4.43   |      47.92 $\\pm$ 2.01 | \n| R-EDL        |  17.13 $\\pm$ 0.47  |  29.85 $\\pm$ 1.05  |  45.06 $\\pm$ 0.74   |   63.75 $\\pm$ 0.51  | 101.74 $\\pm$ 1.21\t    |      51.50 $\\pm$ 0.79 |\n| DA-EDL       |   20.61 $\\pm$ 2.75  |  35.56 $\\pm$ 5.31   |  51.09 $\\pm$ 8.38  |   72.29 $\\pm$ 11.23  |  112.83 $\\pm$ 14.53   |      58.47 $\\pm$ 8.44 |\n| **Ours**     | **8.70 $\\pm$ 0.35** | **14.80 $\\pm$ 0.45** | **21.90 $\\pm$ 0.81**|**35.06 $\\pm$ 1.81** | **66.52 $\\pm$ 7.37** | **29.40 $\\pm$ 2.16**|      \n\n## Reponse to Comment 3\nWe thank the reviewer for this constructive suggestion. \nTo address this, we expanded our evaluation to include ImageNet-200 as the ID dataset and conducted OOD detection experiments on five challenging benchmarks: SSB-Hard, NINCO, iNaturalist, Textures, and OpenImage-O. \nThese datasets cover diverse natural image variations and are widely adopted for large-scale OOD evaluation in OpenOOD. \n\nDue to space limitation, for detailed experimental setup and hyperparameter configurations, please kindly refer to Response to Comment 1 of Reviewer Vmcj. Here, we reproduce the main results below:\n\n\n| Method     | →SSB-Hard        | →NINCO           | →iNaturalist     | →Textures        | →OpenImage-O      | Cls Acc         |\n|------------|------------------|------------------|------------------|------------------|-------------------|------------------|\n| KL-PN      | 50.00        | 50.00        | 50.00       | 50.00      | 50.00        | 0.50         |\n| RKL-PN     | 50.00      | 50.00         | 50.00      | 50.00       | 50.00   | 0.50        |\n| EDL        |45.55\t| 46.34| 40.91| 36.08 | 52.42\t |2.30|\n| RED        |77.81| 82.25 | 85.42| 80.32 | 83.72 |86.19|\n| I-EDL     |48.50| 52.49\t| 52.57 | 54.24 | 49.58 |1.09|\n| R-EDL      |50.88| 53.99 | 41.10\t| 53.99 | 55.28 |2.67|\n| Our | **79.80**|  **83.58** | **93.93**| **90.71** | **89.84** |**87.33**|\n\nThe results show that most standard EDL methods either fail to converge or perform poorly in this large-scale setting, confirming their limitations beyond toy datasets. In contrast, our method demonstrates consistently superior OOD detection performance, highlighting its robustness and scalability.\n# Response to Questions\n## Response to Benchmark coverage and misclassification detection experiments\nWe appreciate the reviewer’s thoughtful comments. We have addressed both points in our Response to Weaknesses above, please kindly refer to that section for further details.\n\n## Reponse to terrible performances of some methods in benchmark\nFirst, the terrible performance of KL-PN and RKL-PN on CIFAR-10 has also been reported in prior works, including I-EDL and R-EDL. We suspect this is primarily due to the use of a regularization strategy that imposes a constraint: Dirichlet strength corresponding to incorrect classes are explicitly required to approach zero (e.g., matching a prior distribution). Such a constraint may result in a collapse of feature representations with exponential activation, as the model over-optimizes toward suppressing incorrect class activations rather than learning expressive features when the number incorrect classes dominate the loss computation (i.e. K-1 $\\gg$ 1). \n\nSecond, we observe that EDL variants based on the Brier score loss—such as EDL, R-EDL, and DA-EDL perform particularly poorly on CIFAR-100 and ImageNet-200 in terms of classification accuracy. This limitation of the Brier score in large-class settings has also been highlighted in the RED method [6], which pointed out its inefficacy when applied to tasks with a large number of categories. \n\n\n## Reponse to the question why model would also learn to assign high uncertainty to inputs far from the training samples\n\nWe hypothesize that this behavior arises from the distance-preserving property of level-1 vicinal labels, which better reflect the similarity structure of the input space. Unlike level-0 hard labels that enforce sharp one-hot decisions, vicinal level-1 labels encode continuous relationships between nearby samples. Consequently, EDL models trained on such labels tend to learn feature representations that are more sensitive to distances in the input space, naturally leading to higher uncertainty in regions far from the training data manifold.\n\nIn contrast, when EDL is trained with level-0 hard labels, we observe that the classifier vectors tend to converge to solutions similar to those of a $K$-class SVM. The regions of high uncertainty are enclosed by negative margins of decision boundaries between different classes of a $K$-clss SVM. This phenomenon, known as the implicit bias of neural networks [5], has attracted increasing attention in recent research and is considered particularly intriguing.\n\n\n## Response to uncertainty-scoring function in Figure 1\nAs clarified in lines 267–270, we define uncertainty as the vacuity of evidence, computed as K ⁄ S for consistency, and this is the measure used in Figure 1 as well.\nWe will make this explicit in the caption of Figure 1 for clarity and completeness.\n\n\n\n### Reference\n\n[1] Uncertainty Estimation Using a Single Deep Deterministic Neural Network. van Amersfoort, Joost and Smith, Lewis and Teh, Yee Whye and Gal, Yarin. ICML 2020.\n\n[2] Deep deterministic uncertainty: A new simple baseline.\nMukhoti, Jishnu and Kirsch, Andreas and van Amersfoort, Joost and Torr, Philip HS and Gal, Yarin. CVPR 2023.\n\n[3] On Feature Collapse and Deep Kernel Learning for Single Forward Pass Uncertainty. van Amersfoort, Joost and Smith, Lewis and Jesson, Andrew and Key, Oscar and Gal, Yarin.\n\n[4] Simple and principled uncertainty estimation with deterministic deep learning via distance awareness. Liu, Jeremiah and Lin, Zi and Padhy, Shreyas and Tran, Dustin and Bedrax Weiss, Tania and Lakshminarayanan, Balaji. NIPS 2020\n\n[5] Soudry D, Hoffer E, Nacson M S, et al. The implicit bias of gradient descent on separable data[J]. Journal of Machine Learning Research, 2018.\n\n[6] Pandey D S, Yu Q. Learn to accumulate evidence from all training samples: theory and practice[C]//International Conference on Machine Learning. PMLR, 2023."}]

Author rebuttal:

Meta-review: In the recent literature, EDL has been shown to be flawed in the sense of collapsing to degenerate (Dirac) predictions when trained on standard (level 0) data, unless being regularised as proposed in the original paper by Sensoy et al. Here, to overcome this problem, the authors propose the use of vicinal supervision, thereby enhancing aleatoric uncertainty estimation. In addition, they propose noise-augmented vicinal risk minimization for better epistemic uncertainty estimation. The proposed method is analysed theoretically and evaluated empirically for OOD detection and generalization on several common image datasets.

The reviewers are quite in favor of the paper, and I do agree that the combination of vicinal risk minimization (VRM) and EDL is an appealing idea, although the noise augmentation looks somewhat ad hoc to me. Moreover, the empirical evidence is quite convincing. That said, there are a few points that bother me and that the authors should take into account when finalising their paper. 

For example, the authors observe that "EDL also fails to faithfully capture aleatoric uncertainty under the same empirical risk minimization principle", and they claim this observation as a new finding (Theorem 1). However, it follows directly from Theorem 3.3 in Jürgens et al. (2024). It's true that they don't say it explicitly there, but given the obvious observation that the L_1 loss is 0 for $\delta_y$, it indeed follows immediately. By the way, Jürgens et al. (2024) also state an important technical assumption, namely a universal approximation property, which is missing in the authors' theorem/proof, making it technically incorrect. Without a universal approximation property, the learner might not be able to realize the prediction of a Dirac delata function at every point x_i. Finally, and perhaps most importantly, I would argue that the claim itself is somewhat misleading, simply because it is not a defect of EDL. Exactly the same happens for a standard (first-order) neural network: Without regularisation, it will try to fit the data as much as it can, and if it has the capacity (universal approximation property), it will simply reproduce the data -- that's nothing else than standard overfitting. 

A possible drawback of the proposed method is the dependency of the learner's uncertainty representation on external parameters ($\beta$), which are controlled by the user: Isn't the degree of epistemic uncertainty directly influenced by how "smooth" the first-order "vicinal" distributions are made? Don't we then have the same problem as for original EDL, namely that the uncertainty is controlled externally and hence more or less arbitrary? In EDL, it is controlled by regularising the learner, here it is controlled by "regularising the data".

Theorem 2 looks at lower bounds on the risk, not at upper bounds, which would actually be more interesting. Then, having a lower lower bound for mixing compared to the original bound is interpreted as a result in favour of mixing. This is a least a bit unusual, isn't it? After all, these are only bounds, and we even don't don't know whether they are tight.
Decision: Accept (poster)