Uncertainty Estimation by Flexible Evidential Deep Learning
Authors: Taeseong Yoon
Heeyoung Kim
Venue: NeurIPS 2025 poster ()
Keywords: uncertainty quantification
single forward pass uncertainty
evidential deep learning
flexible Dirichlet distribution
Primary area: general_machine_learning
TL;DR: We enhance the generalizability of evidential deep learning by predicting the flexible Dirichlet distribution over class probabilities.

Abstract: Uncertainty quantification (UQ) is crucial for deploying machine learning models in high-stakes applications, where overconfident predictions can lead to serious consequences.  An effective UQ method must balance computational efficiency with the ability to generalize across diverse scenarios. Evidential deep learning (EDL) achieves efficiency by modeling uncertainty through the prediction of a Dirichlet distribution over class probabilities. However, the restrictive assumption of Dirichlet-distributed class probabilities limits EDL's robustness, particularly in complex or unforeseen situations. To address this, we propose *flexible evidential deep learning* ($\mathcal{F}$-EDL), which extends EDL by predicting a flexible Dirichlet distribution—a generalization of the Dirichlet distribution—over class probabilities. 
This approach provides a more expressive and adaptive representation of uncertainty, significantly enhancing UQ generalization and reliability under challenging scenarios. We theoretically establish several advantages of $\mathcal{F}$-EDL and empirically demonstrate its state-of-the-art UQ performance across diverse evaluation settings, including classical, long-tailed, and noisy in-distribution scenarios.

Uncertainty Estimation by Flexible Evidential Deep Learning
Taeseong Yoon Heeyoung Kim
Department of Industrial and Systems Engineering, KAIST (bigstar0423, heeyoungkin))0kaise
Abstract
Uncertainty quantification (UQ) is crucial for deploying machine learning models in high-stakes applications, where overconfident predictions can lead to serious consequences. An effective UQ method must balance computational efficiency with the ability to generalize across diverse scenarios. Evidential deep learning (EDL) achieves efficiency by modeling uncertainty through the prediction of a Dirichlet distribution over class probabilities. However, the restrictive assumption of diittilet-distibbuteed class probabilities limits EDL's robustness, particularly complex unforeseen situations. To address this, we propose prexisef evidential deep learning (F-EDL). which extends EDL by predicting flexible Dirichlet distribution-4 generalization of the Dirichlet distribution This approach provides more expressive and adaptive representation of uncer- tainty, significantly enhancing UQ generalization and reliability under challenging scenarios. We theoretically establish several advantages F-EDL and empirically demonstrate its state-of-the-art UQ performance across diverse evaluation settings, including classical, long-tailed, and noisy in-distribution scenarios.
class probabilities.
1 Introduction
Machine learning models have achieved remarkable predictive performance in diverse fields, includ- ing computer vision and natural language processing. However, their deployment in high-stakes applications-auuont autonomous driving, medical diagnosis, and manufacturing -enettrrrin-emmminsss limited due to concerns about reliability and overconfident predictions 21 Robust uncertainty quantification (UQ) is essential for ensuring safer, more trustworthy decision-making making in such contexts.
Effective UQ methods must meet two fundamental requirements: (i) computational efficiency, enabling integration into real-time systems, and (ii) generalizability to diverse and unforeseen scenarios. Classical UQ methods-Bayesing neural networks (3), Monte Carlo dropout 44] and deep ensembles (5) are well established but computationally intensive, as they require multiple g have emerged efficient alternatives, among which evidential deep learning (EDL) 11 stands out. EDL predicts Dirichlet distribution over class probabilities to quantify uncertainty, leveraging the conjugate prior structure of the Dirichlet distribution and its flosed-form uncertainty measures. This approach enables computationally efficient UQ and demonstrates strong performance in downstream tasks, such as ou distribution (OOD) detection [2].
forward passes. In response, single forward pass UQ methods 7
Despite these advantages, EDI. may struggle to provide robust uncertainty estimates in complex or unforeseen scenarios, leading to suboptimal UQ performance. This is shown in Figure Ta). which illustrates the epistemic uncertainty distributions for EDL trained on the Dirty-MNIST (DMNIST) dataset [8]. DMNIST combines clean MNIST (blue), representing clean in-distribution (ID) samples, with Ambiguous-MNIIT (AMNIST, green), a noisy ID dataset containing ambiguous samples. Addi- tionally, Fashion-MNEN (FMNIST, red) serves as an OOD dataset. To ensure fair comparison, we
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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DMNIST (ID)
MNIST ID)
AMNIST (Noisy ID]
FMNISS (OOD)
6
1
1
9
Density
PHREST
(a) EDL Epistemic Uncertainty
(b) J-EDD Epistemic Uncertainty
Figure 1: Epistemic uncertainty distributions with DMNIST as the ID dataset. The top presents sample images from three datasets: MNIST, AMNIST, and FMNIST. Panels (a) and (h) display histograms depicting the epistemic uncertainty distributions obtained by the EDL and F-EDL (proposed) models, respectively, across these datasets.
adopt a standard uncertainty metric used in EDL [9. An ideal UQ model should assign progressively higher epistemic uncertainty from clean ID to noisy ID and OOD samples, as intended in the DMNIST benchmark [8]. However, EDL exhibits substantial overlap between noisy ID and OOD distributions, and even assigns high uncertainty to some clean ID) samples, indicating limited generalization in noisy ID scenarios. In contrast, our proposed model, described below, achieves the desired separation among these datasets, as shown in Figure [b), demonstrating robust UQ and improved generalization under noisy conditions. We hypothesize that EDL's limitation stems from its core assumption that the class probabilities for each data point follow Dirichlet distribution. While this assumption facilitates computational efficiency, it restricts the model's ss ability to express complex or ambiguous forms of uncertainty. This motivates the need for more flexible yet efficient UQ methods.
Building upon this insight, we propose flexible evidential deep learning (F-EDL), novel UQ framework that extends EDL by predicting a flexible Dirichlet (FD) distribution 3 over class probabilities. The FD distribution generalizes the Dirichlet, enabling a more expressive representation of uncertainty while retaining the computational efficiency of EDL through its conjugate prior- allowing a single forward pass UQ suitable for real-time applications. This expressiveness improves the model's 's generalizability, enabling robust UQ across diverse and challenging scenarios such as noisy ID data, tail classes, and distribution shifts. Beyond the FD-based structure, J-EDL introduces tailored objective function for uncertainty-awee classification and label-wise variance-based uncertainty measures for robust, generalizable UQ
F-EDL introduces several theoretical ad advancements that strengthen its UQ capabilities. First, we prove that the FD distribution serves conjugate prior the categorical likelihood. Building on this, we prove that F-EDL predicts the posterior FD distribution for each input by employing improper prior with parameters learned from the data, addressing the limitations of fixed priors in traditional EDL. Second, we prove that F-EDL converges to standard EDL under specific parameter settings, establishing it as a generalization of EDL. Third, we prove that F-EDL can generate multimodal class probability distributions on the simplex, allowing it to capture complex and nuanced uncertainty patterns. Fourth, we prove that F-EDL. functions as mixture model that adaptively combines EDL and softmax predictions using input-dependent mixture weights, demonstrating how it leverages the strengths of both approaches to achieve optimal predictions and enhanced UQ. Finally, we prove that F-EDL admits generalized subjective logic interpretation, modeling uncertainty mixture of class-specific opinions that capture the model's 's hesitation among plausible hypotheses.
structured
We empirically evaluate F-EDL across a wide range of UQ-related downstream tasks, including classification, misclassification detection, OOD detection, and distribution shift detection. Notably, F-EDL consistently achieves state-of-ther -hat performance across diverse settings, including classical, long-tailed, and noisy ID scenarios, highlighting its robustness and generalizability. In addition, qualitative analyses show that F-EDL captures interpretable multimodal uncertainty reflecting ambiguity across plausible classes, while demonstrating faithful epistemic behavior that decreases with more data.

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2 Preliminaries
2.1 Evidential Deep Learning
Evidential deep learning (EDL) quantifies uncertainty in classification tasks by predicting Dirichlet distribution over class probabilities. Introduced by Sensoy et al. [9] EDL is grounded in Dempster- Shafer Theory of Evidence (DST) [14 and subjective logic (SL) [1]. DST represents subjective beliefs by assigning belief masses to classes, capturing both class likelihoods and uncertainty. SL extends DST by formalizing these subjective beliefs using Dirichlet distribution enablllnn uncertainty- classification within a principled framework.
The core innovation of Sensoy et al. [9] is using neural network to predict Dirichlet parameters. Let D denote the input dimensionality and K denote the number of classes. For given input x E IR, the class probability distribution is represented a E R are defined as a = 1 + Rf (x)). Here, 1 = (I...., 1) E R is a vector of ones, and fe RP IK represents the neural network parameterized by 0. The objective function of EDL for a given data point (x, y), where y is the one-hot encoded label, consists of two components: the expected mean squared error (MSE) and Kullback-Leibler (KL) divergence penalty. It is formulated as follows:
Dir(x)), where the concentration parameters
L = Diraaaa)
ADRi. [Dir(ā)] Dir(1))
Expected MSE over Dirichlet
divergence penally
where ā = (1 y) y. and À is regularization parameter.
2.2 Flexible Dirichlet Distribution
The flexible Dirichlet (FD) distribution 13
is generalization of the Dirichlet distribution. The FD distribution is derived by normalizing a flexible Gamma (FG) basis. Specifically, the FG basis, denoted as Y (Yi..., Yk), is constructed as follows:
Y = W Zz U, vk E [K],
where WR Gamma(ax), Vk E [K]. are independent Gamma random variables, and U
Gamma(r) is another independent Gamma random variable sharing the same scale parameter as each Ws. Here, the parameters satisfy aę 0, vk E [K] and T > O. The vector Z (Zi..... ....... Zk)
follows multinomial distribution, Z ~ Mu(1, p), where = (PI..... PK). 0 S Pk S 1, Vk E [K],
and E k= PI: = 1. This FG basis is denoted as Y ~ FG (Gg*(a,,pp T). where = (a1,... ... ox) By combining independent Gamma random variables with a shared random component, the FG basis introduces flexibility, effectively modeling dependencies among components.
The FD distribution, denoted as X ~ FD (a, P. is defined as X = (X Xx) with
X₁ = Yx ////> -k= Ys, 4k E [K]. Due to its ability to capture complex dependencies and multimodal behavior, the FD distribution well-suited for compositional data analysis [3]. While previously unexplored for UQ, its flexibility makes it a compelling candidate for modeling uncertainty.
3 Flexible Evidential Deep Learning
The F-EDL framework introduces an efficient and generalizable UQ methodology by leveraging the FD distribution to model a distribution over class probabilities. The adoption of the FD distribution in F-EDL represents a principled generalization of EDL, rather than an ad-hoc extension of model flexibility. By introducing additional parameters (p, r), the model governs how evidence is allocated across classes (via p) and concentrated in magnitude (via r), thereby enabling structured and inter- pretable uncertainty may be simultancously plausible. Furthermore, the FD-based formulation offers an input-adaptive mechanism for evidence extraction, capturing how uncertainty evolves dynamically in response to the characteristics of each input. A detailed theoretical discussion is provided in Appendix D
for ambiguous inputs where multiple class hypotheses
To operationalize this formulation within a neural framework. F-EDL consists of three key compo- nents: (i) a distinctive model structure (Section (ii) a tailored objective function (Section 3.2)..
and (iii) label-wise variance-based uncertainty measures (Section 3.3
We refer to this distribution as FD, omitting the superscript when the dimensionality K is clear from the context.
3

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(a) Graphical Model
(b) Generative Process
2 fo(x)
exp(1en (z))
P = softmax (gobz (z))
x
Z
α
π
y
softplus(ges (z))
π FD(a,,,
Cat(
Figure (a) Graphical model and (b) generative process of the proposed F-EDL framework.
3.1
Model Structure
Let x RD denote input with dimension D, and let fo(x) I denote its feature representation, where fo RD + R is a feature extractor parameterized by O and is the feature dimension. From the model predicts the parameters of the FD distribution: concentration parameters a E R allocation probabilities p E K-1, and dispersion T E R4, which together define a distribution over class probabilities π E A K-1 from which the label y is sampled. These parameters are computed by three neural heads: I6. : IR I2 Jø. 1 IR and Jds. R R. parameterized by Φ1, Φ2, and Φ3. respectively.
To ensure non-negativity and stability a, we use exp activation, following prior work [77 To map p onto the probability simplex, activation for smoother gradients and improved empirical performance. We also apply spectral normalization [8 to fe and Pø, to improve the robustness of and enhance the quality of UQ 12 This normalization further enforces Lipschitz continuity and constrains the magnitudes of network outputs, thereby implicitly regularizing without requiring explicit penalty terms.
the softmax function. For
the softplus
The overall structure is illustrated Figure 2 process in panel (b).
with the graphical model in panel (a) and the generative
Objective Function
The F-EDL framework is optimized using objective function consisting of two components: (i) the expected MSE with respect to the FD distribution, following standard EDL practice for uncertainty-awan classification, and (ii) Brier-score-bar input-dependent calibration and prevent degenerate solutions.
regularization term for p, promote
For given data point (x, y), where y is the one-hot encoded representation of the label y, the objective function defined as follows:
C E Eall - π
+ yy
31
Expected MSE
Regularization term
F-EDL enables analytic training due the closed-form moments of the FD distribution, eliminating the need for sampling. also simplifies optimization by reducing sensitivity to hyperparameters, unlike recent EDL methods that require careful tuning [10
3.3 Label-Wise Variance-Based Uncertainty Measures
After training, the predicted label for a testing example x* is determined by assigning the class with the highest expected class probability under the FD distribution: ((x*) argmax [E=FD((.. [TR], where a. and are predicted parameters x*.
To quantify uncertainty, we adopt a label-wise variance-based approach [9], replacing the Dirichlet with the FD distribution, enabling robust and interpretable UQ consistent with established axioms
20 For each class & E [K]. the predictive uncertainty x", denoted by TU (x*(x)), defined the variance of the class label: TU+(x*) = Var()x(x') This quantity is decomposed via the law of total

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variance into aleatoric and epistemic components, denoted by (x) and EU&(x*), respectively:
Var(ys|x*)
En [Var(yn Var (B 7., [T]
(UR(x*)
Axx(x*)
BUx(x*)
The total, aleatoric, and epistemic uncertainties for x* are computed by summing the respective class-wiss uncertainties and substituting the moments of the FD distribution.
K
F
Tk (1 E [T]])
p2 - pk)
EU(x') =
(ag + 1)
(ag + (
k=
TU(x*) Σ
o [Tx])
AU(x*) TU(x*) -FU((x")
where org = > OKk and [Tx]
O++TPE 00+T
Vk E [K].
Detailed derivations of the closed-form expressions for the objective function and uncertainty mea- sures are presented in Appendix A
while the algorithms for F-EDL are provided in Appendix B
Theoretical Analysis
F-EDL offers several theoretical advancements that enhance its UQ generalizability. First, we prove that the FD distribution is conjugate prior to the categorical likelihood (Lemma [41). Building on this, we prove that the class probability distribution for a given input x, derived from F-EDL, corresponds the posterior FD distribution for π,, obtained with improper prior (x) ~ FD(0, Pprior Px)rio(X), Tprior [x)) where the prior parameters Pprior (x) and Tprior (x) are learned from the data (Theorem 4.2 By introducing improper priors with input-dependent parameters, F-EDL eliminates the need for manually specified priors required in traditional EDL.
Second, we prove that the class probability distribution for a given input derived from F-EDL is equivalent to that of standard EDL. under specific parameter settings: T == and Pk = ax/ Ex= O for all & [K] (Theorem 4.3). This establishes EDI. special case of F-EDL, demonstrating that our framework retains EDLS strengths while enabling greater expressiveness when needed.
Third, we prove that the class probability distribution for testing example derived from F-EDL forms multimodal distribution over the simplex, represented as a mixture of Dirichlet distributions, with the number of distinct modes determined by the cardinality of the allocation probability vector p. denoted as ||Plo (Theorem 4.4). This allows F-EDL to capture complex, multimodal uncertainty- particularly useful for handling ambiguous inputs.
Fourth, we prove that the predictive distribution for a testing example derived from F-EDI can be decomposed into contributions from EDI. and softmax, with input-dependent mixture weights (Theorem 4.5). This demonstrates that F-EDI functions as mixture model of EDL and softmax, adaptively integrating their complementary strengths to achieve optimal performance in both pre- diction and UQ. Specifically, the EDL component tends to dominate for clean ID data, whereas for ambiguous or OOD inputs, the model interpolates between the two, maintaining robust performance across varying uncertainty levels.
Finally, we prove that F-EDL admits a generalized subjective logic interpretation, modeling mixture of K class- class-specific subjective opinions (i.e., hypotheses), each defined by shared base evidence a and dispersion parameter and weighted by the hypothesis selection probabilities p. P This perspective reveals that F-EDL performs UQ via principled evidence aggregation grounded in subjective logic [15]. rather than simply adding flexibility.
Lemma 4.1. (Conjugacy of FD Prior) Let yt Cat(I)), Vi E [N], where (D(a,,,n) T). The posterior distribution of ग given { is also an FD distribution, π KN FD(a', P, where a' = (a1....,a'g) and o/2
1N
EN (w:=k), Yk E [K].
"See Appendix C.2 for a more detailed explanation the Bayesian interpretation of F-EDI..
"See Appendix C.3 for more detailed explanation of the subjective logic interpretation of F-EDL.
5

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Theorem 4.2. (Bayesian Interpretation of F-EDL) The class probability distribution for given input x, derived from F-EDL, equivalent to the posterior FD distribution of π., with likelihood Σ [mck)) = (exp(go, (fo(x)))) Vk [K], and an improper prior π(x) ~ FD Pprior (x); (prior(x)) where Pprior(X) and Tprion (x) are uyput-dependens parameters learned from the data.
Theorem 4.3. (Generalization of EDL) The class probability distribution for a testing example x", derived from F-EDL, is equivalent to the class probability distribution of EDL when T and Pk = 0 Ek=1 O22, Vk E [K].
K
Theorem 4.4. (Multimodaliity of F-EDL) class probability distribution for a testing example x*, derived from F-EDL, is expressed as a mixture of Dirichlet distributions:
K
PF. (D.
Σ Pk Dir(Tr/o +
k=1
where es represents the k-th standard basis vector in RKK The number of distinct modes is determined by the cardinality of p, i.e, [|P||o-
Theorem 4.5. (Predictive Distribution Decomposition) Let Prn (ylm,yy)x") and psM(y)(*) denote the predictive distributions for a testing example x", derived from the EDL and softmax models, respectively. The predictive distribution for x", obttained by the F-EDL
decomposed
into the contributions of the EDL and softmax models as follows:
PF-EDL x*) "EDDL(x)) x PEDL xx") + wSM(x*) x (ss(y)x");
where WEDL((x)) = 00/((o + T) and WSM (x*) = (000 + T) are input-dependent mixture weights.
Proposition 4.6. (Subjective Logic Interpretation of F-EDL) F-EDL admits generalized subjec- tive logic interpretation, representing K class-specific subjective opinions (i.e., hypotheses) defined by shared base evidence ax dispersion parameter т, and hypothesis selection probabilities p.
For the proofs and additional insights about the theorems, refer to Appendix C
Related Works
EDL methods. EDL. (21] refers to family of UQ methods that predict Dirichlet distribution over class probabilities to estimate uncertainty in classification. KL-PN (22) encourages concentrated
Dirichlets for ID data and uniform Dirichlets for OOD data via KL divergence, while RKL-PN (23) uses reverse KL to mitigate issues with unintended multimodal distributions. However, both rely on OOD data, which often unavailable in practice. To mitigate the reliance on OOD data, PostNet
24 and NatPN 25 leverage feature space density estimated using Normalizing Flow (26] to predict the posterior Dirichlet distribution. However, their performance heavily depends on accurate density estimation, which remains challenging in high- -ing--iimnsional settings.
Building on EDL 99, several methods have attempted to address its limitations. I-EDL 10
incorporates Fisher information to account for varying levels of data uncertainty. R-EDI. 11 treats a prior parameter as tunable hyperparameter and simplifies the loss by removing the variance term. thereby relaxing EDL's restrictive assumption-4 motivation shared with our approach. DAEDL [12 uses feature space density during prediction with an alternative parameterization. However, all these methods remain constrained by the Dirichlet assumption, limiting their generalization to complex scenarios. Separately, distillation-bar estimation but typically require multiple forward passes, reducing their practicality. Beyond standard classification, EDL methods have also been extended diverse tasks, including regression [30] domain adaptation [31], semantic segmentation
approaches [27
28 29
improve epistemic uncertainty
32
33
calibration of large language models 34
and multi-vies learning
36
On the other hand, recent studies have questioned the ability of EDL to represent epistemic uncertainty. Bengs al. [39 40) showed that its second-order loss is not a strictly proper scoring rule, leading to non-vanishing uncertainty. Jürgens et al. [4 found that regularized EDL enforces a fixed uncertainty
'Adapted from Proposition 4.1 of 33

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Table I: UQ-related downstream task results are reported using the 11FAR-10 and CIFAR-100 datasets as the ID dataset. "Test.Ace." denotes the test accuracy, "Conf." denotes the AUPR scores for misclassification detection based on aleatoric uncertainty, and "SVHN/ CC-10"" and "SVHN TIN" indicate AUPR scores for OOD detection based on epistemic uncertainty. where the former corresponds to using CIFAR-10 as the ID dataset with SVHN and CIFAR- 100 as the OOD datasets, and the latter corresponds to using CIFAR- 100 as the ID dataset with SVHN and TinylmageNet as the OOD datasets. Baseline results for CIFAR-10 10 are sourced from existing literature
TO 11
12
Bold numbers indicate the best performance for cach metric.
ID: CIFAR-10
ID: 100
Method
Aect.Ace.
Conf.
SVHN C-100
Test.Ace.
Conf.
SVHN TIN
Dropout
82.84 10.1
97.15
10.0
51.39 :0.1 45.57 11.0
65,94
10.6
92.00
10.3
71.83 #2. 74.93
EDI.
83.55 :0.6
97.86
:02
79.12 #17 84.18 +0.7
45.91
15.6
91.28
+08
56.21
#31 70.13 =200
I-EDL
89,20 1.03
98.72
10.1
82.96 #22/ 84.84
10.6
66.38
10.5
92.84
1..1
67.51
#2,9 75.86 =0.3
R-EDL
90.09
10.3
98.98
1:.1
85.00
#12 87.73
10.3
63.53
10.5
92.69
10.
61.80
=3. / 69.78
=1.3
DAEDL
91.11
#.0.2
99.08
#:0.0
85.54
114 88.19
+0.11
66.01
#:26
86.00
#:0.3
72.07
#4.1 / 77.40 1.6
F-ED
91.19
+0.2
99.10
10.0
91.20 #1.3 88.37 20.3
69.40
+002
94.01 2:01
75.35
#2.3 / 80,58 =022
Table 2: AUPR scores for distribution shift detection from CIFAR-I0 to CIFAR-10- using alcatoric uncertainty
estimates. E {1, 2, 3, 4, 5) denotes severity levels corruption CIFAR-IO-C averaged across 19 corruption types. Baseline results are sourced from 12]
Method
C=1
C= 2
C 3
C C= 4
= 5
MSP
56.39 007
61.88 #1,1
65,86 +1.3
69,91
#1.
75.01
=1.8
EDL
54.76 =03
59.01
=0.4
62,46
65,87
#0.6
70.21
=088
T-EDL
56.33 =02
61.52
=0.5
65,44
+0.5
69.45
+0.5
74.56
=0.
R-EDL.
57.37
=0.5
62.20
65.74
69.33
119
73.58
-26
DAEDI
57.89
-0.3
63.23
10.4
67.53
72.21 10.
77.74
-0.4
F-EDL
59.01 =0.8
65.11
+0.7
69.48 +0.
73.88 +0.3
78.72
=0.4
budget, yielding relative rather than absolute uncertainty measures. Shen et al. 22 further unified these critiques, revealing that the EDL objective collapses to samplesiiz-inneeeendenl Dirichlet target, behaving more like energy-based OOD detectors than true uncertainty estimators. However, F-EDL empirically mitigates these issues through more expressive FD-based formulation and distinct loss design, yielding better-calibrated epistemic uncertainty in practice.
Deterministic uncertainty methods. Beyond EDL., deterministic uncertainty methods (DUMs) 32 offer computationally efficient UQ single forward pass. DUMs are designed with two primary objectives: (i) applying regularization techniques to learn feature representations that distinguish between ID and OOD data, and (ii) quantifying uncertainty based on these regularized representations. Regularization methods include spectral normalization [6 42 g and gradient penalties
7 For uncertainty estimation, DUMs employ diverse strategies, including Gaussian processes 6 radial basis function networks (7). Gaussian discriminant analysis [8], and kernel density estimation [43]. However, these methods often require significant modifications to the neural network architecture, hindering their practical applicability.
Experiments
We conducted comprehensive experiments to evaluate the generalizability and effectiveness of F-EDL in UQ Our evaluation consists of two parts. First, we performed a quantitative study (Section 6.2) across diverse ID scenarios-ineluding classical, long-tailed, and noisy settings- related downstream tasks. We further conducted an ablation study to assess the contribution of each FD parameter. Second, we performed a qualitative analysis (Section(6.) examining whether F-EDL produces semantically meaningful uncertainty representations for ambiguous samples and whether its epistemic uncertainty decreases with increasing training data, as theoretically expected. The code for our model is publicly available at htttt:///gttubbbc htwprtthubbooooaaaaennnnnggonooe Toso/aacceoooooo/-EEEn
on UQ-
6.1 Overview of Experiments
Tasks. For each scenario, we conducted three primary UQ-related downstream tasks: (i) classification, (ii) misclassification detection, and (iii) OOD detection. In the classical setting, we additionally
7

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Table 3: UQ-related downstream task results are reported using the CIFAR-IO-LT dataset as the ID dataset under both heavily (p = 0.01) and mildly (p- 0.1) imbalanced setrings.
Heavy Imbalance (p = 0.01)
Mild Imbalance (p = 0.1)
Method
Aect,Acc.
Conf.
SVHN / C-100
Test.Ace.
Conf.
SVHN / C-100
Dropout
39.22 #3.1
63.62 #27
33.33 11.7 54.17 #1.1
70.87 #30
89.82 =23
37.37 #14 61.18 #13
EDL
42.62 #27
82.63
+17
51.99
#388 66.86 109
79.09
404
95.36
#0.1
72.18 #2.1 80.09 4.07
Z-EDL
57.88 #1.3
84.10
11.
52.85 #68 69.19 113
84.86
+0.1
97.31
=0.2
79.83 #39 83,50 =04
R-EDL
63,36 =1.0
78.34
+1.0
48.71
#7.1 64.20
+14
85.35
+02
94.35
=02
60.58 #50/ 69.53 #1.6
DAEDL
63.36 +1.4
82.15
+1.0
51.03
#5.6 65,31
#1.2
84.95
+.04
95.22
#.44
69.40
14.5 74.56 +1.7
F-EDL
63.73 #1.4
85.99
+1.7
62.56 #28 70.18
#2.0
85.46
1:0.2
97.60
=0.1
85.36 #1.5 83.64 10.7
Table 4: UQ-related downstream task results are re- ported using the DMNIST dataset as the ID dataset. "FMNIST" denotes AUPR scores for OOD detection based on epistemic uncertainty, with FMNIST as the OOD dataset.
Table 5: Ablation study on F-EDL using the DMNIST dataset the ID dataset. "Fix-p (U), r" and "Fix- p (N), fix p uniform vector or normalized concentration parameters, respectively, and fix r = 1. "Fix-p (U)" and "Fix-p (N)" fix only "Fix-7" fixes only T. The full model learns both.
Method
Test.Acc.
Conf.
FMNIST
MSP
83,90 =0.1
96.01 =0.0
98.10 +0.3
Variant
Test.Ace.
Conf.
FMNIST
Dropout
84.13 +0.1
96.09
=0.0
94.68 10.6
Fix-p (U), r
83.34 402
95.62
10.1
97.22 #100
DDU
84.05 =0.1
82.73
=0.1
98.49 10.4
Fix-p (N),
83.27 #0.1
95.59
#0.4
97.91 #133
EDL
77.37 +5.8
95.19 =0.3
92.23 +1.7
I-EDL
83,46 +0.2
95.58 =0.0
94.11 +0.9
Fix-p (U)
83.39 #0.1
95.60
=0.1
96,94 #1.1
R-EDL
83.41 10.1
95.58 =0.1
90.91 #3.6
Fix-p (N)
83.32 +0.1
95.61
+0.3
97.48 #1.8
Fix-T
83.39 40.1
95.60
1:01
98.46 10.1
DAEDL
84.12 +0.1
95.93
=0.0
99.44 +022
F-EDL
84.28 +0.1
96.17
=0.1
99.76 +0.1
F-EDL
84.28 +001
96.17
1:011
99.76 #0.1
included distribution shift detection. Classification performance was assessed using test accuracy. Misclassification detection was evaluated using the under the precision-recall curve (AUPR) score, with confidence defined as negative aleatoric uncertainty for ID samples (label: correct = incorrect 0). OOD detection was also evaluated using AUPR. based on negative epistemic uncertainty (labels: ID = 1, OOD = 0), assessing whether the model assigns higher uncertainty to OOD examples. Distribution shift detection was formulated as OOD detection by applying varying levels of corruption to a base dataset to simulate shifts. All AUPR scores were normalized to 0-100 scale.
Datasets. In the classical setting, CIFAR-10 10 and CIFAR- 100 were used as the primary ID datasets. For the long-tailed setting, we used CIFAR-IO-LT [46), artificially imbalanced version of 10, as the ID dataset. For the noisy setting, DMNIST, a variant of MNIST containing ambiguous data points, was used the ID dataset. For OOD detection, SVHN 47 and CIFAR- 100 served as OOD datasets when CIFAR-10 or CIFAR- TinyImageNet (TIN) were used as the OOD datasets for CIFAR-I00. FMNIST was used as the OOD dataset for DMNIST. For distribution shift detection, we utilized CIFAR-IO-C [8]. a dataset created by applying continuous distribution shifts to CIFAR-10, as the OOD dataset.
was used as ID. while SVHN and
Baselines, We compared F-EDL against classical and EDL 99. I-EDL [00], R-EDL [11, and DAEDL [1]. To highlight task difficulty, also included Dropout 4 as a traditional UQ baseline. For distribution shift detection, we compared with MSP
art EDL methods, including
48 widely used benchmark. For the noisy setting, evaluated both MSP and DDU g with DDU specifically introduced for DMNIST. All baselines were evaluated following the uncertainty estimation protocols described in their original papers.
Implementation. We adopted VGG-16 (49 for CIFAR- 10 and CIFAR-IO-LT, and ResNet-18 (50] for CIFAR-100. For DMNIST, used lightweight CNN with three convolutional and three dense layers. To predict the FD parameters p and т, we added two shallow MLPs with 1-2 layers depending on the dataset, introducing minimal overhead (e.g.. 1.8% for VGG-16), At inference time, F-EDL remains highly efficient without any post-hoc processing, running only 1.3% slower than EDL yet over 50% faster than DAEDL on CIFAR-10.
For clarity and brevity, we defer additional experimental explanations (Appendix F.. extended results (Appendix F. and supplementary qualitative analyses (Appendix G to the appendix.
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EDL:
EDL: >-EDL:
>-EDL:
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Demety
100
Figure Posterior class probability distributions a DMNIST input (true label: 9), which is visually ambiguous and resembles class 7. The leftmost panel shows the input image. The second and third panels display the marginal distributions of the two most probable classes predicted by EDL (denoted as Te for class k), while the fourth and fifth panels show those predicted by F-EDI..
P-DL
055 70-
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70
4.30.04 04
3.04
3.02
10006 2000 33000 49000 52300 Thaining
10665 20CC SICC aoice 3000
19900 20003 30603 LECOS stece Tasning Siee
Figure 4: Test accuracy (blue, solid line) and epistemic uncertainty (red, dashed line) of EDL (left), DAEDI. (middle), and F-EDL (right) across increasing training set sizes: (500, 1000, 2000, 5000, 10000, 25000, 50000). Accuracy is plotted on the left y-axis, and uncertainty is plotted on the right y-axis each subplot.
6.2 Quantitative Study: UQ-Related Downstream Tasks across Diverse Settings
UQ in Classical Setting. We first evaluate F-EDL the classical setting, standard benchmark for UQ models, using CIFAR-10 and CIFAR- 100 across four UQ-related downstream tasks described in Section[6.]] As shown in Table and Table F-EDL consistently outperforms competitive baselines across all tasks and datasets. These improvements in UQ performance are broadly attributed two key innovations: (i) enhanced expressiveness enabled by generalizing beyond the Dirichlet family (Theorem 4.3) and (ii) the use of input-dependent improper priors that address the prior specification issue (Theorem 4.2). In addition, its strong performance in distribution shift detection reflects the model's s ability to effectively combine the complementary strengths of softmax-bast based (MSP) and evidential (EDL) approaches to achieve optimal UQ (Theorem 4.5)
UQ in Tailed Setting. CIFAR- 10-LT with two levels of imbalance: (i) mild (p 0.1) and (ii) heavy (p = 0.01), across the three tasks described in Sec Effective UQ in longttailed settings requires producing calibrated uncertainty estimates under class imbalance to detect misclassified examples and distinguish rare ID inputs from OOD data. These capabilities were assessed through misclassification detection and OOD detection tasks, respectively. As shown in Table F-EDL. demonstrates strong and consistent performance across all tasks and imbalance levels. This performance highlights F-EDL's ability to provide reliable uncertainty estimates under complex distributional structures a benefit attributed to its flexibility beyond the Dirichlet assumption (Theorem 4.3
To evaluate robustness under class imbalance, we further test F-EDL on
UQ in Noisy Setting. To evaluate robustness noisy ID scenarios, we apply F-EDL to the DMNIST dataset across the three tasks described in Section 6.1] Effective UQ in such settings requires leveraging aleatoric uncertainty to detect ambiguous inputs that are prone misclassification and epistemic uncertainty to distinguish ID from OOD data. These capabilities were assessed via misclassification detection and OOD detection, respectively. As shown in Table 4 F-EDL consistently achieves strong performance across all tasks. This performance highlights F-EDL's improved ability to handle ambiguous inputs more effcctively by representing multimodal opinions over plausible classes in a structured manner (Theorem 4.4 and Proposition 4.6).
Ablation Study: Contributions of p and T. To assess the contribution of additional parameters introduced in the FD distribution, we conduct an ablation study in the noisy ID setting using the DMNIST dataset. Specifically, we compare the full F-EDL model to several simplified variants: two configurations that fix p one to a uniform distribution (p 1/K) and another to the normalized concentration parameters (p- = az a/||a</| ||lxx/)))aandd another configuration that fixes 7 = Combinations where both p and are fixed are also evaluated. As shown in Table 5 all fixed variants degrade in UQ performance, with "Fix-7" showing favorable trade-offs, but no single variant consistently outperforms the others. The full model, which learns both p and achieves the best results,

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suggesting that the performance gains arise from the synergistic flexibility of the FD distribution, rather than merely increased capacity.
6.3 Qualitative Study: Multimodality and Faithful Epistemic Behavior
Multimodal Uncertainty Representations for Ambiguous Inputs. To assess whether the flexibility of J-EDL leads semantically meaningful uncertainty representations, distributions for ambiguous inputs from the DMNIST dataset, on which the model was trained. For each test input, we extract the marginal distributions of the two most probable classes under both EDL and F-EDL. As shown in Figure 3 EDL collapses the prediction into single, overconfident mode, skewed toward one class, failing to capture the underlying uncertainty, with the second most probable class often misaligned with any visually plausible alternative. In contrast, F-EDI produces multimodal distributions with distinet peaks near plausible classes such as "7" and "9" reflectingg the model's 's hesitation in the presence of ambiguity. These qualitative improvements highlight F- EDL's ability to express structured, multimodal representations, in line with its theoretical grounding (Theorem 4.4 Proposition 4.6).
visualize class probability
Faithfulness of Epistemic Uncertainty Estimates. To evaluate whether F-EDL provides faithful estimates of epistemic uncertainty, we examine how uncertainty evolves as the training dataset grows. key characteristic of epistemic uncertainty that should decrease with more data, reflecting increased model confidence as knowledge improves [29]. Following the evaluation protocol proposed in [29], we train models on progressively larger subsets of CIFAR- and compute the average epistemic uncertainty over fixed test set. As shown in Figure 4 F-EDL demonstrates a clear, monotonic decrease uncertainty with more data, unlike EDI. and DAEDL, which show inconsistent trends. This suggests that F-EDL successfully captures the theoretically desirable behavior wherein epistemic uncertainty consistently decreases as more data become mailable--mpiricallll addressing key limitation of standard EDL, whose epistemic uncertainty often fails to exhibit reliable monotonic decline [29].
Conclusion
Summary. We propose F-EDL, a novel UQ framework that maintains the efficiency of EDL while enhancing its generalizability by predicting a more expressive FD distribution over class probabilities. Theoretically, J-EDL. possesses several desirable properties that support reliable and robust UQ under complex conditions. Empirically, achieves state-of-the-art performance across a variety of UQ-related downstream tasks in diverse and challenging ID scenarios, supported by qualitative evidence of interpretable multimodal predictions and faithful epistemic uncertainty.
Limitations & Future Directions. Despite its improved flexibility, F-EDL faces several open challenges. First, it is currently limited to classification; extending to regression, for instance, by building on evidential regression models [30], is natural next step. Second, although F-EDL provides a variance-based decomposition of uncertainty, does not fully disentangle aleatoric and epistemic components highlighting the need for further work structured disentanglement, longstanding challenge in UQ [51] 52). Third, while F-EDL empirically alleviates several theoretical limitations of EDL., it still relies external regularization control epistemic uncertainty (39 suggesting the need for an intrinsically stable training objective.
Beyond these aspects, F-EDL opens multiple promising research avenues. First, it can serve as a foundation for developing UQ methods tailored to specific ID scenarios such as long-tailed classification [53] 54| 55 [6], where combining F-EDL with logit adjustment [57, 58, 59] could improve calibration under imbalance. Second, its modular architecture supports plug-am andd-ay integration into existing EDL-based frameworks, allowing the Dirichlet components be replaced with the more expressive FD formulation in downstream applications-stin as trusted multi-view learning (35 through the design of compatible evidence-fusse fusion mechanisms.
10

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Acknowledgments and Disclosure of Funding
This work was supported by the National Research Foundation Korea (NRF) grant funded by the Korea government (MSIT) (2023R1/422200555,, RS-2023-00218813)
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Appendix for the Paper
"Uncertainty Estimation by Flexible Evidential Deep Learning"
A
Additional Explanations about Objective Function and Uncertainty Measures
A.I
Objective Function
A.2
Uncertainty Measures
Algorithm
C
Additional Explanations about Theorems
C.1 Proofs for Theorems
C.2 Bayesian Interpretation of F-EDL (Proposition 4.6
C.3
Subjective Logic Interpretation of F-EDL (Proposition 4.6
Interpretation of FD Parameters and Justification of Using the FD Distribution in UQ
D.1
Semantics of FD Distribution Parameters
D.2
Justification for Using FD Distribution UQ
D.3
Evidence Extraction in F-EDL.
Additional Explanations about Experiments
E.1
Datasets
E.2
Implementation Details
Additional Results for Quantitative Study on UQ-Related Downstream Tasks
Classical Setting
Long- Tailed Setting
Noisy Setting
Ablation Study
Additional Explanations and Results for Qualitative Study
G.1 Toy Experiment in Figure
G.2 Multimodal Uncertainty Representations for Ambiguous Inputs.
22

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A Additional Explanations about Objective Function and Uncertainty Measures
In this section, we present the closed-form representations for the formulas discussed in the main text. First, we derive the closed-form representation of the objective function for F-EDL, as provided in Section 3.2 Next, we introduce the formulas for total, aleatoric, and epistemic uncertainty using variance-basee based uncertainty measures and derive the corresponding closed-form representations for F-EDL,, as outlined in Section 3.3
A.1 Objective Function
Let y (y),- (y1.-....K UK) denote the one-hot encoded labels, and let (..,... ,K) represent the class probabilities. Furthermore, let a (a1,..., ..K. P = (Pi... Pk), and T denote the parameters of the FD distribution. The objective function is defined as:
Em-FD((a, Ily π
lly pll
essn
where the first term, MSE represents the expected MSE over the FD distribution, and the second term, _ is Brier-score-batt based regularization term for p.
Expected MSE over FD. In the first term, the expected MSE can be expressed as:
MSF
E
!lly π 122)
Expanding the squared difference gives:
MSE 1 2(E [TEm + E. [|]]
Rewriting this as summation over A:
MSE
( - [TK]UK + Ex (m)))
Using the relationship E [+]] = (E [T])) 2 + Var (TT)). this simplifies to:
MSE
(un Ex [T&])²
Var (mx).
The mean of the FD distribution, Ex [Tx]. is given by:
[m[Tx] = a% + TPk
ou + vk E [K],
where ogp = Ek Osk-
The variance of the FD distribution, Var [TT) is given by:
Var (TT)) (at + <Px) (Pp)((a0 - as + 7(1 (())
T Tppx(1 - Ph))
+
Wk E [K].
(ao + (a0 + +
(ao + (a0 + + +
By substituting the mean and variance of the FD distribution, MSE becomes:
LMSE
Oi + P&T
(ax TPp) aq + 7(1 pk)) (ao + ()0 +T+1)
7 T2 Pk))
yk
0g +
(an + () + r +
Regularization Term.
The second term, representing the regularization term for p. is given by:
K Ukk A=1
m ly
pl
px)
23

---

Combined Objective Function.
Combining the two terms, the complete objective function be-
comes:
C MSE
+ C
yk ak + P&ST
(au + TPp) ax + T -Pa) K
T T2px -Pk) Pk) (an 7)( T
00
+
(o0 + ( + 7 ++
yk pt))
A.2
Uncertainty Measures
The closed-form expressions for the total, aleatoric, and epistemic uncertainties for a testing example x be derived using the variance-based uncertainty measures and the moments of the FD distribution. Below, we present the derivations for these uncertainty measures.
Total Uncertainty. The predictive uncertainty for each class k E [K], for testing example x*, denoted as (x",,"), is defined as the variance of the label for that class:
(x*(x") Var(yx
The total uncertainty for denoted TU(x*), ). is obtained by summing the class-wise uncertainties:
TU(x*) Var(yx(x").
Using the law of total variance, this can be decomposed as:
K
K
TU(x*) Σ En [Var( /,, xx)x Σ
Var, (E yR T x*).
Since ys Ber(TT), Wk: [K]. have:
TU(x') Σ En [TT [[]]]]]] + Σ
Var (TTx).
km
Expanding the terms:
K TU(x*) 2
[R [TT](T]]
Ex (m]m)
Ee [Tk]
(Ex (T)))
k=1
Simplifying:
K Σ
TU(x*)
Ex [TT]
(E [TT)))
km
Since Σx Ex [TT]] 1, the total uncertainty simplifies to:
TU(x*) = Σ EE (x)))
where Ex (TT)
aLTTPE optr
Vk [K].
Epistemic Uncertainty. The epistemic uncertainty for testing example x*, denoted EU(x*). can be expressed as:
K EU(x*) = Var (Tx).
Substituting the expression for the variance of the FD distribution, we have:
Ex [T E (mx]) + (oo + () + 1)¹
7'px pk)
FU(x*)
On + +
where ag >K 1= ag and Ex [TT]
actIPs VK E [K]. 1o+t
24

---

Aleatoric Uncertainty. The aleatoric uncertainty for a testing example x", denoted as AU(x", is obtained by subtracting the epistemic uncertainty from the total uncertainty:
AU(x*) TU(x") EU(x*).
B Algorithm
We present the training algorithm and the prediction and UQ procedure for F-EDL in Algorithm and Algorithm 2
|
respectively. These algorithms are straightforward to implement, requiring no sensitive hyperparameter tuning or reliance on additional techniques.
Algorithm F-EDD Training
Training data Du. {(x, ) initial model parameters {0, Φi, 22, Φ3 maximum epoch T max. learning rate 17 for i = to maxx do Update the parameters using the Adam optimizer and F-EDL objective function:
Input:
(1) (0 (W 00 J Φ1
(1) (W
Φi Φe J1 Φ2, Φ3) Adam V FFELD n))
Apply spectral normalization the parameters & and Φ1:
(1) ( Ha (w Hz SpectNorm( (1) | (W 1/2
end for Output: Trained model parameters {0, क1 62 63)
Algorithm 2 F-EL Prediction and Uncertainty Quantification
Input: Testing example x* and trained model parameters {Ô, 21. 62. E2}
Step 1: Compute FD Parameters:
a(x*) exp (0h (fo(x*)))
p(x") softmax (b.. (6((*))) r(x*) = softplus (9 (fo(x'))
Step 2: Prediction:
((x") argmax Em[ where E, [mT]] (x(xx) + T (px(x*) kE[K]
Ak E [K].
(x(xx) + T(x*)
Step 3: Uncertainty Quantification:
K
Total Uncertainty: TU(x') =
EE [Tx]))
k=
E [
(mb)
Epistemic Uncertainty:
EU(x*) =
aul (x*) T(x*)
1=
72 72(x*) pp (x' (1 pxx(x*)))
(oo + (xx))) (x)(x*)))))))))))))))))) + 7 (xx) + +1)
Aleatoric Uncertainty:
AU(x') TU(x*) -U((x*)
Output: Predicted label j(x*) and uncertainty estimates (TU((*), EU(x*). AUXxx))
25

---

C Additional Explanations about Theorems
In this section, expand on the theoretical foundations of F-EDL (Section 4). First, we present detailed proofs for Lemma 4.1 through Theorem 4.5 Second, we provide further insights into the Bayesian perspective underlying F-EDL and elaborate Theorem 4.2 Finally, we offer an extended
explanation of the subjective logic (SL) interpretation associated with Proposition 4.6
C.1 Proofs for Theorems
We prove Lemma
Theorem
4.2 Theorem
4.3
Theorem and Theorem 4.5 sequentially.
Proof of Lemma
The posterior distribution of ग given { can be expressed as: x PTD or a, x pl (1) (
Expanding each term, we have:
I(ax) T(ax +
N
K
p(T (u) x
[v.=k)
Th
Pk
T
k=1
Simplifying the terms yields:
K
K
I(at)
p
II
T on
(vc=A) -1
Pk T(ax +
Th.
k=1
Recognizing the resulting form, we see that the posterior distribution of π retains the form of an FD distribution, i.e.,
p( pFd ( P).),
where a' (di,..., a(k) and each a' is given by:
at = a + Σ (. Ak E [K].
Thus, the FD distribution is conjugate to the categorical likelihood.
Proof of Theorem 4.2
Assume that the prior distribution for a given input x is specified as: (x) ~ FD (Aprior (x), Pprior (x), (prroo(x)).
where the parameters defined as follows:
O'prior (x) = 0, Pprior = softmax(go
[xrior()) = softllss(go, (fo(x)))
Here, aiprior (x) = O represents an improper prior as the prior parameter a is zero. The prior parameters for p and r are learned for each input using a neural network.
The likelihood is estimated using neural network as:
N
[x-= (exp(s>) (f(()))) vk E [K].
Using Lemma 4.1 the posterior distribution of π for given input x is obtained as:
π (x) D FD (ep((s. (fe(x))), softmax(gs.. (.fe (x))) softll {
This posterior distribution corresponds to the predicted FD distribution of π for given input x, as described in Section 3.1 Consequently, we conclude that the class probability distribution for a given input x, derived from -EDL, is equivalent the posterior FD distribution of π, obtained with an improper prior with input-dependent parameters learned from the data.
'We omit the proof for Proposition 4.6 as it provides conceptual interpretation rather than a formal mathematical statement. detailed discussion is included in Appendix C
26

---

Proof of Theorem 4.3 F-EDL, is expressed as:
The class probability distribution for a testing example x", derived from
I(ao + 7) π -= [(ak))
I(ax) T(an + r) tko
Σ
"
Ph
k=1
where ao = = By substituting T = =1 and ak/ao. Ak [K], the expression simplifies as follows:
PFD(TT; a) = T(ag + +1) I I(ox)
K
I(ax) r'ax + 1) T
0
k=1
Since [(ax + 1) ag (av:) this reduces to:
PTD(TT;)) T(ag +
II
Tk
k T(ou)
ax
Because Σ = the expression becomes:
D(on +
K
(TD(T:0) 0))
m
as
-
I== [(ax).
000
km
Rewriting using [(ao +) (oT(ao):
T(oo)
K 1I
PED(T;; (T; a)
I ((ox).
x
This is precisely the probability density function of the Dirichlet distribution, i.e.,
PTD(T;; a:) (Dir(T:: aT)...
Therefore, conclude that the class probability distribution for x*, derived from F-EDL, is equivalent to the class probability distribution of EDL when 7 1 and ps: = ag/ao. Ak E [K].
Proof of Theorem 4.4
For this proof, we rely on the definition and notation of the FG basis and FD distribution, as detailed in Section 2.2
To establish that the class probability distribution for testing example x*, derived from F-EDL, represented as a mixture of Dirichlet distributions and exhibits multimodality, we derive the probability density function of the FD distribution, starting from its FG basis representation.
Since Mu(1, p), the random variable Z₂ follows the distribution:
1 with probability pk:
with probability 1 -
The conditional distribution Ys given Zμ is:
Wx + U Zx = 1,
Z
We if Zx = 0,
where W₁ ~ Gamma(ax), Vk E [K]. and U Gamma(7) are independent Gamma random variables sharing the same scale parameter. Consequently, Wi + U is also Gamma-distributed: Wa + U ~ Gamma(ax +ox++ TT.
Let (T....., TK) denote the class probability vector, where
YE
Vk
[K].
ER= Yv
By the definition of the FD distribution, it follows that:
~ (D(a.p,7).
27

---

where a (a1...., p ppt.aa.p...............
Using the conditional distribution of Yu, the relationship between and Yx. Vk: [K], and the property that normalizing a set of independent Gamma random variables yields a Dirichlet distribution, the conditional distribution of ZZ is given by:
Dir(a1,..... ag ...... axk if Zs = 1 for some l. Dir(ai,.... ogk if Z for all k.
Z
Since Z ~ MM p), the case Z2 for all k is impossible. Consequently, the distribution of π is derived as:
p(T) =
PiZ = for kth class) x Dir(ar,...,, + T, , .ak
km1
pa Dir(a + Tew),
k=1
where C2 represents the k-th standard basis vector in RK
Thus, the class probability distribution for testing example x*, derived from F-EDL, is expressed as a mixture of Dirichlet distributions. The number of distinct modes in this distribution is determined by the cardinality of p. i.e., ||P|o.-
The predictive class probability for a testing example x belonging each class k Vk [K], derived from J-EDL, can be represented as follows:
Proof of Theorem 4.5
PF- (y k(x*) f ply k|T) a p. x. )dT.
Using the fact that p(y = k| π = kR. vk E [ this integral simplifies to:
Ps- EDL (y= = k|x'' Em=FF((a.p.)) [m*)-
Substituting the expression for the expectation of Ta& and expanding the terms, obtain:
ago ak
EF- = k|x ag +
+
+ x Pk-
ag ag
On the other hand, for the EDL framework, the predictive distribution for x is given by:
PEDL (y = k|x*) = Dir = Clj
For the cestmaxx-aseed model, the predictive distribution for x* is given by: PSM (u k|x*) Pk-
Define the input-dependent mixture weights as: WEDL(x")
Og
WSM
0gg +
an +
Using the predictive distributions of EDL and softmax models and substituting these weights, the predictive distribution of F-EDL can be expressed as:
Pr- PJ.EDL (y) x") ) (EDDI(x)) PEDL(y) + uxs((x)) x ())x") Thus, the predictive distribution for x*, obtained by the F-EDL framework, is decomposed into the contributions of the EDL and softmax models.
PEDL
C.2 Bayesian Interpretation of F-EDL (Theorem 4.2
In three stages, we provide an additional explanation about the Bayesian interpretation of F-EDL, outlined Theorem 4.2 First, establish Bayesian framework termed the input-dependent FD FD-Categorical model, extending the input-dependent Dirichlet Sricclet--Catggriice model commonly used interpret EDL methods 22 [2]. Second, we introduce prior specification problem, which has been a significant challenge in traditional EDL methods. Third, we introduce an additional theorem (Theorem C.I that bridges the gap between the prior specification F-EDL and traditional EDL methods. By leveraging this theorem together with Theorem 4.2 from the main text, we clarify how F-EDI. resolves the prior specification problem using an improper prior and input-dependent prior parameters.
28

---

Input-Dependens FD-Categorical Model. Let x represent given imput, π (x) the corresponding class probabilities, and {ūr the of pseudo- oobuuddtobbervttios.. Then, the deput-deeendent FD- Categorical model can be described as Bayesian model as follows:
Prior
T (x) ~ FD (rrirr (x), Pprior (x), Tprior [prior(x)),
Likelihood
M Cat(Tr())),
Posterior
T { (D(()po(())) Prost (x), Tpost (x)),
where Orprio (x), Pprior (x), and [ are the prior parameters, and (x)s(())) Ppost(x); and Tpost are the posterior parameters.
Leveraging the results in Lemma the relationship between the prior and posterior parameters can be expressed as follows:
N
(k) aposs (x)
(k) "pror (x)
1 1o Vk E [K],
Ppost (x) Pprior (x),
Tostt
(x) [prior(X)
This formulation reveals the intrinsic connection between F-EDL and the input-dependent FD-Categorical model. The fundamental concept of F-EDL, predicting FD distribution over class probabilities for a given input x, is inherently equivalent to predicting the posterior distribution the mechanism of F-EDL is equivalent predicting psendo-observations {1% رز= point x using a neural network. Specifically, F-EDL predicts the counts of the pseudo-observations for each class, i.e., pseudo-counts, utilizing neural network So and gø, follows:
class probabilities, given the pre-specified prior distributions. In particular, " for each data
Σ (ū=k)) x exp(go. (fo(x)))k.
Using this, the posterior parameter, Oppost, is expressed as:
Oppott (x) = appiio (x) + exp(go, (fo(x))).
Prior Specification Problem of EDL Methods. Standard EDL methods face the prior specification
problem, as highlighted in recent works, including R-EDL I and DAEDL [1]. This issue arises because EDL methods employ a uniform prior, prior (x) Dir(1), when predicting the posterior distribution under the input- dependent Birichlet-Categorice model framework (24 12). Notably. the inpudedependent Dirichlet-Caten Categorical model analogous the inpur-dependent FD- DD-Categlrical model discussed earlier, differing only in its use of the Dirichlet distribution in place of the FD distribution. address the prior specification problem under this framework, R-EDL 11 treats the prior parameter an adjustable hyperparameter, adopting prior of the form Triior (x) Dir(A1), where a tunable hyperparameter. DAEDL [12] eliminates the prior parameter, effectively utilizing an improper prior priior (x) Dir(O). For a more detailed discussion of the prior specification problem EDL methods and their proposed solutions, we refer the reader to the respective papers
Table 6: Comparison of prior specification scheme across EDL 9 I-EDI 10, R-EDI. 11), DAEDI 12,
and F-EDL.. Here, À IR denotes tunable hyperparameter, E R is vector of ones, 0 R. is vector of zeros, and Parior RR and Tprine (x) denote the prior parameters dynamically learned for given imput x.
Method
I-EDL
R-EDL
DAEDL
F.EDL (Ours)
Prior
(xx) ~ Dir(1)
((x) ~ Dir(A1) (()) Oi
Txx) ~ FD(O, Pprion (x), Tpriox (x)
While these methods partially address the prior specification problem, both approaches have notable limitations. First, R-EDL requires manual tuning of the additional hyperparameter X, which can be both challenging and time-consuming, particularly when applied to new tasks datasets. Since the
29

---

choice of À has non-negligible impact on UQ performance, precise tuning is critical for achieving optimal performance. Second, while DAEDL effectively mitigates the prior specification issue through its parameterization scheme, its success depends heavily on accurate density estimation to represent uncertainty. This dependence can become problematic in complex IID scenarios, where obtaining high-quality density estimates is particularly challenging. Thus, there remains need for an EDL model that addresses the prior specification problem without introducing additional computational overhead or relying precise density estimation. Such model would enhance the robustness and generalizability EDL methods across complex and unforeseen scenarios.
provides comprehensive comparison of the prior specification scheme used by representative EDL methods and F-EDL.
Resolving the Prior Specification Problem with F-EDL. F-EDL presents novel approach the prior specification problem by utilizing an improper prior with learned parameters. Before exploring this further, we introduce an additional theorem that bridges the gap between F-EDL and EDL.
Theorem C.1. (Variant of Theorem 4.2 The predictive distribution for given input derived from F-EDL, is equivalent to the predictive distribution derived from EDL, utilizing the prior (x)) Dir Dir(Apriorxx)) where Xprior (x) are the ieput-dependent prior parameters learned from the data.
Proof of Theorem C.1 expressed as:
The predictive distribution for given input x, derived from F-EDL, is
P ED- = k|(x) = 0K + TPk Wk E [K].
Ek=1 ak +
Let us define a' = a'g); where
de = TPk: Vk [K].
Using this definition, the predictive distribution for F-EDL can be rewritten as:
PF. FDI k|x) k= o
This expression is equivalent the predictive distribution derived from the standard EDL model, where the concentration parameters a'' are parameterized as follows:
a' Aprior (x) + e(x).
with the components defined as:
Aprior (x) softplus(ge,(fo(x))) softmax(g@., (fo(x)), e(x) exp(ge, (x))).
Within the input-dependent Birichlet-Categonie model framework, EDL with this parameterization can be interpreted predicting the posterior Dirichlet distribution for æ, where the likelihood (evidence vector) c(x) defined as described earlier, and the prior distribution is specified as:
π (x) Dir (x)).
where Aprior (x) is the input-dependent prior parameters learned from the data.
In essence, Theorem 4. demonstrates that F-EDL can be interpreted as predicting the posterior FD distribution for given input, under the input- dependenn FD-Categorical model framework. Furthermore, Theorem establishes that the predictive distribution of F-EDL is equivalent that of EDL, with an input-dependen prior, i.e., π (x) ~ Dir (Aprior(x)).
From these theorems, we can infer that the prior specification scheme of F-EDL demonstrates distinct advantages two different aspects compared to the conventional approaches. First, F-EDI. utilizes an improper prior by setting (xpioo(X) 0. This addresses the limitation of uniform priors, which
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Table 7: Subjective logic interpretation of EDL 11. R-EDL [.I and F-EDL. For EDL and R-EDL, a single SL. triplet (b, u, a) represents single opinion, where b & R E IR, a E A K=1 represent belief mass, uncertainty, and base rate, respectively. In contrast, F-EDL yields K distinct SL triplets, each encoding a class-specific opinion.
Property
EDL
R-EDL
F-EDL (Ours)
Distribution
Dirichlet
Dirichlet
Flexible Dirichlet
SL Triplet (b, a)
a-1 |al1
K Iolls
1)
a- Tall
AK
1 X
aallerr Iaalitr: 9j) Vy é [K]
Jall
Hypothesis assignment
over K class-specific opinions
Projected Probability
Same
a+Tp
aali
|aalier
Multimodality
No
No
Yes (Dirichlet mixture)
Epistemic source
magnitude
Same
p (inter-opinion) a, (intra-opinion)
often degrade classification performance due to the difficulty in balancing the prior parameters with the pseudo-likelihond [12] By adopting this approach, F-EDL avoids the challenges associated with using fixed uniform priors. Second, F-EDI employs learnable prior parameters for p and Specifically, the optimal prior parameters for each data point x are estimated using neural networks. Compared to R-EDL, which requires manual tuning of prior parameters, F-EDL offers significant advantage by adaptively determining these parameters, enabling both optimal predictive performance and robust UQ Additionally, compared to DAEDL, F-EDL reduces the dependence density estimation to achieve optimal UQ, enabling effective generalization to the challenging scenarios where obtaining high-quality density estimates is infeasible.
C.3 Subjective Logic Interpretation of F-EDL (Proposition 4.6
In this section, we elaborate on Proposition 4.6 clarify the subjective logic (SL) perspective of F-EDL. We begin by revisiting the motivation of F-EDL through the SL lens. Next, we formalize its multimodal predictive structure as a mixture of class-specific Dirichlet-distributte opinions. We then introduce generulized subjective logic interpretation, highlighting how it extends traditional EDL formulations. Finally, we explain how the proposed variance-based uncertainty measures naturally align with this interpretation, enabling faithful UQ.
Motivation: Limitations of Single-Opinion SL in EDL Most EDL methods 10 12 can
be interpreted through the lens of SL [15), where each prediction corresponds to single subjective opinion-a Dirichlet distribution encoding belief and uncertainty over class probabilities. However, this one-opinion-pert -er--nputttttttttteee framework restricts the model's ability to express the ambiguity between multiple plausible hypotheses. Such limitations are especially evident for inputs with overlapping features (e.g., ambiguous digits such "1" vs "7" in (OIST)-whise commonly arise in high-risk, real-world MI. applications.
F-EDL Mixture of Opinions. To address the limitation of representing only single opinion, F-EDL models uncertainty as mixture of class-specific subjective opinions. Each component corresponds to class-specific hypothesis, represented by Dirichlet distribution biased toward that class, and weighted by hypothesis selection probability. For a test input x*, the resulting predictive distribution is:
K
p(T(x*)=
Pk Dir(a + re&),
where a is the shared base concentration vector, controls the strength class-specific bias in each opinion, p .. .P)) denotes the mixture weights, and eş is the k-th standard basis vector. This defines an FD distribution-a structured Dirichlet mixture capable of expressing multimodal beliefs over class probabilities.
Generalized Subjective Logic Interpretation F-DDI. This formulation aligns F-EDL with generalized SL framework that accommodates multiple competing opinions, rather than collapsing
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them into single fused belief. Each Dirichlet component Dir(a + rey) encodes class-specific opinion, while the hypothesis selection probabilities p represent epistemic uncertainty over which opinion to endorse. Unlike classical EDL models, which aggregate all evidence into a single unimodal Dirichlet, F-EDI. maintains the structure of uncertainty across hypotheses. This allows it to capture ambiguity between plausible iiternatives-enaliii more expressive and faithful representation of model belief.
Table 7 summarizes this interpretation by comparing F-EDL with representative EDL variants from the SL perspective. Unlike conventional EDL models that yield SL triplets per input, F-EDL produces K distinct SL triplets- each representing class-specific subjective opinion. These triples share the belief and uncertainty masses (b, u), but differ in the base rate a, reflecting the class-dependent bias. The overall opinion is expressed as a mixture of these K opinions, weighted by categorical distribution p that is learned per input. This structured formulation enables principled UQ through evidence aggregation grounded in subjective logic.
In particular, it is noteworthy that, unlike EDL variants that yield single SL triplet per input, F-EDL produces K distinct SL triplet, sharing the belief and uncertainty massess (b, u) but differing in the base rate a due class-dependent bias. The overall subjective opinion is distributed through these K subjective opinions, following the categorical distribution parameterized by which is also learned per input. These allow principled uncertainty quantification through the structured evidence aggregation grounded by subjective logic.
Variance-basee Uncertainty Measures in F-EDL Classical SL metrics (e.g., inverse total evi- dence) are not directly applicable to this mixture. Instead, we quantify epistemic uncertainty using the total variance of the FD distribution:
EU(x)) Σ (xr(=x(x*)).
This formulation captures both inter-hypothesis ambiguity, reflected in the hypothesis selection probabilities (i.e., allocation probabilities) p, and intra-hypothesis variability, driven by the evidence parameters a and dispersion т. By explicitly modeling these two sources of uncertainty, F-EDL offers more faithful and interpretable estimates of epistemic uncertainty, especially in ambiguous or complex prediction scenarios.
D Interpretation of FD Parameters and Justification for Its Use in UQ
This section provides additional explanation for adopting the FD distribution for UQ, demonstrating that its use represents principled generalization rather than an arbitrary increase in flexibility. First, we clarify the semantics the FD parameters (a, P. Second, we justify the suitability of the FD distribution for UQ. particularly in cases involving multiple plausible class hypotheses. Third, we describe how evidence structurally extracted in F-EDL.
D.1 Semanties of FD Distribution Parameters
In the FD distribution, the parameters a, P. and jointly characterize different aspects of uncertainty. The vector represents the total amount of evidence supporting each class, analogous to the concentration parameters in a standard Dirichlet distribution. However, unlike the Dirichlet case, the FD introduces two additional parameters and that disentangle how this evidence is distributed and how sharply it is expressed.
The parameter p specifies how belief is allocated among class-specific hypotheses. A sharp p. where one component dominates, reflects a decisive belief in particular class; conversely, diffuse p indicates competing hypotheses and greater ambiguity in class assignment. Thus, p governs the directional aspect of uncertainty- belief is distributed across -pecific hypotheses.
The scalar modulates the intensity or concentration of each hypothesis. A smaller yields more concentrated, confident distributions for each hypothesis, while larger produces flatter, more dispersed components, effectively tempering overconfidence in uncertain ambiguous regions. By dynamically adapting т, the FD can represent varying degrees of epistemic caution response to data complexity.
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Together, p and T provide structural flexibility that extends beyond the magnitude of evidence encoded by a. This decomposition allows the FD to express both how much evidence the model has and how that evidence is organized across each competing hypothesis-leading to more faithful and principled UQ under complex or unforeseen data conditions.
D.2 Justification for Using the FD Distribution for UQ
We employ the FD distribution to address a central limitation of traditional EDL-it its inability to produce reliable uncertainty estimates complex or ambiguous scenarios where multiple class hypotheses may be simultaneously plausible.
For instance, when input resembles both "7" and "9", standard EDI, which relies single Dirichlet distribution, produces a unimodal belief that tends to overcommit to the dominant class (see Figure second and third panels). This representation fails capture the inherent ambiguity of the input and leads to overconfident predictions. In contrast, F-EDL employs a structured mixture of class-specific Dirichlet hypotheses, where each component encodes the belief this image corresponds to class k. All components share a common base evidence a, while the parameters p and т respectively determine the mixture weights and concentration of each component. This formulation enables the model to represent multimodal beliefs in a principled way, naturally capturing conflicting hypotheses and reducing overconfidence in uncertainty regions of the input space.
D.3 Evidence Extraction in F-EDL
For each input, F-EDL extracts and structures evidence through three conceptually interpretable stages:
The model first estimates the base evidence a, analogous standard EDL. This represents the initial strength of belief across classes and may be overconfident in ambiguous cases.
(1) Base Evidence Extraction:
(2) Construction of Class-Specific Hypotheses: Using a, F-EDL constructs a Dirichlet hypothesis for each class, treating each plausible explanation of the input. This enables the model to represent multiple competing class-level beliefs.
(3) Input-Adaptive Mixing via p and т: The parameters p and govern how these hypotheses integrated. p assigns mixture weights to each class-specific Dirichlet- peaked for clean inputs, dispersed for ambiguous samples, and near-uniform for OOD data-reflccting how belief is distributed across hypotheses. modulates the sharpness each component: it increases for ambiguous inputs to reduce spurious confidence, and remains low for clear inputs to preserve certainty.
This three-stage process aligns naturally with the mixture-basee formulation of the FD distribution, emphasizing that J-EDL's architecture arises from principled probabilistic design rather than an ad-hoc extension of EDL.
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E Additional Explanations about Experiments
In this section, present a comprehensive explanation of our experiments. First, we describe the datasets used in our study. Second, we outline the implementation details.
E.1 Datasets
In line with recent works in EDL 24 10 12), we used CIFAR-10 45 as the primary ID dataset for our evaluations. To assess UQ in more complex scenarios, we also employed CIFAR-100 45 as an additional ID dataset.
For CIFAR-10, we considered the Street View House Numbers (SVHN) and CIFAR-100 as OOD datasets. When CIFAR-100 was used as the ID dataset, its corresponding OOD datasets included SVHN and Tiny-ImageNett2000 (TIN). To investigate the generalizability of F-EDL in long-tailed ID scenarios, we employed CIFAR-10-LT [46], version of CIFAR-10 with artificially imbalanced class distributions. Specifically, we evaluated two levels of imbalance: mild (p 0.1) and heavy (p = 0.01), using the same OOD dataset as CIFAR-10 10. For noisy ID scenarios, we utilized Dirty-MNIST (DMNIST) [8], a noisy variant of MNIST [60], with Fashion-MNIST (FMNIST) [6]] as its OOD dataset. Additionally, for assessing distribution shift detection capabilities, we adopted CIFAR-10-4 10-C 488, which introduces controlled perturbations to CIFAR- 10. Detailed descriptions of these datasets are provided below.
CIFAR-10 [45 is one of the most widely used image classification datasets in the UQ literature. It consists of color images categorized into 10 classes, representing various animals and objects: airplane, automobile, bird, cat, deer, dog, frog, horse, ship. and truck. The dataset contains 50,000 training images and 10,000 testing images, with each image represented as tensor of shape 3x32 32 corresponding RGB channels and spatial dimensions. The dataset is balanced, meaning each class contains an equal number of samples in both the training and testing splits. For our experiments, the training set further divided into training and validation subsets with a ratio of 0.95 0.05. CIFAR-10 served as the primary ID dataset and was also used to create CIFAR- 10-LT, a long-tailed variant.
Street View House Numbers (SVHN) 47 is a dataset comprising cropped images of house numbers obtained from Google Street View. It contains 73,257 training images, 26,032 testing images, and 531,131 additional images, each represented as tensor of the shape 3 x 32 x 32. In our experiments, the test set of SVHN was employed as the OOD dataset when CIFAR- 10, CIFAR-1O-LT, or CIFAR- 100 served as the ID dataset.
more detailed and challenging version of CIFAR-10, comprising 100 classes, each representing a distinct animal and object category. It includes 50,000 training images and 10,000 testing images, with each image represented as a tensor of the shape 3 32 32. In our experiments, the training set was further divided into training and validation subsets with a ratio of 0.95 0.05. CIFAR-100 was used the ID dataset in the classical setting. Additionally, its test set was employed as an OOD dataset when either CIFAR-10 or LIFAR-IO-LT served as the ID dataset.
CIFAR R-00 45
Imag-eeageNet-200 (TIN) 662 is a smaller version of the ImageNet (63) dataset, comprising 200 classes, each representing a distinct category. It contains 100,000 training images, 10,000 validation images, and 10,000 testing images, with each image represented In our experiments, the test set of TIN was used as the OOD dataset when CIFAR- 100 served as the ID dataset. Due to the mismatch in the image sizes, resized TIN images to 3 32 32 for compatibility.
tensor of the shape 3 x 64 x 64.
CIFAR-10-LT [64] is a long-tailed version of CIFAR-10, characterized by artifically imbalanced class distributions. The imbalance severity is controlled using an imbalance factor (p), defined as the ratio of the number of samples in the head class to the tail class. In our experiments, CIFAR-IO-LT datasets with two imbalance factors, p = 0.1 and = 0.01, were used to validate the generalizability of F-EDL in long-tailed ID scenarios.
CIFAR- 1TAAR-10-C 10-C 66 a corrupted version of the CIFAR-10 dataset, designed to evaluate the robustness of image classification models against various types of corruption and noise. It introduces 19 types of real-world corruptions, including Gaussian noise, shot noise, impulse noise, defocus blur, glass blur, motion blur, zoom blur, snow, frost, fog, brightness, contrast, elastic transform, pixelate, jpeg
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Table 8: Comparison of the total parameters and the additional parameters introduced by the MLPs across different architectures. "# Params," " refers to total number of parameters the base model architectures (e.g.. ConvNet, VGG-16, and ResNet-18). "# of Additional Params." denotes the number of parameters added by the MLPs. "Proportion" indicates the percentage increase in the total parameter count due to the addition of the MLPs.
Dataset
Architecture Params.
# of additional params. Proportion (%)
DMNIST CIFAR-IO, CIFAR-IO-LT
ConvNet
248,330 14,857,546
6,347
2.56 %
VGG-16 ResNet-18 18
266,507
1.79%
CIFAR-100
11,046,308
289,367
2.62% %
Table 9: Average batch inference time (batch size 64) on the CIFAR-10 dataset using the VGG- 16 backbone. The mean and standard deviation are calculated over five independent runs.
Metric
EDL
DAEDL
F-EDL
Inference Time (s) 1.262 0.06 2.683 = 0.05 1.279 + 0.08
compression, speckle noise, Gaussian blur, splatter, and saturate. Each corruption is applied at five different severity levels E {1,2, 4,5), resulting in a total of 950,000 corrupted images (10,000 images X 19 corruptions 5 severities). In experiments, CIFAR-IO-C was used for distribution shift detection tasks. Specifically, models trained on CIFAR- 10 performed OOD detection using CIFAR-10- the OOD dataset to assess their ability to capture distribution shifts through uncertainty measures.
Dirty-MNIST (DMNIST) g is noisy variant of the MNIST (60) dataset, containing ambiguous data points. It generated by integrating Ambiguous-MNIST (AMNIST) { which includes artificially synthesized MNIST samples with varying entropy levels, into the original MNIST dataset. DMNIST contains 120,000 training images (60,000 from MNIST and 60,000 from AMNIST) and 70,000 testing images (10,000 from MNIST and 60,000 from AMNIST). In our experiments, DMNIST was used to validate the generalizability of F-EDL in noisy ID scenarios.
FaSSiOn-MNIST (FMNIST) 61 is modern replacement for the MNIST dataset, consisting of grayscale images from 10 classes that represent various fashion items, including clothing, footwear, and accessories. The dataset contains 60,000 training images and 10,000 testing images, each represented as a tensor of shape 1 x 28 x 28, similar to MNIST. In our experiments, FMNIST served as the OOD dataset when DMNIST was used as the ID dataset.
E.2 Implementation Details
To ensure a fair comparison, we adopted VGG-16 49 as the model architecture when CIFAR-10 served as the ID dataset, consistent with recent studies 24 10 utilized when CIFAR-1O-LT was employed as the ID dataset. For CIFAR- 100, ResNet-18 [50] was employed the model architecture, while simple convolutional neural network (ConvNet) consisting of three convolutional layers followed by three dense layers was implemented for DMNIST.
12]
11
12] The same architecture was
To compute the parameters p and we added two shallow MIPs into each architecture. These MLPs consisted of a single layer when DMNIST was the ID dataset and two layers for the other dataset. The additional model complexity introduced by these MLPs was minimal, as demonstrated in Table 8 Moreover, the added overhead becomes negligible for larger architectures, indicating that F- scales efficiently. For instance, with WideResNet-28- 10 (36.5M parameters) on Ten the extra heads introduce only 346K parameters (0.95%). This relative overhead further diminishes as the model capacity increases.
In terms of inference efficiency, F-EDL enables fast prediction and UQ without sampling or post-hoc steps (e.g.. DAEDL's density estimation). As shown in Table the average batch inference time, including FD parameter prediction and uncertainty computation, was 1.279s 0.08-only 1. 1.35% slower than EDL (1.262s + 0.06) and 52.34% faster than DAEDL (2.683s =0.05).
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Table 10: Implementation details and hyperparameler settings for experiments. Here, max. B, and v denote the maximum number epochs, batch size, and learning rate, respectively. Additionally, L represent number of layers and hidden dimensions of the MLPs, respectively.
ID Dataset
Architecture
Optimizer
Scheduler
Taxx
B
Step size
L
H
DMNIST
ConvNet
Adam
StepL
50
64
5 x 10 4
20
1
64
CIFAR-
VGG-16
Adam
StepLR
100
64
x 10
30
2
256
CIFAR-IO-LY VGG-16
Adam
StepLR
100
x 10-4
2
256
CIFAR-100
ResNet-18
Adam
StepI.
100
64
10 4
30
2
256
Based on the validation loss, early stopping was applied to mitigate overfitting across all experiments. fixed batch size of = 64 was used in all settings. Training was conducted for up to 50 epochs on DMNIST and 100 epochs on other datasets. The Adam optimizer [66] and the StepL. scheduler were consistently employed. Hyperparameters for both our proposed method and the baselines were selected through grid search. Importantl, J-EDL eliminates the need for hyperparameter tuning in its objective function, significantly simplifying the training process. The detailed experimental setups and hyperparameter configurations provided in Table 10 All experiments were implemented in PyTorch. Depending on availability, we used either an RTX 4060 GPU with 8GB memory or TITAN V GPU with 12GB memory.
Table 11: Comprehensive results for UQ-related downstream tasks in the classical setting using CIFAR-10 the ID dataset, with additional results from PostNet [24] and DUQ 77.
Method
Test.Acc.
Conf.
SVHN/ CIFAR-100
Dropout
82.84 +0.1
97.15
10.0
51.39 =0.1 +1.0
PostNet
84.85 #0.0
97.76
+0.0
77.71 =03 / 81.96 #0.8
DUQ
89.33 10.2
97.89
10.3
80.23 =34/ 84.75 =1.1
EDL
83.55 10.6
97.86
10.2
79.12 137/ 84.18 10.7
I-EDL
89.20 10.3
98.72
#0.1
82.96 #22/ 84.84 #0.6
R-EDL
90.09 10.3
98.98
10.1
85.00 #12/ 1033
DAEDL
91.11
#0.2
99.08
+0.0
+0.1
F-EDL
91.19 +0.2
99.10
#0.0
91.20 =1.3 88.37 10.3
Table 12: AUROC scores for OOD detection using epistemic uncertainty estimates in the classical setting.
Results reported using CIFAR-I0 as dataset with SVHN and CIFAR-100 (C-100) OOD datasets, CIFAR-100 as dataset with SVHN and TinylmageNet (TIN) as OOD datasets.
Method
ID: CIFAR-10
ID: CIFAR-100
OOD: SVHN / C-100
OOD: SVHN TIN
EDL
81.06 +4.5 80.63 +
63.95 #34/ +2.3
T-EDL
86.79 82.15 10.5
77.85 73.3/ 10.3
R-EDL
87.47 85.26 10.4
77.06 #22/71.10 +1.0
DAEDL
41.24410 10.1
81.07 43.0/55.04 +1.3
F-EDL
93.74 86.374...
81.59 #18/79.24 10.2
F Additional Results for Quantitative Study UQ-Related Downstream Tasks
Classical Setting. Table 11 presents comprehensive results for UQ-related downstream tasks in the classical setting using CIFAR- 10 the ID dataset, comparing F-EDL with additional basclines: DUQ
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Table 13: AUROC scores for OOD detection using epistemic uncertainty estimates in the long-tailed setting. Results reported using CIFAR-10-H the IDD dataset under mild (p 0.1) and heavy (p 0.01) imbalance, with SVHN and CIFAR- 100 (C-100) as OOD datasets.
Method
ID: CIFAR-IO-LT (p = 0.1) OOD: SVHN C-100
ID: CIFAR-IO-LT (p 0.01)
OOD: SVHN C-100
EDL
80.36 #1.5/ #0.6
58.25 14.4 61.05 +0.7
Z-EDL
84.65 #38/ 00.
61.47 #74 65.12 #1.
R-EDL
77.39 +53 +0,9
62.24 +2. 64.03 106
DAEDL
80.34 #29 #1.
64.21 / 63.49 +0.6
F-EDL
89.22 / 10.6
71.62 11.9 67.74 +18
Table 14; UQ-related downstream task results are reported using the DMNIST datasct as the ID dataset. "Brier." denotes the Brier score for calibration. "FUR and "FMNIST.AURG" indicate AUPR and AUROC scores for OOD detection using epistemic uncertainty, with FMNIST the OOD dataset.
Method
Test.Acc.
Conf.
Brier.
FMNIST.AUPR.
FMNIST.AUROC.
MSP
83,90 10.1
96.01 10.0
2.38 10.0
98.10 10.3
89.30 11.5
Dropout
84.13 10.1
96.09
10.0
2.44 10.0
94.68 10.6
68.97 128
DDU
84.05 =0.1
82.73
#0.1
2.35 #0.0
98.49 +04
90.16 +2.1
EDL
77.37 15.8
95.19
10.3
3.25 10.5
92.23 +17
61.51 16.6
I-EDL
83,46
+0.2
+000 95.58
2.74 1000
94.11 10.9
68.70 +4.0
R-EDL
83,41
10.1
95.58
10.1
2.80
90.91 15.6
59.45 1139
DAEDL
84.12 +0.1
95,93
+0.0
2.78 #0.1
99.44 #0.2
96.34 #14
F-EDL
84.28 10.1
96.17
10.1
2.32 10.0
99.76 10.1
98.46 10.7
Table 15: Ablation study results on F-EDL using the CIFAR-100 dataset as the dataset. "Fix-p (U), T" fixes to uniform vector (1/ K) and F to 1, while "Fix-p (N), 7" fixes p to the normalized concentration parameters (a/||a]]1) and r to 1. "Fix-p (U)" and "Fix-p (N)" fix but allow to be learned. "Fix-r" fixes 1 while learning p. The full "F-EDL" model learns both p and
Variant
Test.Acc.
Conf.
SVHN TIN
Fix-p (U), 5
63.54 #0.8
90.95 +0.4
71.49 #4.2 77.57 +0.4
Fix-p (N), 5
64.03 =0.5
91.27 +0.4
71.73 +1.7 77.53 +0.4
Fix-p (U)
63.74 10.3
91.04 10.2
69.06 / 10.6
Fix-p (N)
63.77 1.3
91.22
+0.6
69.59 +2.0/ 77.60 #0.5
Fix-T
65.68 +0.8
92.60 +0.5
73.38 14.6 78.76 +0.4
F-EDL 69.40 +002
94.00 #:01
75.40 +2.3 80.60 +.02
Table Ablation study on the effect of different regularization terms F-EDL using CIFAR-10 as the ID dataset. "No Reg." denotes training without regularization, "KL-Div." indicates KL-based regularization applied to the Dirichlet parameters. "CE." refers to cross-entropy regularization applied on and "Brier" represents the Brier-based regularization adopted in F-EDL.
Regularization
Aec
Conf.
SVHN / C-100
No Reg.
91.13 +022
99.02 +04
89.32 +0.1 87.65 +1.0
KI.-Div.
34.61 #27.8
50.69
134.7
45.83 +21.3 / 58.93 113,
CE
91.11 +0.1
98.87 +0.1
89.83 40.7 87.77 +0.2
Brier (Ours)
91.19 4-0.2
99.10 4:0.0
91.20 +1.3 88.37 4-0.3
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7 and PostNet (24. DUQ represents deterministic uncertainty methods, while PostNet exemplifies a density-based EDL approach. F-EDL outperforms both by significant margin, underscoring its robustness. Results on CIFAR-100 are omitted due to the high computational cost and training instability these baselines face when applied larger label spaces. Table reports AUROC scores for OOD detection in the classical setting, confirming that F-EDL maintains strong performance across different datasets and metrics.
Long-Tailed Setting. Table presents additional OOD detection results in the long-tailed setting using the CIFAR- 10-LT dataset, evaluated with AUROC. The results show that F-EDL consistently outperforms its competitors regardless of the evaluation metric.
Noisy Setting, Table 14 reports the full results for UQ-related downstream tasks in the noisy setting using the DMNIST dataset. Metrics include the Brier score [67], a standard measure of calibration, and AUROC scores for OOI) detection based epistemic uncertainty estimates. For the Brier score, lower values indicate better calibration. The results show that F-EDL consistently outperforms all competing methods across all metries.
Ablation Study. Table 15 presents the additional ablation study results using CIFAR-I00 as the ID dataset. The results demonstrate that the trend observed in the ablation study using the DMNIST dataset holds consistently: fixing either p or T leads to degraded performance, while the full model that jointly learns both components yields the strongest results. This confirms that the observed benefits are not dataset-specific but arise from the principled generalization enabled by the FD distribution.
In addition, Table T6 presents the ablation study on the regularization term using CIFAR-10 10 as the II) dataset. The results indicate that the Brier-based regularization yields the best empirical performance, owing to its improved training stability and better-calibrated allocation class probabilities.
G Additional Explanations and Results for Qualitative Study
In this section, we provide additional details the toy experiment from Jnd the posterior multimodality analysis from Section 6.3 including extended motivation, minor experimental settings, and supplementary figures.
G.1 Toy Experiment in the Figure
We provide additional explanations about the toy experiments described Section First, we outline the experimental setup, including the procedures followed and the uncertainty measures utilized. Second, we present the results for the aleatoric uncertainty distributions, along with corresponding figures and explanations, to complement the epistemic uncertainty distribution results shown in Section
Details on Experimental Setup. To investigate the challenges EDL models face in generalizing to difficult scenarios and assess whether F-EDL addresses these limitations, we conducted toy experiments compare their uncertainty distributions. These experiments use DMNIST, noisy variant of MNIST, as the ID dataset. The objective is to evaluate the ability of EDL and F-EDL to generalize noisy ID scenarios and reliably distinguish among three types of data: (i) clean ID data (MNIST), (ii) noisy ID data (AMNIST), and (iii) OOD data (FMNIST).
We trained both EDL and F-EDL using the training of DMNIST. After training, we computed the aleatoric and epistemic uncertainties for data points in the testing sets of DMNIST and FMNIST. For EDI., the aleatoric uncertainty (AU(x*)) and epistemic uncertainty (EU(x*)) for testing example x* are defined as follows:
K
AU(x') =1 max E Din(a) EU(x") = ke[K]
00
where 0g- = >K Og and Ex-Dir(a)) [T]] = =at/ap-
For F-EDL, we utilized the aleatoric and epistemic uncertainty measures outlined in Section 3.3 To
enhance the clarity the figures, we normalized the uncertainty estimates. For epistemic uncertainty, logarithmic transformation was applied. Both uncertainty estimates were then scaled to the range
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DMNIST (ID)
MNIST (Clean ID)
AMNIST (Noisy ID)
6
1
Density
60
(a) EDL Alcatoric Uncertainty
(b) F-EDL Alcatoric Uncertainty
Figure 5: Alcatoric uncertainty distributions with DMNIST as the dataset. The top row presents sample
images from MNIST and AMNIST. Panels (a) and (b) display histograms depicting the aleatoric uncertainty distributions obtained by the EDL and J-EDL (proposed) models, respectively, across these datasets.
[0, as follows:
AU(X') + AU(x*) )- - AUmin
log log(EU(x*)) - log (EUmin)
EU(x*) +
AU AUmin
log(EU (EUmaa)) log EUmin))
Here, AU max and mu denote the maximum and minimum values of aleatoric uncertainty, while EU max and EU min represent the maximum and minimum values of epistemic uncertainty. These values were computed using the combined test sets of DMNIST and FMNIST to ensure consistent scaling across both datasets.
Results for Aleatoric Uncertainty Distributions. The middle panel of Figure 5 illustrates the aleatoric uncertainty distributions generated by the EDL model. Ideally, a UQ model should assign higher aleatoric uncertainty to noisy samples, effectively distinguishing them from clean samples.
The results highlight the limitations of EDL in handling aleatoric UQ in noisy IDD scenarios. First, substantial portion of the MNIST test set exhibits high aleatoric uncertainty, nearing the maximum value shown in the figure. This suggests that EDL struggled effectively train on the noisy ID dataset consisting of ambiguous data points. Second, there is a significant overlap between the uncertainty distributions of MNIST and AMNIST, suggesting that the model incorrectly assigned low aleatoric uncertainty to AMNIST test points, despite their artificial construction to exhibit higher aleatoric uncertainty than MNIST.
In contrast, the bottom panel of Figure 5 which presents the aleatoric uncertainty distributions EDL in generalizing noisy ID
produced by F-EDL, demonstrates notable improvements scenarios and delivering robust UQ for ID data. Specifically, AMNIST demonstrates significantly higher aleatoric uncertainty compared MNIST, with minimal overlap between their uncertainty distributions. The small overlap observed between distributions is expected, as certain AMNIST data points with relatively low aleatoric uncertainty may naturally align with the uncertainty levels of MNIST.
G.2 Multimodal Uncertainty Representations for Ambiguous Inputs.
Motivation. We investigate whether the theoretical flexibility of F-EDL to represent multimodal class probability distribution translates to semantically meaningful and interpretable uncertainty in practice. While Theorem 4.4 shows that F-EDL can express predictive distributions mixture of Dirichlet components, it crucial to assess whether this capacity emerges perceptually ambiguous inputs. In particular, we aim to verify whether the model reflects structured hesitation between multiple plausible class hypotheses instead of defaulting to single, potentially overconfident mode (Proposition
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Figure 6: Additional examples supplementing Figure showing posterior class probability distributions for ambiguous DMNIST inputs. Fach subfigure includes: the input image, followed by the marginal distributions over the prohable classes as predicted by EDL (second and third panels) and by F-EDI. (fourth and fifth panels).
Experimental Setup. We analyze test examples from the DMNIST dataset that exhibit visual ambiguity. For each input x*, identify the top two predicted classes based on the expected class probabilities:
Vi argmmax Ee [m&(x*)]; y kE[K]
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We visualize the marginal distributions 7 (x*) and p(T (x*). Under the FD distribution, each marginal is a mixture of Beta distributions, parameterized by shared base evidence a, allocation probabilities p. and dispersion parameter
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Additional Results. In Figure 6 we present five additional examples of posterior multimodality on visually ambiguous inputs, extending the results from Figure 3 Each examples display the class
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probability distributions predicted by EDL and F-EDL for images with overlapping or unclear digit structures. For both models, we extract the marginal distributions of the two most probable classes, denoted as Ty; and Ty. EDI. produces unimodal Beta distributions that average over competing hypotheses, often concentrating mass on single class. This unimodality results in overconfident medictions-esseecilly problematic in ambiguous cases. In contrast, F-EDL produces multimodal marginals with distinct peaks, each reflecting a plausible interpretation as "7" and "9" for the hybrid digit. By retaining structured multimodality, F-EDL preserves input ambiguity rather than collapsing into single smoothed belief.
This ability to model structured multimodal beliefs offers key advantages. First, it improves in- terpretability; multimodal outputs reveal not only that the model is uncertain but also why, by highlighting plausible alternatives. This property is particularly valuable in safety-critics domains like medical diagnosis, where knowing which alternatives the model considers plausible can inform downstream decisions. Second, this flexibility contributes stronger UQ performance. Structured multimodality helps avoid overconfident beliefs on ambiguous inputs, thereby reducing misclassifica- tion and enhancing robustness in OOD detection [68]. Intuitively, EDL is biased toward collapsing uncertainty into a single mode. While this may be sufficient for downstream tasks when the dominant mode aligns with the true class, it risks failure when it favors incorrect one-leading to confident misclassification or false OOD rejection ambiguous ID inputs. contrast, F-EDL preserves a mixture of multiple plausible hypotheses, preventing premature convergence and allowing the model to gradually learn and better represent the underlying uncertainty. As demonstrated in our quantitative evaluations (Section 6.2). J-EDL consistently outperforms existing methods, particularly under ambiguity (e.g., AMNIST) !') or data imbalance (e.g., CIEAR- 10-LT tail classes).
41


Reviews:
[{"review_id": "ZeRAH1tc3M", "reviewer": "Reviewer_bX31", "summary": "This paper proposes Flexible Evidential Deep Learning (F-EDL), which extends EDL by predicting a flexible Dirichlet (FD) distribution parameterized by concentration parameters, allocation probabilities, and dispersion. The objective function combines expected MSE over the FD distribution with a Brier score-based regularization term, with a closed form. The advantage of the proposed method is demonstrated by both theoretical analysis and empirical experiments. It is proved the F-EDL is a generalization of traditonal EDL while enabling greater expressiveness.", "questions": "1. Can the authors explain the complementary strengths of softmax-based (MSP) and evidential (EDL) approaches? It is not clear when the softmax-based approach dominates and when EDL dominates, and how F-EDL leverages the complementary strengths to achieve optimal UQ.\n2. Fig. 4a is weird. Can the authors confirm the accuracy drops when training size increases from 30k to 50k for EDL?", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "# Strengths:\n1. It is novel to introduce the Flexible Dirichlet Distribution to evidential deep learning.\n2. It provides closed form formulation of total, aleatoric, and epistemic uncertainty. \n3. The form of objective function is elegant.\n4. The theoretical analysis is solid and experiments are comprehensive.\n\n# Weaknesses:\nThis is pretty solid paper. Honestly I did not find much signficiant weaknesses.", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your careful review of our paper and for the insightful and constructive comments. Please find our detailed answers to your comment below. \n\n**1. (When does the softmax-based approach or EDL dominate, and how does $\\mathcal{F}$-EDL combine its strengths to achieve optimal UQ?)**\n\nIn short, on clean in-distribution (ID) data, EDL dominates, while for ambiguous or out-of-distribution (OOD) data, neither EDL nor softmax alone is sufficient—$\\mathcal{F}$-EDL learns the optimal combination between the two predictions for each input.\n\nSpecifically, on clean ID data, the total evidence $\\alpha_{0}$ tends to be large compared to the dispersion parameter $\\tau$, causing the predictive distribution to be dominated by the EDL component. Since EDL provides reliable uncertainty estimates in such regions, this results in well-calibrated predictions.\n\nHowever, in more challenging scenarios such as ambiguous or OOD data—where EDL may be less reliable due to its restrictive Dirichlet assumption—$\\alpha_{0}$ is often not significantly larger than $\\tau$. In these cases, $\\mathcal{F}$-EDL learns to interpolate between the EDL and softmax predictions on a per-input basis. Specifically, it learns an adaptive combination of EDL ($p_{\\text{EDL}}$) and softmax ($p_{\\text{SM}}$) predictions, where the combination ratio is determined by the relative strengths of $\\alpha_{0}$ and $\\tau$. \n\nTheorem 4.5 formally characterizes this behavior, demonstrating that $\\mathcal{F}$-EDL functions as an input-dependent mixture of the EDL and softmax models. This formulation allows the model to adaptively adjust its predictions based on the properties of each input, providing robust UQ across diverse data conditions. We will include an intuitive explanation of Theorem 4.5 in the main text to enhance clarity. \n\n**2. (Why does EDL accuracy drop as training size increases in Fig. 4a?)**\n\nThis is an empirical pattern that emerged repeatedly across several runs. We attribute this counterintuitive behavior to the following contributing factors:\n\n * **Unstable learning dynamics:** EDL is known to exhibit unstable training behavior under its original formulation. As the training data size increases, the model may fail to calibrate the predicted evidence accurately, leading to degraded performance. This issue has motivated the development of several variants designed to improve the training dynamics of EDL, including $\\mathcal{I}$-EDL [1], R-EDL [2], and DAEDL [3]. For example, DAEDL (middle panel of Fig.4) applies spectral normalization to the feature extractor and adopts an alternative parameterization of the concentration parameters—both of which likely contribute to its comparatively more stable behavior. \n\n* **Misrepresentation of uncertainty:** As illustrated by the red dashed line in Fig. 4a, EDL’s average epistemic uncertainty *increases* as the training set size grows from 30k to 50k—an undesired behavior. Ideally, epistemic uncertainty should *decrease* as the model observes more data and gains confidence. This miscalibration suggests that the model becomes *more uncertain* despite increased supervision, which may account for the observed drop in test accuracy. \n\n\nWe will include a brief discussion of this observation in the main text alongside the explanation of Fig. 4 to improve clarity. \n\n\n**References**\n\n[1] Deng et al., \"Uncertainty estimation by fisher information-based evidential deep learning.\", ICML 2023\n\n\n[2] Chen et al., \"R-edl: Relaxing nonessential settings of evidential deep learning.\", ICLR 2024\n\n\n[3] Yoon & Kim., \"Uncertainty Estimation by Density Aware Evidential Deep Learning.\", ICML 2024"}, {"review_id": "1raOJxFmbT", "reviewer": "Reviewer_a4Mo", "summary": "This study argues that the restrictive assumption of Dirichlet-distributed class probabilities undermines the robustness of EDL. To address this limitation, the authors propose Flexible Evidential Deep Learning (F-EDL), which generalizes EDL by predicting a flexible Dirichlet distribution. Theoretically, we establish several advantageous properties of F-EDL, and extensive empirical evaluations demonstrate its SOTA uncertainty quantification performance across diverse experimental settings.", "questions": "See weaknesses.", "limitations": "Lately, Deep Evidential Learning (EDL) has been criticized for lacking a sound theoretical foundation. In particular, the work by Jürgens et al. [2024] (ICML '24, pre-print: arXiv:2402.09056) has analyzed many EDL methods and found that they all fail to represent epistemic uncertainty faithfully. In my opinion, this manuscript should address these issues and argue how the proposed changes overcome/bypass them.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "1. (Strength) Relaxing the restrictive assumption of Dirichlet-distributed class probabilities presents an interesting and meaningful direction for enhancing the UQ capability of EDL. This work shares a similar motivation with Relaxed-EDL (R-EDL), as both aim to improve EDL by relaxing existing restrictive assumptions. However, the proposed method adopts a fundamentally different implementation, offering novel insights that could significantly benefit the EDL community.\n2. (Strength) The paper is well-structured and clearly written, with rigorous theoretical analysis. The experimental design is comprehensive, and the results demonstrate strong performance across diverse settings. Overall, this is a high-quality contribution to the field of EDL.\n3. (Question) To my knowledge, I-EDL was the first to interpret EDL as a generative process and represent it via a graphical model. The graphical model and generative process presented in Figure 2 appear to differ in form from those in I-EDL. Could the authors clarify whether there are any substantive differences between the two formulations?\n4. (Weakness) Theorem 4.5 (Predictive Distribution Decomposition) is somewhat challenging to follow. I recommend that the authors provide a more intuitive and accessible explanation in the main text to improve readability.\n5. (Weakness) The rationale behind the choice of regularization in Section 3.2 remains unclear. Why does F-EDL employ the proposed regularization form instead of traditional KL-divergence-based regularization or no regularization (as seen in some other EDL-related works)? A more thorough theoretical and empirical analysis of how regularization selection impacts F-EDL's performance would strengthen the paper.\n6. (Weakness) Recent critiques, particularly the work by Jürgens et al. [2024] (ICML ’24, arXiv:2402.09056), argue that many EDL methods lack a solid theoretical foundation and fail to faithfully represent epistemic uncertainty. While the authors briefly discuss this issue in \"Faithfulness of Epistemic Uncertainty Estimates,\" their analysis relies primarily on empirical observations (e.g., uncertainty trends with varying training dataset sizes) rather than addressing the theoretical concerns raised by Jürgens et al. A more rigorous theoretical response would significantly enhance the manuscript's robustness.\n7. (Suggestion) I suggest that the authors include a brief discussion in the Related Work section comparing the motivation of this work with that of R-EDL. Since both approaches aim to enhance EDL by relaxing restrictive assumptions, such a discussion would help readers better contextualize the contributions of this work without diminishing its novelty.", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your careful review of our paper and for the insightful and constructive comments. Please find our detailed answers to your comment below. \n\n**1. (Can you clarify the differences between the graphical models of $\\mathcal{I}$-EDL and $\\mathcal{F}$-EDL?)**\n\nAt a high level, both models follow a similar generative process: (i) compute the parameters of the distribution over class probabilities, (ii) sample class probabilities, and (iii) generate the observed label. \n\nHowever, there are several key differences:\n* **Parameterization:**  $\\mathcal{F}$-EDL models the class probability distribution using three parameters—$\\boldsymbol{\\alpha}$, $\\mathbf{p}$, and $\\tau$—whereas $\\mathcal{I}$-EDL only predicts $\\boldsymbol{\\alpha}$. \n* **Label generation:**  $\\mathcal{I}$-EDL assumes that the label $\\mathbf{y}$ is sampled from a multivariate Gaussian distribution whose mean corresponds to the class probabilities and maximizes the corresponding log-likelihood. This formulation may be intended to motivate their use of Fisher information. However, it is functionally equivalent to assuming that $\\mathbf{y}$ is drawn from a Categorical distribution and minimizing the expected MSE, which aligns with our approach. \n* **Conditional dependence:** In $\\mathcal{I}$-EDL, the label $\\mathbf{y}$ is conditionally dependent on both the predicted class probabilities $\\boldsymbol{\\pi}$ (or $\\mathbf{p}$ in their notation) *and* the Dirichlet parameter $\\boldsymbol{\\alpha}$, reflecting an additional dependency related to Fisher information. In contrast, $\\mathcal{F}$-EDL adheres to the standard EDL formulation where $\\mathbf{y}$ depends only on $\\boldsymbol{\\pi}$. \n\nWe also note notational differences: $\\mathcal{I}$-EDL adopts plate notation, possibly for clarity, and $\\mathcal{F}$-EDL denotes deterministic variables (e.g., $\\mathbf{z}, \\boldsymbol{\\alpha}, \\mathbf{p}, \\tau$) with rectangular nodes, following standard conventions. \n\n\n**2. (Theorem 4.5 is challenging to follow; please provide an intuitive explanation in the main text.)**\n\nTheorem 4.5 aims to provide intuition for how $\\mathcal{F}$-EDL behaves as an adaptive mixture of EDL and softmax predictions, interpolating between them based on input characteristics. \n\nFor example, on clean in-distribution (ID) data, $\\alpha_{0}$ is typically much larger than $\\tau$, so the predictive distribution is dominated by the EDL component. Since EDL provides reliable uncertainty estimates in such regions, this results in well-calibrated predictions.\n\nFor ambiguous or out-of-distribution (OOD) data, neither EDL nor softmax alone is sufficient—$\\mathcal{F}$-EDL learns to optimally combine two, adjusting the balance for each input. Theorem 4.5 formally characterizes this behavior, showing that $\\mathcal{F}$-EDL functions as an input-dependent mixture of the EDL and softmax models.\nWe will include a clearer, more intuitive explanation in the revised version.\n\n**3. (Why does $\\mathcal{F}$-EDL use Brier-based regularization instead of KL-divergence or no regularization?)**\n\nWe adopt a Brier-based regularization term for both theoretical and empirical reasons: it encourages input-dependent calibration and demonstrates strong empirical performance. \n\nTheoretically,  Brier-based regularization promotes calibrated allocation probabilities $\\mathbf{p}$. To ensure that $\\mathbf{p}$ remains both interpretable and aligned with the true label, it should be softly encouraged toward the one-hot-target—without collapsing into overconfident assignments. The Brier score naturally supports this goal: it penalizes the squared deviation from the ground-truth label, thereby encouraging calibrated, smooth distributions. Compared to cross-entropy (CE)—which often drives sharp and overconfident outputs—Brier provides softer gradients suited for learning well-calibrated $\\mathbf{p}$. \n\nEmpirically, we chose Brier-based regularization for the following reasons: \n\n* **No regularization** led to unstable training and poor calibrated $\\mathbf{p}$ due to lack of guidance.\n* **KL-based regularization** as used in standard EDL is ill-suited, as we do not explicitly model the Dirichlet distribution parameterized solely by $\\boldsymbol{\\alpha}$. Moreover, computing the KL divergence between FD distributions is analytically intractable, making direct regularization infeasible. \n* **CE-based regularization** yielded less stable optimization and inferior empirical performance. In particular, it led to poorer misclassification detection, likely due to its tendency to produce less calibrated $\\mathbf{p}$.\n\n\nAmong these options, Brier-based regularization consistently yields the most robust and reliable UQ performance across tasks. To support this claim, we report the ablation study results on UQ-related downstream tasks using CIFAR-10 as the ID dataset. Table 1 summarizes the results averaged over five runs, comparing the effects of different regularization strategies: none, KL, CE, and Brier.  \n\n\n#### **[Table 1. Ablation study results on regularization term on CIFAR-10]**\n| Regularization   | Test.Acc. ↑       | Conf. ↑           | SVHN / C-100 ↑                     |\n|------------------|-------------------|-------------------|----------------------------------|\n| No Reg.          | 91.13 ± 0.2       | 99.02 ± 0.4       | 89.32 ± 0.1   / 87.65 ± 1.0      |\n| KL-Div           | 34.61 ± 27.8      | 50.69 ± 34.7      | 45.83 ± 21.3 / 58.93 ± 13.9      |\n| CE               | 91.11 ± 0.1       | 98.87 ± 0.10      | 89.83 ± 0.7  / 87.77 ± 0.2       |\n| **Brier (Ours)** | **91.19 ± 0.2**   | **99.10 ± 0.0**   | **91.20 ± 1.3 / 88.37 ± 0.3**    |\n\n\nThese results demonstrate that Brier-based regularization yields superior empirical performance, attributable to improved training stability and more calibrated allocation probabilities. \n\n\nWe will include the rationale for our choice of regularization in the main text and provide additional ablation study results in the *Appendix* of our revised version. \n\n\n**4.  (How does $\\mathcal{F}$-EDL respond to recent critiques of EDL, particularly those raised by Jurgens et al. [2024]? Does the proposed method overcome these issues or bypass them?)**\n\nIn essence, we argue that $\\mathcal{F}$-EDL bypasses most of the core theoretical critiques while also empirically overcoming their practical implications. Below, we elaborate further on the critique by Jürgens et al. [2024], as well as more recent and intuitively framed criticism by Shen & Ryu [NeurIPS 2024, arXiv:2402.06160]. \n\n**(On Jürgens et al.)**  This paper claims that an ideal second-order learner (such as EDL) should approximate a *reference second-order distribution*—obtained by bootstrapping first-order predictors trained on resampled datasets. They demonstrate that standard EDL fails to match this distribution in practice. \nHowever, this argument does not directly apply to $\\mathcal{F}$-EDL for two reasons:\n* **(i) Ill-defined first-order distribution**: $\\mathcal{F}$-EDL introduces additional parameters ($\\mathbf{p}$ and $\\tau$), which fundamentally change the hypothesis space. As a result, the corresponding first-order learner is no longer well-defined in the classical sense, rendering the notion of a *reference second-order distribution* assumed by Jürgens et al.—ambiguous or inapplicable. \n* **(ii) Lack of regularization on the second-order distribution**: The core theoretical result in Jürgens et al. relies on explicit regularization of the second-order distribution, typically via a KL divergence term. In contrast, $\\mathcal{F}$-EDL applies regularization only to the parameters $\\mathbf{p}$, not the second-order distribution itself—thereby violating a key assumption required for their analysis to apply. \n\nMoreover, the foundational assumption—that second-order learners *should* be aligned with a bootstrapped reference distribution—is a strong modeling choice, not a universally accepted principle. We view Jürgens et al. as valuable in identifying potential limitations of standard EDL formulations, but not as establishing a definitive criterion that all EDL variants must satisfy. \n\n**(On Shen & Ryu.)** This paper claims that standard EDL models are implicitly constrained to match a *fixed target meta-distribution*. As a result, they claim EDL acts more as an energy-based OOD detector than as a faithful quantifier of epistemic uncertainty—particularly in the sense that uncertainty should decrease as more data becomes available. \nWhile thought-provoking, their analysis assumes KL or reverse-KL divergence-based losses, which do not directly extend to alternative formulations like Brier-style (expected MSE) loss used in $\\mathcal{F}$-EDL. Furthermore, extending their notion of a *fixed target meta-distribution* to our FD family is non-trivial, and a closed-form is not readily available. \n\nIn summary, while these theoretical critiques offer valuable insights, $\\mathcal{F}$-EDL falls outside their core assumption due to its more expressive parameterization and distinct objective. Moreover, given the difficulty of theoretically certifying faithful epistemic uncertainty, we take an empirical approach—showing that $\\mathcal{F}$-EDL’s uncertainty estimates decrease as more data becomes available (Fig. 4). \n\nWe will clarify how $\\mathcal{F}$-EDL relates and bypasses the critiques highlighted in prior work in the revised version. \n\n**5. (Comparison with R-EDL in Related Work will help contextualize contributions.)**\n\nIn the revised version, we will include the following sentence in the *Related Work* section when discussing R-EDL: \n>R-EDL and $\\mathcal{F}$-EDL share the common goal of improving EDL by relaxing restrictive modeling assumptions. \n>R-EDL addresses this by introducing a tunable prior parameter and removing variance terms from the loss function. \n>In contrast, $\\mathcal{F}$-EDL generalizes the Dirichlet assumption itself by adopting a more flexible distribution, thereby enhancing the expressiveness and generalizability of UQ."}, {"review_id": "VgiOiTK4hU", "reviewer": "Reviewer_jKtY", "summary": "This paper discussed the limitations of using a Dirichlet Distribution on compositional data analysis, i.e., classification with all probabilities summing to one. By introducing a Flexible Dirichlet, the EDL framework is enhanced not only in classification accuracy but also in Uncertainty Quantification, i.e., in-domain and out-of-domain detection.\nThe authors provided detailed explanations and derivations in the main text and the appendix. The idea of replacing conventional Dirichlet with a Flexible Dirichlet is interesting, and the authors proposed a data-driven approach, i.e., neural head, to learn the parameters of a Flexible Dirichlet.", "questions": "If my understanding is correct, based on your inference algorithm, the label prediction process is like a post-adjusting process on allocation probability p using learned concentration parameter \\alpha and dispersion \\tau. \n1. The regularisation term is intuitively trying to align with the GT one-hot encoding by the Berier score from the scoring rule, which is similar to the L2-norm. Is there any ablation compared to CE loss/KL loss?\n2. The effectiveness and reliability of learned concentration parameter \\alpha, as in one of your baseline I-EDL, which discussed the learning of \\alpha. It would be better to pick some test instances to analyse the learned \\alpha\n3. The EDL has also been widely used in Multiview/Multimodal Learning. Given the existing works, which are mainly based on Subjective Logic to combine different views’ Dirichlet, have you considered how to combine multiple FDS of each modality/view？Is it still eligible to use the Dempher-Shafer combination rule, or is a new combination method needed?\n\nI would like to kindly request similar experiments on other datasets, at least one more dataset, for example, a text-based dataset, a different vision domain or a larger one with more class labels, to better support the effectiveness of your work.", "limitations": "The authors have discussed the limitations at the end, together with future work.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths\n* The paper introduces Flexible Dirichlet (FD) as a generalisation of the Dirichlet distribution, and provides rigorous theoretical support (e.g., conjugacy, multimodality, subjective logic interpretation). These generalisations are non-trivial and well-motivated.\n*   Using an expected MSE over the FD and a Brier-score-like regularizer provides an elegant and interpretable way to align predictions with the one-hot ground truth, improving calibration without requiring OOD data.\n*  The proposed F-EDL reduces to standard EDL under specific settings, thus maintaining backward compatibility while enabling greater expressiveness.\n* The empirical results on CIFAR-10, CIFAR-100, CIFAR-10-LT, and DMNIST demonstrate improvements across various tasks (classification, OOD detection, misclassification detection, and distribution shift). The proposed method outperforms strong EDL baselines.\n* The visualisations of multimodal uncertainty and the faithful trend of decreasing epistemic uncertainty with more data lend strong qualitative evidence of F-EDL's effectiveness.\n* The contribution of FD-specific parameters (p and τ) is well-analysed, confirming that the performance gains are not merely due to model capacity but due to the expressive modelling capability of FD.\n\nWeaknesses\n*  While the results on vision datasets are comprehensive, the paper lacks evaluation on non-vision domains (e.g., NLP). This raises concerns about cross-domain robustness.\n*  Since α is a central component of both EDL and F-EDL, an in-depth analysis of the learned α values across different examples (e.g., low-confidence vs. high-confidence samples) is missing. Such an analysis would shed light on its reliability and distinguishability from prior EDL methods like I-EDL.\n* While a subjective logic interpretation is provided, the paper doesn’t discuss the potential of combining FD parameters from multiple views/modalities, which is an active area in multiview UQ. It's unclear whether Dempster-Shafer-like fusion remains valid for FD or new fusion rules are needed.\n* The paper uses a Brier score–like term instead of the CE/KL loss commonly used in EDL and other methods. However, no ablation study is included to compare the effects of using Brier vs. CE/KL losses.\n* Although the overhead is said to be minor, the architectural complexity introduced by three output heads (for α, p, τ) and the FD computation could be clarified further in terms of inference time and scalability to larger models.\n\nI believe the trusted multiview learning is part of EDL methods, so recommended the authors adding some discussions and references to those works in the Related Works section.\nSome recommended references:\n 1. Han, Zongbo, Changqing Zhang, Huazhu Fu, and Joey Tianyi Zhou. \"Trusted multi-view classification with dynamic evidential fusion.\" IEEE transactions on pattern analysis and machine intelligence 45, no. 2 (2022): 2551-2566.\n 2.  Xu, Cai, Jiajun Si, Ziyu Guan, Wei Zhao, Yue Wu, and Xiyue Gao. \"Reliable conflictive multi-view learning.\" In Proceedings of the AAAI conference on artificial intelligence, vol. 38, no. 14, pp. 16129-16137. 2024.\n3. Liang, Xinyan, Pinhan Fu, Yuhua Qian, Qian Guo, and Guoqing Liu. \"Trusted Multi-View Classification via Evolutionary Multi-View Fusion.\" In The Thirteenth International Conference on Learning Representations. 2025.\n4. Lu, J., Buntine, W., Qi, Y., Dipnall, J., Gabbe, B. and Du, L., 2024. Navigating Conflicting Views: Harnessing Trust for Learning. arXiv preprint arXiv:2406.00958.", "was_revised": true, "final_justification": "The authors have addressed most of my concerns. Therefore, I decided to raise my rating.", "rebuttal": "Thank you for your careful review of our paper and for the insightful and constructive comments. Please find our detailed answers to your comment below. \n\n**1. (Including the results on non-vision or large-scale datasets would enhance effectiveness.)**\n\nWe agree that evaluating on non-vision or large-scale datasets further strengthens the evidence for the effectiveness of our approach. Accordingly, we conducted additional experiments on the following benchmarks:\n\n* **UCI Adult (Tabular)**: Binary classification dataset with 48,842 examples, each consisting of 14 demographic features. We used a 3-layer MLP as the backbone.\n* **AG News (Text)**: 4-class news categorization dataset, containing 120k train / 7.6k test samples. We subsampled 10k train / 2k test examples for efficiency and used the DistilBERT encoder with linear classification heads. \n* **TinyImageNet-200 (Large-Scale Vision)**: 200-class image classification dataset containing 100k training and 10k validation images of size 64 $\\times$ 64 $\\times$ 3. We used a modified ResNet-18 as the backbone.\n\n\nIn all settings, we compared $\\mathcal{F}$-EDL with softmax and EDL, and included either R-EDL or DAEDL as a recent baseline. \nFor tabular/text tasks, we evaluated in-distribution (ID) UQ using test accuracy, misclassification detection AUPR and AUROC, Brier score, and expected calibration error (ECE). For vision tasks, we focused on uncertainty-based OOD  detection, measured by AUPR scores across diverse OOD datasets.\n\n\n\n\n\n#### **[Table 1. UQ-related downstream task performance on UCI Adult]**\n\n| Method          | Test.Acc. ↑       | Brier ↓          | AUROC ↑         | AUPR ↑          |\n|-----------------|------------------|------------------|------------------|------------------|\n| Softmax         | 82.41 ± 0.4     | 13.30 ± 0.2     | 84.88 ± 0.6     | 68.38 ± 0.3     |\n| EDL             | 82.40 ± 0.5     | 15.17 ± 0.3     | 84.76 ± 0.5    | 68.20 ± 1.6     |\n| R-EDL           | 82.40 ± 0.5     | 15.35 ± 0.3     | 84.77 ± 0.5     | 68.23 ± 1.5     |\n|**$\\mathcal{F}$-EDL** | **82.51 ± 0.4** | **12.41 ± 0.1** | **85.83 ± 0.6** | **70.86 ± 1.5** |\n\n\n\n\n\n#### **[Table 2. UQ-related downstream task performance on AG News]**\n\n| Method           | Test.Acc. ↑ | AUPR ↑ | ECE ↓ | Brier ↓ |\n|------------------|------------|--------|--------|-----------|\n|Softmax                              |   84.80       | 80.40      |    15.15  |  30.27  |\n| EDL                                  | 88.25      | 76.15  | 16.81  | 5.41      |\n| R-EDL                              | 86.75      | 90.76  | 12.45  | 6.26      |\n| **$\\mathcal{F}$-EDL** | **89.85**  | **96.99**  | **10.20**  | **4.73**  |\n\n\n#### **[Table 3. AUPR scores for OOD Detection on TinyImageNet]**\n\n| Method           | Places | SVHN  | FMNIST | Avg.    |\n|------------------|--------|-------|--------|--------|\n| Softmax          | 74.08  | 71.06 | 67.23  | 70.79  |\n| EDL              | 69.41  | 68.41 | 67.85  | 68.56  |\n| DAEDL            | 85.69  | 79.87 | 86.01  | 83.86  |\n| **$\\mathcal{F}$-EDL** | **86.65** | **92.06** | **90.01** | **89.57** |\n\nTable 1-3 shows that $\\mathcal{F}$-EDL consistently outperforms the baselines across all datasets and tasks, demonstrating its robustness across modalities and scalability to large-scale vision settings.  \n\nWe will formalize these results, include additional baselines, and include them in the *Appendix* of the revised version. \n\n**2.  (In-depth analysis of the learned $\\alpha$ across different examples would enhance the distinguishability.)**\n\nWe agree that analyzing the learned $\\boldsymbol{\\alpha}$ values—along with $\\mathbf{p}$ and $\\tau$—can improve the distinguishability of our model. Thus, we report the FD parameters for clean (e.g., digit “9”) and ambiguous samples (e.g., digit resembling both “4” and “9”), and compare them with those of EDL.\n\nAs shown in Table 4, $\\mathcal{F}$-EDL captures structured multimodality for ambiguous inputs by assigning large values of $\\alpha$ and $p$ to the plausible classes (“4” and “9”), along with a large $\\tau$ to reflect uncertainty. In contrast, EDL tends to exhibit bias toward a single dominant class, failing to reflect the semantic ambiguity of such inputs. \n\n\n#### **[Table 4. Parameter values for clean vs. ambiguous sample]**\n| Model              | Input               | Dirichlet Parameters $\\boldsymbol{\\alpha}$                            | Flexible Parameters $(\\mathbf{p}, \\tau)$                              |\n|--------------------|---------------------|-----------------------------------------------------------------------|------------------------------------------------------------------------|\n| EDL     | Ambiguous (“4” vs. “9”) | $\\alpha_9 = 14.71,\\ \\alpha_4 = 1.00$    | —                                                                      |\n| EDL                | Clean (“9”)           | $\\alpha_9 = 139.09,\\ \\alpha_4 = 1.00$                                 | —                                                                      |\n| $\\mathcal{F}$-EDL  | Ambiguous (“4” vs. “9”) | $\\alpha_9 = 161.05,\\ \\alpha_4 = 157.65$                               | $p_9 = 0.54,\\ p_4 = 0.32,\\ \\tau = 323.27$                              |\n| $\\mathcal{F}$-EDL  | Clean (“9”)           | $\\alpha_9 = 10048.80,\\ \\alpha_4 = 374.79$                             | $p_9 = 0.98,\\ p_4 = 0.01,\\ \\tau = 55.16$                               |\n\nWe will formalize these by adding more test instances and include them in the revised version. \n\n**3. (Regarding the comments on trusted multi-view learning)**\n\nWe agree that *trusted multi-view learning* is a timely and promising research direction in EDL. We will add a brief discussion in *Related Work*, citing the recommended references as follows:\n> Recently, EDL methods have been widely applied for trusted multi-view learning by fusing Dirichlet opinions across views using subjective logic and Dempster-Shafer fusion rules [1-4]. \n\n**On the applicability of Dempster-Shafer fusion in $\\mathcal{F}$-EDL.** \nWhile many multi-view EDL methods rely on the Dempster-Shafer (D-S) combination under the subjective logic (SL) framework, such strategies do not directly extend to our setting. $\\mathcal{F}$-EDL uses the FD distribution, which models a structured mixture of class-specific SLs via additional parameters $\\mathbf{p}$ and $\\tau$, alongside $\\boldsymbol{\\alpha}$. This richer structure renders classical D-S rules incompatible, highlighting the need for new fusion mechanisms tailored to this setting.\n\n**Potential for leveraging FD for trusted multi-view learning.** \nLeveraging FD-based subjective opinions for multi-view UQ opens up promising avenues that go beyond traditional fusion strategies. Below, we outline three potential directions:\n* **(i) FDs $\\rightarrow$ FD fusion**: Develop novel fusion rules to combine multiple view-specific FD distributions into a unified FD. This may involve extending D-S combination principles, incorporating learnable gating mechanisms, or applying attention-based modules to adaptively weigh each view based on its reliability. \n* **(ii) FDs $\\rightarrow$ Dirichlet fusion**: Fuse view-specific FD distributions, learned via $\\mathcal{F}$-EDL, into a unified Dirichlet using $\\mathbf{p}^{(v)}$ and $\\tau^{(v)}$ as reliability indicators for each view $v \\in [V]$—enabling informed fusion.\n* **(iii) Dirichlets $\\rightarrow$ FD fusion**: Starting from view-specific Dirichlets, construct a fused FD distribution, in which $\\mathbf{p}$ and $\\tau$ are learned to encode inter-view ambiguity and conflict—enabling more expressive representation of fused belief. \n\nGiven $\\mathcal{F}$-EDL’s strong UQ performance under challenging ID scenarios and across diverse modalities, we believe it provides a solid foundation for advancing *trusted multi-view learning*. We will briefly mention this in the *Conclusion* and include a detailed discussion in the *Appendix*. \n\n**4. (Why does the paper use a Brier-score-like term in the loss/regularization instead of CE/KL? Any ablations?)**\n\n**Regarding the loss:** We adopt the expected MSE (i.e., a Brier-style loss) over the FD distribution due to its tractability and empirical effectiveness. This choice is consistent with the original EDL (Sensoy et al., 2018) and is widely used in recent variants ($\\mathcal{I}$-EDL, R-EDL, and DAEDL). \nIn contrast, expected CE is intractable under FD, and KL-based losses lack a closed-form solution and a well-defined target distribution. \n\n**Regarding the regularization:** We adopted a Brier-style regularizer to promote input-dependent calibration and empirical performance. To support this choice, we conducted an additional ablation study. For further details of the rationale behind this regularization and the corresponding ablation results, please refer to our response to **Reviewer a4Mo, Comment 3**.\n\n\n\n\n**5. (Clarify the inference time and scalability to larger models.)**\n\nWe clarify the architectural and computational overhead of $\\mathcal{F}$-EDL below:\n\n**Scalability.** We predict $\\boldsymbol{\\alpha}$ using the backbone’s existing MLP head and add two lightweight MLPs for $\\mathbf{p}$ and $\\tau$. The added overhead is minimal for especially for larger architectures—e.g., for WideResNet-28-10 (36.5M parameters) on TinyImageNet-200, the extra heads add only 346K parameters (\\~0.95\\%). This relative overhead further diminishes as model capacity increases. \n\n**Inference Time.** $\\mathcal{F}$-EDL supports efficient prediction and UQ without sampling or post-hoc steps (e.g., DAEDL’s density estimation). On CIFAR-10 with VGG-16, the average batch inference time (including FD parameter prediction and uncertainty computation) was 1.279s $\\pm$ 0.08—only 1.35\\% slower than EDL (1.262s $\\pm$ 0.06) and 52.34\\% faster than DAEDL (2.683s $\\pm$0.05). \n\nWe will add this analysis to the main text, including quantitative comparisons of the inference time and a discussion of scalability for larger models."}, {"review_id": "dmCWa4t6Sx", "reviewer": "Reviewer_jVmC", "summary": "The paper introduces flexible evidential deep learning (${\\cal F}$-EDL) as an extension to evidential deep learning (EDL) by introducing a generalization to the Dirichlet distribution that the coin the flexible Dirichlet distribution.  The paper provides theory to prove that the flexible Dirichlet distribution is still a conjugate prior for a multinomial likelihood and the distribution can be viewed as mixture of Dirichlet so that it can capture multimodal distribution with better control of adjusting aleatoric and epistemic uncertainty.  The experimental section provides a comprehensive demonstration that ${\\cal F}$-EDL is more effective for misclassification detection and out-of-distribution detection that other state-of-the-art uncertainty aware methods including EDL and more recent variants thereof.", "questions": "The key questions include:\n\n1) Why would minimization of the cost function not require that  $\\alpha_0,\\tau \\to \\infty$? \n\n2) What is the semantics behind the evidence that leads to a likelihood function adjusting the values for $p_k$ and $\\tau$?\n\n3) What justifies how evidence is extracted from training data relative to the image under test to justify the flexible Dirichlet distribution?", "limitations": "The conclusions do point out the limitations of the current approach.  These appear comprehensive.\n\nAs basic research for uncertainty quantification, I do not see an potential negative societal impact.  Rather, the techniques discussed in the paper help to identify when not to trust the machine learning model.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "The flexible Dirichlet distribution is an interesting generalization of the Dirichlet distribution, and the experimental results are comprehensive and demonstrate consistent improvement via the ${\\cal F}$-EDL. I especially liked the idea to separate out aleatoric and epistemic uncertainty for misclassification and OOD detection, respectively. \n\nThe main weakness of the paper is that it is not clear how ${\\cal F}$-EDL achieves its performance gains via the cost function that is minimizes.  The overall cost function, which is not in the main paper but in the supplementary appendix A.1 (line 848).  Unlike most variants of EDL, it does not include a KL-like regularization term to temper the values of the Dirichlet strength $\\alpha_0$ or the other strength parameter $\\tau$.   The problem is that the middle two terms in the cost function go to zero as $\\tau,\\alpha_0 \\to \\infty$. The  two squared error terms can be minimized where $\\alpha_k \\propto p_k$.  In other words, minimization of the cost function as stated in the paper would indicate that the two strength parameters should go to infinity.  What is actually tempering these two values?  Why would minimization of the two squared error terms require different values for $p_k$ and $\\frac{\\alpha_k}{\\alpha_0}$? \n\nFigure 3 is really nice to illustrate that the flexible Dirichlet distribution is multimodal.  Perhaps there is an argument that this multimodal nature is necessary in terms of reducing the squared error terms in the cost function. However, I really do not understand it at this moment.\n\nAnother issue is that while the flexible EDL is a conjugate prior of the multinomial likelihood, observations from the multinomial distribution only affect the values of $\\alpha_k$.  What is the semantics behind the evidence that leads to the value of $p_k$ and $\\tau$? Perhaps this could give insights into what notions of uncertainty are captured by the flexible distribution and how it helps the performance of ${\\cal F}$-EDL. \n\nThe paper hypothesizes that the limitations of EDL is the used of the Dirichlet distribution to express the second order uncertainty. This reviewer believes that the limitation with EDL is more about how it extracts an approximation of evidence for the Dirichlet distribution.  The original version of EDL conflates aleatoric and epistemic uncertainty.  More recent EDL variants try to address this issue by altering the cost function.  Somehow, EDL should capture how the training data serves as evidence for various class labels for the input (usually an image) under test. As more training data is used, the evidence should increase accordingly and the Dirichlet strength should decrease.  Using the original cost function for EDL, it has been demonstrated that this is not the case. I am not convinced that a generalized version of the Dirichlet distribution fixes this problem.\n\nThe weaknesses discussed above are driving my current recommendation.  A few additional minor points are provided below. \n\nOne particular method to address the problem to disentangle aleatoric and epistemic uncertainty is the generative EDL [R1].  I am very curious how it would fair in the paper's experiments.  Nevertheless, I do not view this generative EDL as a required baseline for publication. \n\nGeneralized Dirichlet distributions make sense if there is likelihood function that provides for an argument for its use.  This gets to the semantics questions from above. For instance some papers have looked as heretical set classification to address blurry images where the model might not be able to label the image as a \"dog\" but can label it as an \"animal\". In that case, hyper-Dirichlet and group Dirichlet distributions naturally emerge, which are generalizations of the Dirichlet distribution. What justifies how evidence is extracted from training data relative to the image under test to justify the flexible Dirichlet distribution?\n\n[R1] Sensoy, Murat, Lance Kaplan, Federico Cerutti, and Maryam Saleki. \"Uncertainty-aware deep classifiers using generative models.\" In Proceedings of the AAAI conference on artificial intelligence, vol. 34, no. 04, pp. 5620-5627. 2020.", "was_revised": true, "final_justification": "The authors have answered my most serious concerns about grounding the loss function through regularization.  I am not convinced that the authors have the final story of why the flexible Dirichlet is able to better capture epistemic uncertainty than the less expressive Dirichlet distribution. Nevertheless, in light of the strong empirical results, the EDL research community can try to answer this question moving forward. \n\nOverall, I have moved my recommendation up to an accept decision.", "rebuttal": "Thank you for your careful review of our paper and for the insightful and constructive comments. Please find our detailed answers to your comment below. \n\n**1. (Why would minimization of the cost function not require that $\\alpha_{0}$ and $\\tau \\rightarrow \\infty$?)**\n\nWe clarify this issue step by step, addressing both the reviewer’s primary question and underlying concerns. \n\n**KL regularization is not suitable for $\\mathcal{F}$-EDL.**\nWe omit KL regularization as it is ill-defined for the FD distribution, intractable to compute, and empirically ineffective. See our response to **Reviewer a4Mo, Comment 3** for details. \n\n\n**Loss minimization does *not* always imply that $\\alpha_{0}$ and $\\tau \\rightarrow \\infty$.**\nFor a given batch $B$, the core loss consists of squared error terms:\n$$\n\\sum_{\\mathbf{x} \\in B} \\left(\\sum_{k=1}^{K} \\left( y_k(\\mathbf{x}) - \\frac{\\alpha_k(\\mathbf{x}) + \\tau(\\mathbf{x}) p_k(\\mathbf{x})}{\\alpha_0(\\mathbf{x}) + \\tau(\\mathbf{x})} \\right)^2 + \\sum_{k=1}^{K} (y_{k}(\\mathbf{x}) - p_{k}(\\mathbf{x}))^{2} \\right).\n$$\n\nThis is minimized when: \n$$\n\\frac{\\alpha_{k}(\\mathbf{x})}{\\alpha_{0}(\\mathbf{x})} = p_{k}(\\mathbf{x}) = y_{k}(\\mathbf{x}) \\quad \\text{for all} \\  \\mathbf{x} \\in B, k \\in [K].\n$$\n\nIn this ideal case—when predictions are confident and correct for all samples in the batch— increasing $\\alpha_{0}$ and $\\tau$ toward infinity further minimizes the loss. \nHowever, if the model is uncertain or incorrect—i.e., \n$$\n\\text{argmax}_k \\left( \\frac{\\alpha_k(\\mathbf{x})}{\\alpha_0(\\mathbf{x})} \\right) \\neq y \n\\quad \\text{or} \\quad \n\\text{argmax}_k \\ p_k(\\mathbf{x}) \\neq y,\n$$\nwhere $y$ denotes the true class index for $\\mathbf{x}$, then increasing $\\alpha_0$ and $\\tau$ amplifies overconfidence in the wrong direction, thereby *increasing* the loss. This naturally discourages the unbounded growth of these parameters unless the predictions are exactly correct. \n\nIn practice, achieving perfect predictions across a batch is rare. Analogously, in logistic regression, weights diverge under perfect separability, but for non-separable data, they remain bounded. Similarly, in our setting, imperfect predictions across batches act as an implicit regularizer—preventing $\\alpha_{0}$ and $\\tau$ from growing unbounded, even without explicit penalties. \n\n**Spectral normalization implicitly regularizes $\\alpha$.**\nWe apply spectral normalization to both the feature extractor and the head predicting $\\boldsymbol{\\alpha}$, which enforces Lipschitz continuity and constrains logit magnitudes, which in turn bounds $\\alpha_{0}$. As a result, the model is inherently prevented from producing arbitrarily large $\\alpha_{0}$, even without explicit regularization. \n\n**Multimodality emerges to minimize loss on ambiguous inputs.**\nFor instance, consider an input resembling both “7” and “9” but labeled as “7”. Due to inductive biases or limited evidence, the model may initially favor incorrect classes (e.g., $\\alpha_{9} > \\alpha_{7}$). To reduce the squared error, it must adjust $\\alpha_{7}$, $p_{7}$, and $\\tau$, gradually shifting belief toward the correct class.\nThis naturally induces *multimodality* over plausible classes, helping the model avoid overconfident local optima—highlighting that multimodality emerges as a natural consequence of minimizing loss under ambiguity. \n\n**Edge cases can be handled via clamping.** \nIn early-stage or synthetic scenarios where perfect classification is attainable, $\\alpha_{0}$ and $\\tau$ may grow rapidly. While we did not observe this in practice, *clamping* can be applied as a safeguard if needed. \n\n**2. (What is the semantics behind $\\mathbf{p}$ and $\\tau$ in terms of evidence, given that only $\\boldsymbol{\\alpha}$ is directly affected by multinomial observations).**\n\n**Clarification on Bayesian Interpretation.**\nWe clarify that in $\\mathcal{F}$-EDL, the parameters $(\\boldsymbol{\\alpha}, \\mathbf{p}, \\tau)$ are learned directly for the input $\\mathbf{x}$ via a neural network, not inferred through Bayesian updates. The Bayesian formulation (Theorem 4.2) serves as a *conceptual lens*, not a generative model. The *observations*  used in the conjugate analogy are *pseudo-observations*, not actual samples (see Appendix C.2 for details).\n\n**Semantics of $\\mathbf{p}$ and $\\tau$.** While $\\boldsymbol{\\alpha}$ captures the *amount* of evidence for each class, $\\mathbf{p}$ and $\\tau$ govern its *alllocation* and *concentration*:\n* $\\mathbf{p}$ controls how belief is distributed across class-specific hypotheses. A sharp $\\mathbf{p}$ indicates a strong preference for a single class; diffuse $\\mathbf{p}$ reflects ambiguity or uncertainty across multiple classes. \n* $\\tau$ controls the concentration within each hypothesis. A low $\\tau$ yields peaked, confident components; a high $\\tau$ flattens them, mitigating overconfidence—especially in ambiguous cases.\n\nTogether, $\\mathbf{p}$ and $\\tau$ capture structural uncertainty beyond $\\boldsymbol{\\alpha}$, enabling better ambiguity modeling and generalization. \n\n**3. (The limitation of EDL may lie more in its evidence approximation mechanism than in the Dirichlet itself. How does the FD distribution resolve this limitation?)**\n\nWe address this with three key points:\n\n**$\\mathcal{F}$-EDL empirically exhibits faithful epistemic uncertainty.**\nAs training data increases, $\\mathcal{F}$-EDL consistently shows a decrease in uncertainty—aligning with the desirable behavior noted by the reviewer. This trend is clearly demonstrated in Fig. 4 (Section 6.3).\nImportantly, the *variance of the FD distribution*—epistemic uncertainty measure in $\\mathcal{F}$-EDL—serves as a natural generalization of *inverse Dirichlet strength* used in standard EDL. Thus, the observed decrease in FD variance with more training data directly supports that $\\mathcal{F}$-EDL faithfully captures epistemic uncertainty—suggesting that its evidence approximation mechanism is effective in practice.\n\n**Structural generalization is a principled path beyond EDL.** \nThe key innovation in $\\mathcal{F}$-EDL lies in its structurally richer representation, which we view as a principled and underexplored direction for improving EDL. \nStandard EDL struggles with multimodal uncertainty, as a *single* Dirichlet distribution—optimized using one-hot labels—cannot represent *multiple* plausible hypotheses. While refining *evidence approximation* under such constraints may offer marginal gains, it does not address the core representational bottleneck. In contrast, the FD distribution enables a *structured mixture of class-specific beliefs*, allowing the model to represent *multimodal* uncertainty naturally—without overcommitting to a single explanation. \n\n**$\\mathcal{F}$-EDL bypasses key theoretical critiques of EDL’s learning mechanism.**\nSimilar to the reviewer’s observation, recent works have argued that the core limitation of EDL stems from its inherent learning mechanism. However, $\\mathcal{F}$-EDL falls outside the scope and assumptions of these critiques due to its flexible UQ structure. For a detailed discussion, please refer to our response to **“Reviewer a4Mo, Comment 4”**\n\n**Final Takeaway.** \nWhile disentangling whether EDL’s limitation stems from *evidence approximation* or *representational structure* is challenging, we argue that structural expressiveness is the deeper bottleneck. Our results suggest that richer representational families like FD offer a promising path forward.\n\n**4. (Regarding generative EDL)**\n\nGenerative EDL is a promising direction, but it is not directly comparable to our approach, as it relies on auxiliary generative modules that introduce substantial computational and training overhead. Still, it could complement $\\mathcal{F}$-EDL—e.g., via applying generative outlier exposure to $\\mathcal{F}$-EDL for improved UQ. \n\n**5. (What justifies the use of FD distribution, and how does it relate to how evidence is extracted from the training data for a given input?)**\n\nWe adopt the FD distribution to overcome a key limitation of EDL: its inability to provide robust uncertainty estimates in *complex or unforeseen scenarios*—particularly when multiple class hypotheses are plausible.\n \nFor instance, when an input resembles both a “9” and a “7”, a standard EDL—relying on a *single* Dirichlet—produces a *unimodal* belief that may overcommit to the dominant class (see Fig. 3, panels 2-3), failing to capture the input’s intrinsic ambiguity. \n\nIn contrast, $\\mathcal{F}$-EDL models a *structured mixture* of class-specific Dirichlet hypotheses, where each component encodes the belief *“this image corresponds to class $k$”*. All components share a base evidence  $\\boldsymbol{\\alpha}$, while $\\mathbf{p}$ assigns mixture weights and $\\tau$ modulates their concentration. This structured design enables natural modeling of *multimodal* beliefs and mitigates overconfident predictions in ambiguous settings. \n\n**Evidence Extraction in $\\mathcal{F}$-EDL:** \n\nFor each input, $\\mathcal{F}$-EDL extracts and structures evidence through three conceptually interpretable stages. \n\n* **1. Extracting Base evidence**: The model estimates the base evidence $\\boldsymbol{\\alpha}$ as in standard EDL. This may initially reflect an erroneously overconfident belief in ambiguous cases.\n\n* **2. Hypotheses Construction**: A Dirichlet hypothesis is constructed for each class using $\\boldsymbol{\\alpha}$, treating each as a plausible explanation of the input.\n\n* **3. Input-Adaptive Mixing via $\\mathbf{p}$ and $\\tau$**:\n    \n    - $\\mathbf{p}$ assigns mixture weights to each class-specific hypothesis:  \n    peaked for clean inputs, spread for ambiguous ones, and near-uniform for OOD.\n\n   - $\\tau$ adjusts component sharpness: increasing in ambiguous inputs to reduce erroneous overconfidence, and remaining low for clean inputs to preserve confident beliefs. \n\nThe resulting model aligns with the *mixture formulation of the FD distribution*, highlighting it as a principled modeling choice rather than an ad-hoc generalization."}]

Author rebuttal:

Meta-review: The authors elaborate on the evidential deep learning (EDL) approach to uncertainty quantification, in which a second-order distribution is modeled by a Dirichlet distribution. In order to increase expressivity, they propose to replace the Dirichlet by the Flexible Dirichlet (FD), which is a mixture of Dirichlet distributions. They propose a (regularised) loss to me minimised, analyse formal properties of the new model, and conduct experimental studies.

The reviewers are quite positive about the paper and highlight a several strengths, such as novelty, solid theoretical analysis, and especially the impressive empirical results. They also raised a number of critical comments and concerns, though most of them could be clarified during the discussion phase. So there is no doubt that this is a good paper. 

During the discussion phase, I asked the authors a question regarding a promised improvement of their ICML submission. The answer was not really to the point. I was mainly referring to the problem that the second-order loss of EDL, without regularisation, is minimised by a degenerate solution (a Dirac delta function putting all mass on a single first-order probability), thereby suggesting zero epistemic uncertainty (Bengs et al. 2022, 2023); consequently, epistemic uncertainty is only incorporated "externally" through regularisation, and the amount of epistemic uncertainty expressed by the model is (artificially) controlled by how much one regularises. Instead, the authors answers referred to other issues, e.g., the problem of non-vanishing epistemic uncertainty, approximation of a reference distribution, etc.  

Regarding the requirement to regularise the loss to keep the epistemic uncertainty from going to zero, it seems that this issue was also subject to discussion between the authors and a reviewer. Apparently, the flexible EDL proposed in the paper still has this problem, and the way to tackle it is via the spectral normalisation. The reviewers seem to be fine with this, though I have to say that control of epistemic uncertainty through regularisation of the loss still looks dubious to me. 

Admittedly, I am also still unconvinced by the very motivation behind the generalised Dirichlet distribution, in particular the motivation for multimodality. For example, looking at the illustration in Fig. 3, why should a probability of 0.6 for digit 9 be plausible, and a probability of 0.8, too, but a probability of 0.7 not? Frankly, this doesn't make much sense to me, and it violated the commonly accepted property of coherence of beliefs. 


Bengs et al. Pitfalls of epistemic uncertainty quantification through loss minimisation. NeurIPS 2022.

Bengs et al. On second-order scoring rules for epistemic uncertainty quantification. ICML 2023.
Decision: Accept (poster)