Are Uncertainty Quantification Capabilities of Evidential Deep Learning a Mirage?
Authors: Maohao Shen
Jongha Jon Ryu
Soumya Ghosh
Yuheng Bu
Prasanna Sattigeri
Subhro Das
Gregory W. Wornell
Venue: NeurIPS 2024 poster (2024)
Keywords: Uncertainty Quantification
Evidential Deep Learning
Out-of-distribution Data Detection
Bayesian Learning
Epistemic Uncertainty
Primary area: interpretability_and_explainability
TL;DR: We affirmatively answer the question in the title by providing comprehensive analyses and empirical evidence.

Abstract: This paper questions the effectiveness of a modern predictive uncertainty quantification approach, called *evidential deep learning* (EDL), in which a single neural network model is trained to learn a meta distribution over the predictive distribution by minimizing a specific objective function. Despite their perceived strong empirical performance on downstream tasks, a line of recent studies by Bengs et al. identify limitations of the existing methods to conclude their learned epistemic uncertainties are unreliable, e.g., in that they are non-vanishing even with infinite data. Building on and sharpening such analysis, we 1) provide a sharper understanding of the asymptotic behavior of a wide class of EDL methods by unifying various objective functions; 2) reveal that the EDL methods can be better interpreted as an out-of-distribution detection algorithm based on energy-based-models; and  3) conduct extensive ablation studies to better assess their empirical effectiveness with real-world datasets. 
Through all these analyses, we conclude that even when EDL methods are empirically effective on downstream tasks, this occurs despite their poor uncertainty quantification capabilities. Our investigation suggests that incorporating model uncertainty can help EDL methods faithfully quantify uncertainties and further improve performance on representative downstream tasks, albeit at the cost of additional computational complexity.

Are Uncertainty Quantification Capabilities of Evidential Deep Learning a Mirage?
Maohao S. Jon Ryu'" Soumya Ghosh Yuheng Bu", Prasanna Sattigerf', Subhro Das?, Gregory W. Wornell'
'Department of EECS, MIT, Cambridge, MA 02139 MIIT-IBM Watson AI Lab, IBM Research, Cambridge, MA 02142 DDparrment of ECE, University of Florida, Gainesville, FL. 32611
{maohao, , {ghoshoso, prasammaas imm com, subhro. wew.ddds@bbiii com,
edu,
buyuheng@ufl.edu
Abstract
This paper questions the effectiveness of modern predictive uncertainty quantifi- cation approach, called evidential deep learning (EDL). in which single neural network model trained learn a meta distribution over the predictive distribu- tion by minimizing a specific objective function. Despite their perceived strong empirical performance on downstream tasks, a line of recent studies by Bengs al. identify limitations of the existing methods to conclude their learned epistemic un- certainties are unreliable, e.g., that they are non-vanishing even with infinite data. Building and sharpening such analysis, we 1) provide sharper understanding the asymptotic behavior wide class of EDL methods by unifying various be better interpreted
objective functions; 2) reveal that the EDL methods an oi-distribution detection algorithm based on energy-based based--odeell and 3) conduct extensive ablation studies to better assess their empirical effectiveness with real-world datasets. Through all these analyses, we conclude that even when EDL methods are empirically effective downstream tasks, this occurs despite their poor uncertainty quantification capabilities. Our investigation suggests that incor- porating model uncertainty can help EDL methods faithfully quantify uncertainties and further improve performance on representative downstream tasks, albeit at the cost of additional computational complexity."
1 Introduction
Accurate estimation of uncertainty in the prediction becomes more crucial to enhance the reliability of predictive model, especially for high-stake applications such medical diagnosis [1, 2]. Among several approaches proposed, a class of uncertainty estimation methods under the category of evidential deep learning (EDL) has recently gained attention [3], due their claimed advantages over other methods. EDL methods typically learn single neural network that maps input data to the parameters of meta distribution, which is distribution over the predictive distribution. The EDL methods generally claim the following advantages. (1) Computational efficiency: they bypass the expensive sampling costs associated with Bayesian ensemble-based methods by training
"The two authors contributed equally this work.
This author is currently affiliated with Merck and can be reached at Soumya. mom.oomerck.com 'The code to replicate the experiments is available on tttpps://githi htmm//achhas2/EELLLLLA Mirae..
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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single neural network and estimating uncertainty with a single forward pass. (2) Promising empirical performance: they achieve superior results on downstream uncertainty quantification (UQ) tasks, particularly in detecting out-of-distribution (OOD) data. (3) Ability to quantify different uncertainties: EDL methods can quantify distributional uncertainty versus aleatoric uncertainty, by representing them as the spread and mean of an estimated meta distribution over the prediction, respectively.
Despite the above benefits, a line of recent works has reported theoretical limitations and pitfalls of uncertainties learned by EDL methods, including the issue of non-vanishing distributional un- certainty [4], a possibility of non-existence of proper scoring rules for meta distributions [5], and a gap between learned uncertainty and an ideal meta distribution [6]. While these works call for the attention of the UQ community for the raised issues, a comprehensive theoretical understanding of the learned uncertainties from this type of UQ model is still lacking, as the existing analyses focus on a restricted subset of objective functions in the literature. Moreover, they did not explain the empirical success of the EDL. methods at downstream tasks, such as OOD detection, despite such issues.
In this paper, we provide a simpler and sharper theoretical characterization of what is being learned by representative EDL methods, and re-examine the empirical success of the EDL methods based on the analysis. More concretely, our contributions are threefold:
1. Theoretical Analysis: In Sec. 4, we provide a unifying perspective on several representative EDL methods proposed in the literature, establishing an exact characterization of the optimal meta distribution defined by existing methods. This reveals that existing methods enforce the meta distribution to fit sample-size-indeeeddeer target distribution (Theorem 5.1). This analysis covers a wider class of objective functions and modalities, and is sharper than the prior analyses [4, 6].
2. Empirical Investigation: In Sec. 5, we further provide empirical evidence to point out the fundamental limitations of the learned uncertainties by EDL methods, and present several findings showing that existing EDL methods are essentially OOD detectors and hence exhibit their pitfalls.
3. Insights and Solutions: In Sec. 6, we explain why model uncertainty seems inevitable for faithful UQ and how we can improve the EDL method accordingly. We propose a new model uncertainty based on the idea of bootstrap, and demonstrate that an EDL. model can well distill its behavior and achieve superior UQ downstream task performance compared to the existing methods.
2 Related Work
We summarize the literature that is closely aligned with the scope of this paper. We refer the reader to Sec. B in Appendix for an overview of classical UQ literature and recent survey paper [3] for comprehensive review of EDL methods.
EDL Methods. EDL methods, mainly applied in classification settings, utilize a single neural network to model Dirichlet distributions over label distributions, which can be divided into three categories. (I) NOD-data-dependeer
methods: Earlier works such as Forward Prior Network [7], and Reverse Prior
Network [8, 9]. proposed to train a model to output sharp Dirichlet distribution for in-distribution (ID) data and flat Dirichlet distribution for OOD data. (2) DOD-data-fres methods: Subsequently. several methods without OOD data were proposed with various training objectives, including the MSE loss with reverse Kullback-eebllrr (KL) regularizer [10, 11], the "VI" loss [12, 13, 14], and the "UCE" loss [15]. (3) Distillation based methods: are motivated by training a single model to mimic the behavior of classical UQ approaches, including END2 [16] that emulates ensemble method, and
S2D [17] that emulates (Gaussian) random dropout method. EDL methods have also been explored for regression problems [18, 19, 20].
Critiques of EDL Methods Recently, several works have raised concerns about the quality of uncertainty learned by EDL methods. Bengs et al. [4] pointed out the learned distributional uncertainty does not vanish even in the asymptotic limit of infinite training samples. Bengs et al. [5] provided further theoretical arguments for why proper scoring rule for learning the meta distribution might not exist. More recently, Jürgens et al. [6] argues that the learned uncertainty by EDL methods is inconsistent with reference distribution. In a similar spirit to these critiques, we offer a sharper analysis to characterize the exact behavior of EDL methods. Our analysis can subsume, generalize, and simplify the existing analyses. See Appendix A for more in-depth review of these works.

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3 Problem Setting and Preliminaries
In the predictive modeling, we aim to learn the label distribution p (z) over y using data D = {<x,, (e)ssy4)>> } drawn from an underlying data distribution p(z)p(y((>) over X' x y. Here, y is label set; y = [c] == (1.....Cr C} for classification for some C 2 and ) = R for regression. A AC-1 denotes the probability simplex over [C]. In addition to accurately learning the conditional distribution fly (ar) := p) we wish to quantify "uncertainty" of the learned prediction. We focus on classification in this paper, but some of our analyses also apply certain existing EDL methods for regression [19, 20J. For the sake of clarity, we defer all related discussion on regression and beyond to Appendix F.
Bayesian and Ensemble-based Approaches. Different UQ methods define predictive uncertainties based on different sources of randomness. The Bayesian approach is arguably the most widely studied UQ approach, in which a parametric classifier p[y|æ, xb) is trained via the Bayesian principle, and inference is performed with the predictive posterior distribution p(y (z, D) == J p xr ())()))))))))))))))))))))))))) du, where p( is the model posterior distribution, induced by the prior p(v) and likelihood p(D) v-). Ensemble-based methods assume different distribution p(v (D) to generate random models given data, e.g., training neural networks with different random seeds. As alluded earlier, both Bayesian and ensemble approaches are computationally demanding due to the intractability of p(v) (D) and the need for computing the integration over s.. Moreover, p(y| D) can capture aleatoric uncertainty (or data uncertainty), and the amount of spread over p(ylar, v/) induced by the model uncertainty of p(1/) (D) is regarded as epistemic uncertainty (or knowledge uncertainty) for its prediction.
EDL Approach. The EDL approach further decomposes the predictive posterior distribution as ply x. x5 = f p(T [z. v p)y( TT) dr, where p(ær [r, v) is meta distribution (or called second- order distribution [4, 5]) over the prediction x, and p(y(T)) ) is a fixed likelihood model. For classification, π E A C--1 is probability vector over C' classes, p(y π = Ty is the categorical likelihood model, and p(T[zz, 15) is a distribution over the simplex A C-1, Oftentimes, p( (z, 1)) chosen as conjugate prior of the likelihood model p(y(TT) like the Dirichlet distribution for classification [7. 8, 13, 15, 20], but sometimes not [10, 18, 11]. Given the full decomposi- tion, p(y xx D) = IJ p(y π p((T) z. ())(() du drt, the uncertainty captured in p(y) RT) called aleatoric uncertainty. in p(TT 2. 1b) is called distributional uncertainty. a kind of epistemic uncertainty. However, EDL methods often assume the best single model v/* learned with data D, without any randomness in P[x) or formally setting it to be 8( - xb*) [7, 3]. It then aims to train the meta distribution y0 (< (r. 3/) under certain learning criteria so that encodes less uncertainty for ID points and more for OOD points. While this simplification allows its computational efficiency over the classical methods, as we will argue later, no randomness assumed in the model p( / (D) lets all the methods in this framework learn spurious distributional uncertainty. Since a single model vb is assumed, we use a frequentist notation Pa ( ( (z) instead of p( [x, vb) in that context.
4 New Taxonomy for EDL Methods
In this section, we propose a new taxonomy to understand different EDL methods (for classification) in systematic way. Ignoring the choice of base architectures, we identify that the key distinguishing features in EDL methods are (1) the parametric form of the model and (2) the learning criteria. A wide class of EDL methods can classified with the taxonomy as in Table 1. Several theoretical implications and empirical consequences of the new taxonomy will be investigated in the next section.
Table 1: New taxonomy of representative EDL methods. C(w) in Eq. (6) subsumes these special cases.
Method (name of loss)
likelihood
D(.-)
prior Okn
%ood
(c.( parameterization
FPriorNet (F-KI. loss) [7]
categorical
fwd. KL.
= 1c
0
direct
RPriorNet (R-KL loss) [8]
categorical
rev. KL.
= 1c
> 0
direct
EDL (MSE loss) [10]
Gaussian
rev. KL
1c
0
direct
Belief Matching (VT loss) [12, 13]
categorical
rev. KL.
E ERS
0
direct
PostNet (UCE loss) [15]
categorical
rev. KL.
= =1c
0
density w/ single flow
NaPP (UCE loss) [20]
categorical
rev. KL
= 1cc
-0
density w/ multiple flows
3

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Criterion 1. Parametric Form of Meta Distribution. For classification, one distinguishing feature is the parametric form of agp Dirichlet distribution Pr: (r( Dir((T: (2)))). Earlier works [12, 7, 8, 13, 10] typically parameterize (x) by direct output of a neural network, e.g.. exponentiated logits; we call this direct parameterization. Later, Charpentier et al. [15] brought up a potential issue with the direct parameterization that a can take arbitrary values on the unseen (i.e., OOD) data points. They proposed a more sophisticated parameterization of the form to explicitly resemble the posterior distribution update of the Dirichlet distribution ag (x) < Oxo + N (2)), where, for y E [C]. (N.(2)) == No (fyz (x)))()) (r)] with (0y) == Ny/N, Ny denotes the number of data points with label y. .N : E yyjor Ny. ++ fe, (z) feature extractor, and pv, (zly) a tractable density model such as normalizing flows [21] for each y E [C]] We call this density parameterization. In Sec. 5, we carefully examine the effectiveness of density parameterization.
Criterion 2. Objective Function. The desired behavior of EDL model is to output sharp Ps Ps( (z) it is confident, and fall back to output prior distribution p( π if it is uncertain at an unseen data æ. To achieve this, various objectives have been introduced with different jargon and motivations, e.g.:
1. Prior Networks (PriorNet) [7. 8] aimed explicitly encourage pa pa( (T]2)) ()) to be diffused prior p(TT) for OOD data, and more concentrated Dirichlet distribution Dir(Tr; Okg + vey) for ID data, with Qrg 1 and ey as the one-hot true label, by minimizing
Ep(a.v) (D((ir(T 0ko + vey), Pa (7 [x)) + Yood E pooa()) (D(iiii 0x)) Pe (T [x)]]
(I)
for D(p,g) q) = Div q) (forward KL) in [7], and D(p, g) D || p) (reverse KL) later in [8]. It
known that PriorNet with forward KL requires an additional auxiliary loss log po(aaz) to ensure
high accuracy, and the reverse-KI KL version can outperform without such term.
2. The "Evidential Deep Learning" paper [10] proposed the MSE loss with a reverse KL regularizer:
MSE (1/,,, w) == (mj |T e + xD (Tr (<xx)) || Dir(Tr;
3. Belief Matching [12, 13] proposed VI loss justified by variational inference framework:
(v x, v) == (xle))
log Ty
(T]p) Dir(T; ao)).
4. Posterior Networks (PostNet [15] and NatPN [20]) proposed the uncertainty cross entropy (UCE) loss, motivating it from general framework for updating belief distributions [22]:
(UCE(V: a,, v) = (m))) log 1 Ty
Ah(po(T) (z:))
The hyper-parameter À >>> 0 in objectives 2, 3, and 4 balances the first likelihood term which forces (T((z) to learn the label distribution and the second regularizer term promotes p [[zz) to be close to the prior p(T)). Below, reveal that the seemingly different objectives 1, 3, 4 are exactly cquivalent, while objective Eq. (2) (different likelihood) can also be unified in a single framework.
A Unifying View. We now provide a unifying view of the fairly wide class of representative objective functions. First, for convenience of analysis, we define the tempered likelihood: for > 0, define
p(0) (T. (T..)
p(T) (yl T
(0))
( (v) I pl (,, y) dr'
where (xx) (TT (T, v) ] T((T) E, p"
π dm
Objectives 3, assume the likelihood model is categorical, i.e., p(u(TT) = my. by the conjugacy of the Dirichlet distribution for the multinomial distribution, it is easy to check that pl (Tly) = Dir(m:; Qn + vey), where ey RC is the one-hot vector activated at y E [C]. Objective used Gaussian likelihood model p(y πT) = (N π. Tc),, which does not admit a closed form expression for p(*) (Tlyy) ly). The prior distribution is usually defined as p( (TT)= Dir(T;; ax)) with ao 1 (all-one vector). Other choices of prior were also proposed to promote some other desired property [9].
We now introduce unified objective function
((() : Bp(x.)) (D(p (T (1ly) Pm [x)]] + Yood (Pp
for some divergence function D)..), tempering parameter v >> 0, and an OOD regularization parameter Yood 2 0 with a distribution Pood for OOD samples. The following theorem summarizes how this unified objective subsumes several existing proposals. The proof is deferred to Appendix D.

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Theorem 4.1 (Unifying EDL Objectives for Classification). Let p( Dir(T:: 1c).
1. Let plylT)) Ty and let "lood > O. With D(p. D(p l| q) (forward KL) and D(p, (p.g)
D(q ll p) (reverse KL), ((ú) equivalent to the objective (Eq, (1) forward PriorNet (FPrior- Net) [7] and that of reverse PriorNet (RPriorNet) [8], respectively.
2. Let plylTt) = N(ey; o2)c).. and let Food = 0, [(v) is equivalent to the objective LMSE(V) Ep(as)) (MMEEMM x. y)] (Eq. (2)).
3. Let ply TT = Ty. V X-1 and let "ood = O. With D(p, g) = D(q ll p) (reverse KL), L(6) is equivalent to the objective Lv((+) == Ep (evi I, y)] (Eq. (3)).
Let p(y) TT) = Ty v = x-1, and let "ooo = O. With (p, 9) = D(g | p) (reverse KLJ,, ((1b is equivalent to the objective (UCE(V) Eppe (f(ee))/uCCE( (fucE(1 y)] (Eq. (4)).
We first remark that the reverse KL divergence captures most of the cases, except the forward KL PriorNet. which is known to be outperformed by PriorNet with reverse KL. Hence. it suffices to focus on understanding the reverse KL objective. Second, the VI loss in Eq. (3) and UCE loss Eq. (4) turn out to be equivalent to the RPriorNet objective, where the tempering parameter and the regularization parameter À is realted to reciprocal À = -ー see Lemma D.1. Overall, this theorem reveals that different motivations, such as VI or Bayesian updating mechanism of belief distributions, were not very significant, and those objectives turn out to be equivalent to the simple divergence matching in Eq. (6). Given this, distinguishing features that better classify different EDL methods are the choices of (1) likelihood model, (2) prior, (3) use OOD data, and (4) model parametric form, as shown in Table 1. The recent survey [3] also offers unified view of different objective functions with these features. However, the dichotomy between "PriorNet-type methods" [7, 8] and "PostNet-type methods" [12, 13, 15, 20, 14] in [3] may not effectively contrast different EDL methods compared to our taxonomy.
We acknowledge that there exist other objective functions that might not be covered by this unified view. We defer the discussion on another line of representative work, including the distillation- based methods [16, 17] to Sec. 6. A notable exception that is not subsumed by this unification is FisherEDL [1]], which was recently proposed to use a variant of the MSE loss [10] by taking into account the Fisher information of p (r)::): see Appendix F for discussion. Finally, similar reasoning can be naturally extended to unify different EDL objectives for other tasks such as regression and
count data analysis, including [19, 20]. We defer detailed discussion to Appendix F.
5 Rethinking the Success of EDL Methods
In this section, aim reveal the secrets of EDL methods' empirical success through combination of theoretical and empirical findings. As alluded to earlier, our empirical investigation focuses on the reverse-KL type EDL methods, i.e., RPriorNet [8], Belief Matching (BM) [13], PostNet [15], and NatPN [20], which are representative and widely used in the literature.
5.1 What the "Optimal" Meta Distribution Characterized By The EDL Objectives?
Based on the unification in Eq. (6), we can provide sharp mathematical characterization of the optimally learned meta distribution for a wide class of EDL objectives. A direct and important consequence of the divergence minimization view is that we can now characterize the "optimal" behavior of the learned meta distribution of wide class of EDL objectives, provided that the global minimizer is achieved in the nonparametric and population limit. The following theorem is proved in Appendix E.
Theorem 5.1.
For any prior (T) and likelihood p(yl IT), we have min Ep( (TT (x) (T(T)))) min (P p (+[x))) $
p(T)) exp [lo ((y)T)))
where p² ( (T| ] p( expl ( (xp(l plogooogpppp ply
In words, Theorem 5.1 states that when the model meta distribution p. (T] is trained with the Leverse-LL objective, it is forced to fit fixed target meta distribution p'(T/z).
5

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PostNet
cslatmmic uncortainty
Test acc.
-10-1
0.4
1.0
102
10f sample size
107
10
10
sample size
sample
sample size
(a) Epistemic Uncertainty: CIFARIO
BM
APriorNet
PostNet
NatPN
05ca uncertaity
10-1
10-1
10
10
10-2
20-1
10-110-3
10-2
10-2
10-
10-2 10-1
A
(b) Aleatoric Uncertainty: CIFARIO
Figure 1: Behavior of Uncertainties Learned by EDL methods on Real Data. (a) EDL methods learn spurious epistemic uncertainty, wherein uncertainty does not vanish with an increasing number of observed samples, contrary to the fundamental definition of epistemic uncertainty. (h) Instead of constant, EDL methods learn model-dependent alcatoric uncertainty that depends on hyper-parameter X, contrary the fundamental definition alcatoric uncertainty. Similar hehavior holds 2D Gaussian data (sce Figure in Appendix H.
Example 5.2 (Categorical likelihood). If we consider p(л) = Dir(T: 0o)) and p(y]T) = Ty (cat- egorical likelihood), we have p" (π|x) Dir(T;; cxo #z)) where n7x) Epl x) ... denotes the true label distribution, Theorem 5.1 implies that
min BinE (( (p (T|I)) [ = min (0
+n(r)))) (8)
In particular, with the most common Dirichlet parameterization Pe (T[(z) Dir(Tr; any (z)), this
shows that ag (z) is forced to match the ccleed-and-shiffe version + (7)xx) of the conditional label distribution ()x)) as the fixed target, under the categorical likelihood model. A similar argument still applies to the Gaussian likelihood model of [10], but the fixed target distribution p [x/x in Eq. (7) does not admit closed-for fored-form expression unlike the categorical likelihood.
An immediate consequence of the Theorem 5.1 is that neither epistemic uncertainty nor aleatoric uncertainty quantified by EDL methods is consistent with their dictionary definition.
Implication EDL Methods Learn Spurious Epistemic Uncertainty
First, epistemic uncertainty for ID data, by definition, should monotonically decrease and eventually vanish the number of observations increases. However, Theorem 5.1 implies that, even with infinite data, the Iearned "distributional uncertainty" would remain constant for ID data. We also empirically confirm such behavior; see Fig. (a)) Specifically, we sample data of varying sizes to train the EDL models and evaluate their test accuracy and averaged epistemic uncertainty (mutual information) on held-out test As observed in Fig. I(a), epistemic uncertainties quantified by EDL methods are almost constant with respect to the sample size and never vanish to 0, regardless of the increasing test accuracy. This suggests that practitioners cannot rely on the learned distributional uncertainty to determine if the model is lacking knowledge. We remark that Bengs et al. [4] identified a similar issue in the EDL methods specifically for the UCE loss of PostNet [15, 20], which the reverse-Ki KL objective for a 1 in our view, is not appropriate in similar spirit. Theorem 5. can be understood as more general and sharper mathematical characterization of the behavior of the reverse-Ki KL objective.
Implication EDL Methods Learn Spurious Alcatoric Uncertainty
Second, EDL methods quantify aleatoric (data) uncertainty as (r) T))], where H denotes the Shannon entropy [23]. Eq. (8) reveals the optimal meta distribution as Per (er((T)))) (2)))
'We remark that the proof's and corresponding arguments in [4] are erroneous, where the errors stem from equating the differential entropy of Dirac delta function 0, instead of co;; Appendix A for more details.

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Dir( au + +m(2)), suggesting that the aleatoric uncertainty quantified by EDL methods would also depend on model or algorithm's hypee-parameter (A = v- This is inconsistent with the definition of aleatoric uncertainty, which should be fixed constant capturing the irreducible uncertainty by underlying label distribution p(y]ar). As shown in Fig. (bb, we empirically demonstrate that the aleatoric uncertainty quantified by EDL methods vary with X. In conclusion, our analysis in this section suggests that EDL methods do not behave as a reasonable uncertainty quantifier since they cannot faithfully quantify either epistemic or aleatoric uncertainty.
5.2 EDL Methods Are EBM-Based OOD Detector Rather Than Uncertainty Quantifier
As we have shown so far, the EDL methods with Dirichlet prior and categorical model typi- cally aim to fit the model meta distribution (z) to a fixed target oko + 1/n(z), which is ap- proximately vn( for v sufficiently large. Since the induced model predictive distribution is (m))a
(v(x)) (m)a) (p(()p(())))
(1) (z)), it can be understood that the reverse-Ki KI n(x) = =p(y(=) for accurate prediction, while letting the summation 1 TT (2) large (i.e., (x) v) for ID data, and small for OOD data, e.g..
enforces py (v] to fit
1 C, if there exists explicit OOD regularization.
In the OOD detection literature, there exists an mnerg-baaeed-mode -odel (EBM) based algorithm aim to achieve the exactly same goal [24]. They consider a standard classifier with exponentiated logits Liu et al. [24] related the denominator to a free energy Es (x) == log Il [. and proposed to train model so that pa (ylar)
(z) whose prediction is given as (y(æ)) == ( (z)/1[99
accurately captures p(ylæ), while minimizing the energy (x) for ID x's, and maximizing for OOD,
They proposed to minimize
- px (logp) po x)] + T [Bp(()]mmxx(( Es(x) - ma) + ( Mood - Es )1)
where 0 and Mood > my 0 are hyperparameters. This reveals that this EBM-based OOD framework has an almost identical learning mechanism to the EDL methods; setting Ex(x) =
log (z), maa log 1/ and oood log C' makes the correspondence explicit.
This resemblance suggests that the EDL methods can be better understood as an EBM-based OOD detector with the additional layer of Dirichlet framework for computational convenience rather than statistically meaningful mechanism that can faithfully distinguish epistemic uncertainty and aleatoric uncertainty. Below, we elaborate on two particular implications based on this connection.
Implication EDI Methods Always Prefer Smaller Hyper-Parameter À
Average AUROC Score
PostNet
Average AUROC Score
NatPN
Average AUROC Score
RPriorNet
1.00
1.00
0.75
0.9
0.95
Average AUROC
0.65
0.8
0.90
0.90
0.7
0.55
0.85
0.85
0.6
0.80
0.45
0.5
0.75
0.75
0.35
0.4
10
4 10
10
10
10
10-3
10-2
10
10
10-3
10-2
10
10
10-3
10-2 - A Value
Value
A Value
Value
Figure 2: OOD Detection Performance v.s. Hyper-parameter À on CIFAR1O. The z-axis represents the increasing À value, and the y-axis represents the Average AUROC score of OOD detection tasks. EDL Methods' uncertainty quantification performance are sensitive hyper- parameter X, while generally benefit from small A.
Just as the EBM- bosedd OOD detection algorithm needs to manually define pppr--prr Mood and mid, EDL methods also significantly rely on hyper-parameter tuning; recall the target distribution Dir(TT: Qn (n) with v = X is determined by the regularization parameter X, we aim to explore the dependency of EDL methods' OOD detection performance on the choice of X. Specifically, we directly conduct such an analysis on real data using CIFARIO as ID data, and use four OOD datasets; see Appendix G for details. By varying À across (1e-4, 1,-,,1e-- 1e-3, 1e 1e-1), we train an EDL model and evaluate its OOD detection performance on these OOD datasets. The performance in terms of average AUROC score with respect to the increasing value of X are shown in Fig. 2. The result shows that using smaller values of À always improves the OOD detection performance, suggesting that the EDL model is essentially encouraged to fit its output og (x) to an extremely large target
7

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Qo + (()) so that (z) ~ A-1 > 1 for ID data, and such behavior seems to benefit the downstream task performance.
Implication 4: Impact of Specific Objective on UQ Performance is Less Significant
Given the close resemblance to the OOD detection algorithm, a notable difference is the reverse- KL learning criterion used by the EDL methods. In Appendix H.2, as another ablation study, we investigate if the reverse-KL objective induced by the Dirichlet framework has a significant practical impact, or other saacclee-fammewo---ndeeeeeeed suffice for the downstream task performance.
objectives which promote the same behavior
5.3 Are EDL Methods Robust for OOD Detection?
Lastly, we investigate the robustness of the auxiliary techniques like density parameterization and OOD regularization of EDL methods for the downstream task performance. Note that in the empirical result with real data presented in Fig. 9 in Appendix, neither OOD regularization (RPriorNet vs. BM) nor density parameterization (PostNet/NalPN vs. BM) demonstrates a clear advantage of deploying such techniques. Thus, we further investigate this counterintuitive phenomenon and discover that the performance of these EDL methods on real data is hindered by certain limitations of the auxiliary techniques they use. Namely, we find that OOD-data -etaadddeendeet methods [8] are sensitive to choice of model architectures, and methods [15, 20] using density models may not perform well even for moderate dimensionality. We defer the detailed discussion to Appendix H.3.
EDL Methods Will Benefit from Incorporating Model Uncertainty
Through the series of analyses in the prior section, we attempted to provide comprehensive understanding of EDL methods, revealing their key limitations. A reader then may ask an important question: What is the fundamental issue in these EDL methods that cause all these problems? As we alluded to in Sec. 3, we identify that the issue arises from that all of the EDL methods discussed in Sec. 4 and Sec. 5 assume
model uncertainty p(v( in the decomposition of predictive posterior distribution p(y|æ, v-) = ʃs p(yl p)p(p)/pp (T (r. 1b pp(p eb (D) dub dr. A proper definition of distributional uncertainty should be based
the induced posterior distribution p( (r. D) := J 1((T) Jr, v p) 'D) dıb over the prediction π at x, given the dataset D. Without the randomness in the model, however, ie., setting p(t|D) t 8(0- " the induced distribution p( T [a:, D) becomes degenerate as p( xr, ). This simplification is often rationalized for the computational efficiency, but reveal, it renders the distributional uncertainty inherently ill-defined in this framework. The existing EDL methods thus have no choice but to train the "UQ" model p [T|x)) to fit to an artificial
target Dir( π:; ax + +r))))) Consequently, the learned distributional uncertainty cannot possesses a statistical meaning, and can be better interpreted as free energy in the EBM-based OOD detector.
Model Uncertainty Can Induce A Proper Distributional Uncertainty. This strongly suggests that it is inevitable assume a stochastic procedure pD) properly define the distributional uncertainty p( [z, D). We remark that, to expect the distributional uncertainty p[T|x; D) to exhibit desirable behavior, i.e., getting concentrated for ID data and remaining dispersed for OOD data as (D] + oc, we implicitly assume that the stochastic algorithm (( (D) would behave as follows: as [ + xc. P)) will become supported on subset of models v that agree upon the prediction for ID data while disagreeing on OOD data. Under this assumption, the induced distributional uncertainty will be consistent with the dictionary definition of epistemic uncertainty.
Revisiting Distillation Based Methods. The downside of such an approach is that approximating p(T I2, D) can be computationally intractable due to the high-ddmensiioal integration with respect to the stochastic algorithm PD+ which means that a practitioner needs to generate (or train) and save multiple models to approximately emulate the randomness. In this context, the EDL framework, which aims to quantify uncertainty by a single neural network, can be used to distill the properly defined distributional uncertainty. Indeed, this is what has been explored by another line of EDL literature called distillation-basee methods [16, 17], which were originally proposed to emulate the behavior the ensemble methods. Our analyses strongly advocate that considering a stochastic algorithm p() and training a single meta distribution trying to fit the induced distributional uncertainty to expedite the inference time complexity is the best practice for the EDL
8

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CIFARIO
CIFARI00
RPriorNet
RPriorNeet
8M
BM
PostNet
PostNet-
NatPN
NatPN
EDL
EDL
Fisher-E0 EDL
Fisher-EDL
END2
END2
S20
Beatstrap-Dizt
Boctstrap-Distill
(New)
(New)
0.5
0.7
0.8
1.0
0.4
0.6
0.7
0.8
Average AUROC Score
Average AUROC
Figure 3: Comparison of Different EDL Methods OOD Detection. Distillation based methods. including new proposed Bootstrap-Diise Distill method, demonstrate clear advantage other classical EDL methods. Similar hchavior holds for selective classification task.
CIFARIO
CIFARI0O
RPriorNet
RPrionNet
BM
BM
PostNet
PostNet-
NetPN
NN
EDL
EDL
Fishen-DL
END2
END2
Bootstrap Distil
Bootstrap-Dast (New)
(New)
0.84
0.86
0.88
0.90
0.92
0.80
0.92
0.84
0.86
0.88
AUROC Score
AUROC Score
Figure 4: Comparison of Different EDL Methods on Selective Classification. Distillation based methods, including new proposed Bootstrap-Distil method, demonstrate clear advantage over other classical EDL methods.
framework to faithfully capture uncertainties. More precisely,
fit UQ model po (z)
to p(T lz, D) through the forward-KI KL objective and Monte Carlo samples, i.c., by minimizing Ep(a) (D(((T (z. D) po(T)())) = - Σm En
log logpe(n, (M (+1=;) + (const.) with respect to 6,
where + na (-Iz) is classifier corresponding to a randomly generated model 2o ~ p( ( DD) [16].
Examples of Stochastic Algorithms (D): Old and New. There are two popular proposals for
P()) in the literature: (1) EnD² [16] considers randomness in random initialization and stochastic optimization, which is called ensemble [25], and (2) S2D [17] considers a random dropout [26] applied single network. We propose yet another algorithm based on the frequentist approach bootstrap: given the training dataset D, the procedure randomly samples M different subsets (D,} M
of size without replacement, and train model e based on D, for each j. Unlike the previous proposals of p( ( (D), the bootstrap method aims to leverage the internal consistency among the ID data, beyond the randomness induced by optimization and architectures. We note that this is inspired by concurrent work of Jürgens et al. [6], which recently proposed an ideal meta distribution. The proposed bootstrapping can be understood as a practical method for approximating such behavior with finite samples. In the next section, demonstrate that the new Bootstrap Distillation method performs almost best on both OOD detection and selective classification tasks.
7 Comprehensive Empirical Evaluation
In this section, we conduct comprehensive evaluation of the existing EDL methods, including the new proposed Bootstrap Distillation method, on two UQ downstream tasks: (1) OOD data detection: identify the OOD data based on the learned epistemic uncertainty; (2) Selective classification: identify the wrongly predicted test samples based on total uncertainty, as wrong prediction can occur from either high epistemic or high data uncertainty, or both. These results corroborate several key findings and insights presented in this paper. The detailed experiment setup are provided Appendix G, and additional experiment results included in Appendix H.

---

Baselines. We include a wide range of EDL methods as baselines: Classical methods: (1)
RPriorNet [8], (2) Belief Matching (BM) [13]. (3) PostNet [15]. (4) NatPN [20], (5) EDL [10], and (6) EDDL [11]. Distillation-based method: (1) EnD>² [16] and (2) S2D [17]. All baseline
results are reproduced using their official implementation, if available, and their recommended hyper-parameters.
Benchmark. We consider two ID datasets: CIFARIO, and CIFARI00. For the OOD detection task, we select four OOD datasets for each ID dataset: we use SVHN, FMNIST, TinylmageNet, and corrupted ID data.
We elaborate metrics for quantifying different types of uncertainties in Section C. We evaluate the UQ downstream performance through the Area Under the ROC Curve (AUROC) and Area Under the Precision- Curve (AUPR), where we treat ID (correctly classified) test samples as the negative class and outlier (misclassified) samples as the positive class.
Evaluation Metric.
Results and Takeaway. The result is summarized in Fig. 3 and Fig. 4. We present the average AUROC score across four OOD datasets for the OOD detection task, and the AUROC score for the selective classification task. More numerical results can be found in Table 2, 3, and 4 in Appendix H. This set of evaluations corroborates our key findings and insights in the previous sections.
Firstly, the results show that varying the likelihood model does not significantly impact performance (BM vs. EDL and Fisher-EDL), supporting the discussion in Sec. 4. Secondly, classical EDL methods achieve comparable performance regardless of the auxiliary techniques, such as density parameterization (PostNet, NatPN), or leveraging OOD data (RPriorNet); this validates our finding in Sec. Thirdly, distillation- based methods, particularly EnD² and the new Bootstrap Distillation, demonstrate superior performance over other baselines, especially on CIFAR 100, thereby validating the insights provided in Sec. 6. Moreover, empirical amalysis confirms that the Bootstrap Distilla- tion method can faithfully quantify epistemic uncertainty, as illustrated in Fig. 13 in Appendix H, effectively addressing the limitations identified with other EDL methods in Sec. 5.1. Finally, note that the performance of Bootstrap Distillation comes at a higher computational cost due to training multiple bootstrap models; see Fig. 14 in Appendix H.
8 Concluding Remarks
In this work, we revealed that the uncertainty learned by most of the existing EDL methods bear no statistical meaning. The key theoretical insight is based on the unification of representative EDL objectives in Sec. 5.1. Given this, we suggested that EDL methods can be better interpreted as energy-based OOD detection algorithms, which can explain the reported empirical successes of EDL methods from different perspective. Additionally, we demonstrated that the performance of EDL methods is sensitive to the choice of auxiliary techniques, further raising questions about their robustness. Finally, we identified that the aforementioned issues with EDL arise from ignoring the model uncertainty for computational efficiency, and argued that the distillation based methods could potentially remedy the issues, at the cost of additional complexity.
Overall, this work calls researchers' attention to carefully recxamine the capabilities and limitations of the EDL approach at large. For practitioners, EDL methods can still be utilized as efficient algorithms for specific applications, such as OOD data detection. However, when considering EDL methods to build reliable machine learning agents based on their UQ capabilities, practitioners should be aware of their limitations, which contrast with the common belief that EDL approaches can accurately learn and distinguish between epistemic and aleatoric uncertainty.
We conclude the paper with a few remarks on the theory of the Bootstrap- -istiil method. As we empirically showed, it can resolve the common issues of the EDL methods with improved downstream task performance, and we thus believe that careful theoretical analysis of its behavior would be a fruitful direction. While it is relatively easy to argue that the epistemic uncertainty would vanish when the sample size grows to infinity with the bootstrap procedure, we believe that more sophisticated asymptotic analysis for vanishing epistemic uncertainty could be carried out with overparameterized neural networks, adopting a similar setting in [27]. That is, if the trained model's prediction can be shown to be asymptotically normal in the limit of the sample size, one can argue that the uncertainty captured by bootstrap behaves as Gaussian of vanishing variance in the sample limit. This implies a naturally vanishing epistemic uncertainty. We leave this as future work.
10

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Acknowledgments and Disclosure of Funding
The authors appreciate the constructive feedback from anonymous reviewers, which helped improve the manuscript. MS and JJR thank Viktor Bengs for helpful discussions on the literature. This work was supported, part, by the MIT-IBM Watson AI Lab under Agreement No. W1771646.
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Appendix
In-Depth Review of Recent Critiques on EDL Methods
14
A.I Review of [4] "Pitfalls of Epistemic Uncertainty Quantification Through Loss Min- imisation"
14
A.2 Review [5] "On Second-Order Scoring Rules for Epistemic Uncertainty Quantifi-
cation".
15
A.3 Review of [6] "Is Epistemic Uncertainty Faithfully Represented by Evidential Deep Learning Methods?"
15
B Literature Review for Classical UQ Methods
16
C Definition of Uncertainty and Its Measures
16
D Proof of Theorem 4.1
17
D.1 Prior Network Objective
17
D.2 Variational Inference and EDL Objective
17
D.3 Uncertainty Cross Entropy Objective
18
Proof of Theorem 5.1
18
An Extension For General Observation Models
19
F.1
Example: Classification
20
F22
Example: Regression
20
Experiment Setup
20
G.1
Data Processing
20
G.2
Model Architecture
G.3
Implementation Details
H Additional Experiment Results
23
H.1
Omitted Results in Sec. 5.1
23
H.2
Ablation Study in Sec. 5.2
23
H.3
Ablation Study in Sec. 5.3
24
H.4
Bootstrap Distillation Method Faithfully quantify Epistemic Uncertainty
25
H.5
Bootstrap Distillation Method Benefits from More Samples
26
H.6
Detailed Results on UQ Downstream Tasks
27
A In-Depth Review of Recent Critiques on EDL Methods
In this section, conduct compressive review of the recent critiques of EDL methods [4, 5, 6] that are closely relevant to our paper. We begin by acknowledging the contributions of [4], which pioneered the theoretical study of EDI. methods. However, there several limitations in these works to be noted: (1) Some of the main arguments presented in [4] are invalid given errors in the corresponding proofs. (2) The analysis in these works purely theoretical. They do sufficiently explain the empirical behaviors of EDL methods in practical application.
(3) The analyses do not include some representative EDL approaches in the literature, such as the distillation-basse methods discussed in our paper. (4) While these works effectively point out various limitations of EDL methods, none of these works provide insights concrete solutions address these issues.
A.1 Review of [4] "Pitfalls of Epistemic Uncertainty Quantification Through Loss Minimisation"
In [4], the authors studied the property of meta distribution characterized by minimizing a certain loss such as the UCE loss [15] for classification. The authors proposed a definition of two desirable properties for faithfully learned meta distribution should satisfy [4, Definition 1]: First, for some distributional uncertainty measure, the expected uncertainty captured in the meta distribution over observations should monotonically decrease as the data size increases. Second, as the data size goes
14

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to infinity, the meta distribution needs to converge to a degenerate Dirac delta distribution around single distribution.
They provide three arguments based on the value of regularization parameter: (1) = 0, which means no regularizer in the UCE loss, then the meta distribution becomes Dirac function thus the first is violated [4, Theorem 1]; (2) if À > 0 in the UCE loss is too large, then the second desideratum too small, then the first is
not satisfied [4, Theorem 2]. (3) Also, they claimed that if À violated [4, Theorem 3].
We respectfully point out, however, that the proofs of both Theorem 2 and Theorem 3 are erroneous due to the steps that identify the differential entropy of Dirac delta function as O, instead of -oo. After correction, we realize the statements of [4, Theorem 3] is no longer valid. Instead, the arguments in [4, Theorem 1] and [4, Theorem 2] can be more sharply inferred from our Theorem 5.1. First, when À 0, it is easy show that the optimal meta distribution p (7r (z:) in Eq. (7) will be Dirac function. Second, Theorem 5.1 implies a stronger result than the second claim: for any value of X, the meta distribution converges to a non-Dirac-dee the second desideratum in [4, Definition I] is not satisfied. We also note that we do not need an additional regularity assumption.
distribution as the sample size grows, thus
A.2 Review of [5] "On Second-Order Scoring Rules for Epistemic Uncertainty Quantification"
As follow-up work, Bengs et al. [5] studied an existence of a proper scoring rule for second-order distributions (i.e., meta distributions). They extended the standard definition of proper scoring rules [28] to meta distributions, and provided theoretical arguments why there can be virtually no such proper scoring rule, even if we hypothetically assume the existence of the true second-order distribution.
At facial value, this seems to contradict the fact that we use the forward KL objective in the distillation approach. To resolve the inconsistency, we consider the classification case, and we even omit the dependence on the covariate x for simplicity. On the one hand, Bengs et [5] considered a particular form of objective in the form
L2 Ptarget) := Emsrge (x)) [By~ [[(b) v)ll,
for some scoring rule E(b: On the other hand, in the distillation based method, we use the forward KI. objective:
T<kLL (vi; Ptarget) ==
(Ptargee (T)
(TT)
Even in the standard EDL methods, we use the KLL objective, though the target may not be meaningful:
LiKL Ptarget) == (T) Prarge:(TT)
As the side-by-side comparison reveals, in practice, most existing EDL methods are using either the forward KL objective (distillation based EDL methods) or the reverse-KI objective (classical EDL methods), but the objective function analyzed in [5] assumes particular form that involves the expectation I. and this implies that the negative results are of rather limited applicability. As our Sec. 6 suggests, if we can directly construct reasonable target meta distribution in way that either KL divergence is computable, then we can use such divergence matching to learn the meta distribution in a faithful manner.
A.3 Review of [6] "Is Epistemic Uncertainty Faithfully Represented by Evidential Deep Learning Methods?"
In a concurrent work, Jürgens et al. [6] questions the reliability of distributional uncertainty learned by two types of loss under different settings, including classification, regression and count data: the "inner loss" widely used by evidential regression framework [18, 29], and the "outer loss" which is more commonly applied in classification setting and some regression methods [19, 20] (equivalent to general UCE loss in Sec. F in our view). Jürgens et al. [6] mainly aims to examine whether the leared distributional uncertainty can be consistent to "target" epistemic uncertainty induced by a reference distribution.
First, [6, Definition 3.1] defines such a reference distribution as qN (0(z) = J(xx)) 10Dx
0 dPN, assuming an access to the underlying data generating distribution D P. is available,
15

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with 0 representing the true first-order target for input The term PD denotes the first-order prediction obtained by a model optimized the first-order objective (e.g., log likelihood) using dataset Dy with size N, randomly sampled from P.
Given this reference distribution, [6, Theorem 3.2] suggests that the optimal second-order distri- butions derived from EDL methods using "inner loss" are not uniquely defined. [6, Theorem 3.3] states that when there is no regularizer, i.c., À the optimal second-order distribution derived from optimizing the "outer loss" is Dirac delta function, and [6, Theorem 3.4] demonstrates that for any data generation distribution, there exists a learned by "outer loss" differs from the reference distribution defined in [6, Definition 3.1].
for which the optimal second-order distribution
While [6. Theorem 3.2] falls outside the primary scope of our paper, the arguments from both [6, Theorem 3.3] and [6. Theorem 3.4] can be derived from our Theorem 5.1, which provides an exact characterization of the optimal second-order distribution follows:
p00) exp( p(y 0)])
2 J p(0) exp( ply ()) d0'
For the X = 0 scenario described in [6, Theorem 3.3], where / = A + co, the sample 0" that maximizes (log will exponentially dominate the integral, leading to:
p(0) exp(vEp(oaa) p(y (0 J P()) exp(<Eptr p(y))()))))))))))))))))))))))})) do
21000 5(0 0*).
where 0* arg max, ( [og p(y (2)) For the case when À analyzed in [6, Theorem 3.4], our analysis provides the exact optimal second-order distribution in Eq. (9), which clearly demonstrates that is different from their reference distribution for any specific choice of
For the classification setting with categorical likelihood p(y)0) Ty our analysis in Example 5.2 provides even sharper results. Here, the optimal Dirichlet distribution can be expressed as:
π (z) Dir( ox + vn(x))
When = 0, oc, the optimal Dirichlet distribution converges to Dirac function, i.e., Dir(T: am #n(x)) - 8(π (z)). When À > O, the behavior of the learned second-order distribution can be exactly characterized by Dir( ap + -- (z))).
Hence, both [6, Theorem 3.3] and [6, Theorem 3.4] can be understood as simple corollaries of our Theorem 5.1.
B Literature Review for Classical UQ Methods
Classical UQ methods include Bayesian approaches, which model posterior distributions over model parameters and approximate the Bayesian inference in various ways, including variational inference [30, 31]. MCMC [32, 331, Monte-Carlo dropout [26], and Laplace approximation [34, 35]. Frequentist methods construct uncertainty sets through techniques such as jackknife [36] and bootstrap [37]. Ensemble methods aggregate decisions of several random models to capture predictive uncertainty [25,38]. 38]. These classical methods suffer from expensive computation cost incurred by either multiple forward passes training multiple models. More comprehensive review of classical UQ approaches can be found in survey papers [39, 40].
C Definition of Uncertainty and Its Measures
In EDL, the uncertainty captured by (T(()) and pe (3)2z) are called the distributional uncertainty and aleatoric (data) uncertainty, respectively. distributional uncertainty is also kind of epistemic (knowledge) uncertainty that captures the model's lack of knowledge arising from the mismatch the training distribution and test distribution. That is, the distribution Pv. (r)) should be dispersed if the model uncertain on the particular query point Zo, and sharp otherwise. As the adjective epistemic suggests, this uncertainty is supposed to vanish at points in the support of the underlying distribution, when the model has observed infinitely many samples. contrast, the aleatoric uncertainty captures the inherent uncertainty in p(y]æ), and thus must be invariant to the sample size in principle.
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There are two standard metrics to measure distributional uncertainty: (1) differential entropy (DEnt) he (T(2))) == p (T| log ( (2) mutual information (MI) (y; π|x)
H (pps(vz)) (m]s) ) where (p((a H (p(y) T))] is usually defined as the aleatoric (data) uncertainry. The resulting probabilistic classifier pe (3) captures both epistemic and aleatoric uncertainties without distinction, and the total uncertainty is often measured via two standard met- rics: (I) entropy (Ent) of (y(c), i.e. H (po(yla)); (2) max probability (MaxP) of p (y(()) i.e., maxy Poo (y)
D Proof of Theorem 4.1
We prove the four statements in Theorem 4.1 in the following three subsections: the first (PriorNet objectives) is proved in Appendix D.I, second (EDL/MSE objective) and third (VI/ELBO objective) in Appendix D.2, and fourth (PostNet/UCE objective) in Appendix D.3.
D.1 Prior Network Objective
Prior networks [7, 8] proposed to use the following form objectives:
Bp(aw)) (D(p (T Pv [x)) + %ood E. E (2) (e( (x))))]
Here, vIE {10?, 109)) and Yood are hyperparameters.
The F-KL objective [7] when D(p, q) = D(p q). The in-distribution objective can be written as
Ep(a,)) (D(p")(TTl)) (Tl Pl (T[[))) = Ep(z) [D po (Tlv)) [v)] |l p [x)]] + (const.).
This implies that minimizing the F-KL objective function forces the unimodal UQ model (T]z) to fit the mixture of Dirichlet distributions Ep(pl=) [p())))))) Since the mixture be multimodal when p(y]æ) is not one-hot, equivalently, if aleatoric uncertainty is nonzero, the F-KL objective will drive the UQ model to spread the probability mass over the simplex, which possibly leads low accuracy.
The R-KL objective [8] is when D(p, q) = D(g p). Unlike the F-KL objective, the in-distribution objective can written as
E ")
p (Tlv))]
(T)(() = | (T (x)
Since p.l P (T|x) is being fit to another Dirichlet distribution Dir( a + (z()), it no longer has the issue of the F-KL objective above. We note in passing that Nandy al. [9] proposed an ad-hoc objective such that ag (x) < 1 for OOD r's.
D.2 Variational Inference and EDL Objective
The variational inference (VI) loss was proposed by Chen al. [12], Joo et al. [13] in the following form:
( xr u) = E (TIa) [oog
( p(T)).
(10)
p(y) π
Although the original derivation is bit involved, we can rephrase the key logic in the paper by the following variational relaxation:
Dpp(,, v) Il ((x)px->())) s D(p(x, y)p (-)) l p())p (T.
where pf*) (v) is induced by the tempered distribution (T. v) x () The inequality is simple application of form of data processing inequality [23]. We remark that the original name, "ELBO loss." is a misnomer for this loss function, as the right-hand side simply bounds a constant on the left-hand side, as opposed to the negative ELBO, which bounds the negative log-liiel
Lemma D.1. Let == - Then, we have
min E Ep(a. (evi x, y)] = min E (ppla)))(ppp (T) w))]
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Proof.
Note that by the chain rule of KL divergence, we have on the one hand
D(p(æ, u) (x) p(x)p!))) (x, u)) D(p(z,)) v) [1 (v)) ( (p (x) pl
(T (Tlv)).
Hence,
min R By(wy) (D ( (z) pl)) (π [v)) min D(p(a, (1p(,,))ppp0 (T.))- (11)
V On the other hand, we have
Dip(,, vips (T (x)
p(z)p( (T, v))
log plx vip [ p(x)p(e) (TT, v)
p(n, Po (r) log p(x) ( pp (v πT)
ppcww)e (m)x))
+ log
Ep(a,)) ME (T
log p π
(T ()) ll [((T)) + D(p(z,)) [1 log
px((, [lvi (2/5 v, v)] (p((1, 2) p(x)) + log B.
Here, we let B = E, ((T))p" TT) dæ is the normalization constant, which satisfies pl (T (ylT). This implies that
min Dpp(2, v)p (TTa)) Il ((x)p()) (n, = min Ep (vv(1;;
(12)
Combining Eq. (12) and Eq. (11), we prove the desired equivalence.
D.3 Uncertainty Cross Entropy Objective
Charpentier et al. [15, 20] proposed to use the uncertainty cross-entropy loss, which is defined as
(u) 2, v) po log ply
h(p ( (T
(
In [15, 20), this loss function was originally motivated from a general framework for updating belief' distributions [22].
As alluded to earlier, however, it is special instance of the VI objective of Joo al. [13] when 11, since p(T) Dir( t; f1) B(J (C 1)! and thus
D(p (x) ((T))) (pp() [log (T(()(x)]
p(T))
-- (z:) B (opgrr))) (pp((( log(C 1)!.
Since the VI loss Eq. (10) also only uses au in the original paper [13], this implies that the VI loss and the UCE loss essentially equivalent.
E Proof of Theorem 5.1
We prove a stronger result than Theorem 5.1, allowing any observation model for beyond the categorical case, i.e., y E (C).
Theorem E.1. For choice of prior distribution p(0) and 0, define
((v)) Eptz.nie (0)x) log p(y)0)
(Ep p()))
(13)
== - Define the tempered likelihood (0,u)
p p(0)p"
"0 (0lg) (0, de
where (0,))
]1 p p 0 do
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Then, we have
min [(v) min pp DD p b))
W &
( (0 x x))]
(14)
where
p0) exp(1 log p 0))
p" (0 ()) = J p(0) exp p(ula) log p 0)) do
Proof. Note that we can rewrite the objective function as
p(0) p0)p"mm (y(0)
((v)) = Appr
D (0)(a))))
+
[((O)p (1 0) d0
ABp(r) (p(pu(0=))
exp
Bp(ulx)
log (ul)
C²
f exp
Ep(()=) log T p(00)r ((0)
do
= )Bp(r) (p(p(()=)) ||
p(0) exp (xp((//ppppaaa) pl
p(0) exp( log p(y d0 + C
(15)
Here, v:= and C, C², and C are some constants with respect to vb In particular,
O = pp(x) log
(16)
p(0) exp(vIE ae) [lo ()) d0
This characterizes the optimal pa: (0 x)) under this objective function as
p( (0) exp( pi [(gg(()()))
par (0|x)
(17)
p(0) exp( (/E, ( log (og de
which concludes the proof.
An Extension For General Observation Models
In this section, following the similar exponential family distribution setup, We consider likelihood model of exponential family distribution over a target variable y E IR with natural parameters O E R
((y(0) == hly) exp(OTuly) - A(0)).
Here, h: R IR is the base measure, A: _R IR is the log-partition function, and u E R + R sufficient statistics. Note that the entropy of the parametric distribution given as h(p(y(0)) A()) A(O))(()) Ellog h()).
In EDL methods, distribution over the parameter 0 is assumed capture uncertainty. convenient choice is the conjugate prior p(0)s, n) of the likelihood p(y)0), which is given as
p(0 &. n²) 17(E, n) exp(n(0T A(0)))
Here, & IR_ is the prior parameter, n E Ryo is the evidence parameter, and ny(E,n) is normalization constant.
The benefit of the conjugate distribution is in the easy posterior update. Given N observations y and prior p(0)E. E., na,), the posterior is p0 + n.4,, where
8. noC + EE u(3;)
no + N
n4 + N.
Now, we assume that there exists neural networks that, for each point can approximate the "associated parameters" ny (x) and (x). We then define the target posterior distribution as
no + ny (xr Po (0 [z) = p
n. + no (x())
no + no (x)
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For choice of prior distribution p(0) and > O, define the objective function
( = p(z..))e
log p(y
Al (0|z) || p())))
This is generalization of the UCE loss proposed by NatPN [20], as will be detailed below. We note that Theorem 5.1 remains to hold for this general observation model. That is, by minimizing the UCE loss, Pe (0) a:) is encouraged fit to a fixed target
p0)) lo p 0)])
p0 x) = p(0) exp( flg ply 0)] d0
F.1 Example: Classification
For classification, set 0(= x)) E p(y 0) fy (categorical model), and we have, for ox, 1,
p(0) = Dir(O;
Ps ( x:) Dir(0: (z))a
[lo Dir(O; or ve,))). While the categorical model is typically assumed, few exceptions exist. Sensoy et al. [41] used 0)y = (ey, 0, o² Deng et al. [11] used p(y]0, on (x)) = (ey 0, (z))))))))-) where I(a= denotes the Fisher information matrix of the probability model Dir(ar), i.e,,
p' (0 x²) Dir(O; oo un(x)) x
2 log Oi a:) Ø log Dir(O:
I(a)= Eir()
da
da
In this modified case, the equivalence in Eq. (14) Theorem E.1 does not hold. The analysis breaks down since the constant C in Eq. (15) now becomes dependent on eÙ, and cannot be ignored in the optimization:
="=Ep(s) log
Ep [exp [log P(y)0, ax
Our high-level critique on the EDL. methods remains to hold, since the FisherEDI. still does not take into account any external uncertainty in the model tb. Therefore, the FisherEDI. would fit the meta distribution to some arbitrary target, which could be however potentially better than the simple scaled-and-shifer target. We also note that in our empirical demonstration in Sec. 7, FisherEDL did not show outstanding performance the downstream task performance.
F.2 Example: Regression
For regression, we set (ju, Rx Rzo, ply N(n For a prior, normal-inverse- Gamma (NIG) distribution A AT (1, Ho. X, a, 3) is convenient choice due to its conjugacy to the Gaussian likelihood. Charpentier loss Eq. (D.3). Malinin et al. [19] considered a multivariate regression with Gaussian likelihood together with the Normal-Wishart distribution, which is a multi-dimensional generalization of the NIG distribution. They used the reverse-Ki KL type objective extending their prior work [8]. In our unified view (Theorem E. both objectives are equivalent, and these approaches would suffer the same issue in its UQ ability.
al. [20] proposed use this choice with the UCE
While the "Deep Evidential Regression" objective proposed by Amini et al. [18] cannot be subsumed by our work, Meinert et al. [42] recently investigated the effectiveness the EDL method of Amini al. [18], and also showed that the objective function characterizes degenerate distribution as its optimal meta distribution, suggesting that its empirical success is likely due to an optimization artifact.
G Experiment Setup
G.1 Data Processing
Details about Synthetic Data. We create mixture of Gaussian distribution with mean [-2,3] [0, 0). and [2,3], and variance 0.25. We randomly sample 1000 training data used to train EDi. models and use 2000 test data for the generation of visualization plots in Fig. 6 and Fig. 8.
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Data Split of Real Data. For in nn-disributtio datasets CIFARIO and CIFARI00, we divide the original training data into two subsets: training set and a validation set, using an 80% /20% split ratio. The testing set of these datasets is utilized as in-distribution samples for evaluation. For OOD datasets, we utilize their testing sets as OOD data. To maintain consistency in sample size, we ensured that each OOD dataset contains exactly 10,000 samples, matching the number of in-distribution test samples.
Details about OOD Data. For all the OOD datasets, standardized the dataset images by resizing them to dimensions of 32x32 pixels. Additionally, all gray-scale images were converted to a three-channel format. The following datasets are selected for the OOD detection task:
Fashion-MNIST: Contains article images of clothes. We use the testing set, which contains 10,000 images.
SVHN: The Street View House Numbers (SVHN) dataset contains images of house numbers sourced from Google Street View. We use 10,000 images from its testing set as OOD samples.
TinyImageNet (TIM): As subset of the larger ImageNet dataset, TIM's validation set, containing 10,000 images, is used as OOD samples.
Corrupted: This is an artificially created dataset generated by applying perturbations, in- cluding Gaussian blurring, pixel permutation, and contrast re-scaling the original testing images.
G.2 Model Architecture
Base Model Architecture. To ensure fair comparison, we use the same base model architecture for all EDL methods. We describe the base model architecture for different tasks as follows:
Synthetic 2-D Gaussian Data: We utilize a simple Multi-Layer Perceptron (MLP) model. This model comprises three linear layers, each with a hidden dimension of 64, each followed by the ReLU activation function.
Real Datasets (CIFARIO and CIFAR100): We adopt the existing model architecture VGG16 [43] and ResNet18 [44] the base model.
The base model serves as the feature extractor for the EDL model, which extracts the feature with latent dimension 6 for 2-D Gaussian Data and CIFAR1O, and extracts the feature with latent dimension 12 for CIFAR100.
UQ Model Head. The UQ model of most EDL methods consists of a base model feature extractor and a UQ head. The feature extractor takes the data as input and extracts the latent features. The UQ head then takes the latent feature to parameterize a Dirichlet distribution Dir(T;; On, (x)). As we discussed in Sec. 4, different EDL method mainly differs from the parametric form of (I),, either direct parameterization density parameterization, which is achieved by using different types of UQ head. We describe the model architecture of these two different UQ heads as follows:
Direct Parameterization: the UQ head of direct parumeterization is similar to a typical classification head, which consists of two linear layers with hidden dimension 64 (128 for CIFARI00), equipped with a ReLU activation between the two layers.
Density Parameterization: density parameterization adopts a density model. PostNet [15] uses a radial flow model with flow length of 6 to estimate class-wise density, which results in the creation of multiple flow models corresponding to the number of classes in datasets. NaPPN [20] uses single radial flow model with flow length of 8 to estimate the density of marginal data distribution p(x).
G.3 Implementation Details
For all the EDL methods, we use the same base model architecture discussed previously. We elaborate on their common configurations, method-specific configurations, and
follows.
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1.00
RPriorNet
PostNet
NatPN
epsteric uncertiinty
10-
0.98
Test
10
0.98
-10
102
10a
102
10a
10a
10a
10a
10a
sample size
sample size
sample size
sample size
(a) Epistemic Uncertainty: 2D Gaussian
BM
RPriorNet
PostNet
NatPN
uncertainty
0.2
0.0
10
10-2
10
10->
10-2 10-h
10
10-a
10-2
10-2
10-2 10-
A
A
A
(b) Aleatoric Uncertainty: 2D Gaussian
Figure 5: Behavior of Uncertainties Learned by EDL methods on Toy Data. (a) EDL methods lear spurious epistemic uncertainty, wherein uncertainty does not vanish with an increasing number of observed samples. contrary to the fundamental definition of epistemic uncertainty. (b) Instead of a constant, EDL methods learn model- poddel-depenentt alcatoric uncertainty that depends on hyper-parameter X, contrary to the fundamental definition of alcatoric uncertainty.
Common configurations. The maximum training epochs are set to 50, 100, and 200 for 2-D Gaussian data, CIFARIO and CIFAR 100, respectively. We train models with the early stopping of patience 10. The model is evaluated on the validation set with frequency 2, and the optimal model with the lowest validation loss or highest accuracy is returned. The training batch size is set to 64, 64, and 256 for Gaussian data, CIFARTI 10 and CIFAR 100, respectively. We use Adam optimizer without weight decay or learning rate schedule during model optimization. The learning rates of the optimizer are 1,-3,2.30 2e-4,, 2e- for Gaussian data, CIFARIO and CIFARIO0 respectively. The default hyper-parameter À is set to 1e-4 for those EDL methods with regularizer. All reported numbers are averaged over five runs. All experiments are implemented in PyTorch using Tesla V100 GPU with 32 GB memory.
Method-specific Configurations. We describe those EDL methods with additional training config- urations and syper-parameterd as follows,
RPrionNet [8]: This method requires additional OOD data during training. In alignment with the original paper, CIFARIÓ0 100 serves as the OOD data for CIFARIO, and Tiny Imngemt is utilized as the OOD data for CIFAR 100. We set the OOD regularizer weight %ood 5 and Tood = 1 for CIFARIO and CIFAR 100, respectively. We realize that both the training loss and validation loss fluctuate significantly due to the conflict between in-distribution and OOD data optimization. In this case, we use validation accuracy as the criterion to save the optimal model, making sure the model is well optimized.
NatPN [20]: NatPN uses a single density model with flow length 8 and latent feature dimension 16 to parameterize density. The method also requires "warm-up training" and "fine-tuning" of the density model, where we set warm-up up epochs to be 3 and fine-tuning epochs to 100, as the paper's 's official implementation suggests.
Fisher-EDL [1]]: This method requires regularizers, one is the standard r-KL regularizer, the other their proposed "Fisher Information" regularizer in the training objective. As the paper suggest, set the corresponding hyper-parameter X₂ 5e-2 and X2 5 for CIFARIO and CIFAR100, respectively, and set λ₁ for both datasets.
END2 [16]: To obtain the ensemble samples, we train 100 ensemble models using standard cross entropy loss. In alignment with the original paper, we also use the temperature annealing technique, which the key technique to ensure the distillation training works smoothly. The temperature will be initialized as large value and then gradually decay to 1 after certain epochs. We set the initial temperature for both datasets and set the decay epochs to 30 and 60 for CIFARIO and CIFAR100, respectively.
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S2D [17]: obtain 100 samples. We also use the temperature annealing technique with the same configurations END2 for the distillation training.
We train single model with random dropout and then run inference of the model
Bootstrap Distillation (Ours): First, we randomly sample 100 subsets of training data from the original training set with 80%/20% split ratio. Then, for each dataset, train bootstrap model using standard cross entropy loss, resulting 100 bootstrap models. For the distillation stage, we also apply the temperature annealing technique with the same configurations as END2.
H
Additional Experiment Results
H.1
Omitted Results in Sec. 5.1
The behavior of both epistemic uncertainty and alcatoric uncertainty learned by EDL methods on 2D Gaussian data is shown in Fig. 5. We observe similar behavior as real data. EDL methods cannot quantify either epistemic or aleatoric uncertainty in a faithful way.
H.2
Ablation Study Sec. 5.2
48
-04
HA
400
-12
-32
44
44
(a) r-Kl.
(b) r-Kl, OOD
(c) r-K., multiple flow
(d) r-KL, single flow
(e) MSE
( MSE + OOD
(g) MSE multiple flow
(h) MSE single flow
Figure 6: Visualization of Epistemic Uncertainty 2D Gaussian Data. The first (a)-(d) corresponds differential entropy quantified by EDL models trained using their original proposed training objective (reverse- KL), and the second row (e)-(h) corresponds to epistemic uncertainty quantified by same EDL methods ablated with MSE objective in Eq. (18). This ablation study implics that the specific training objectives has no actual impact on the methods' ability to quantify uncertainty. Other techniques, such as using density estimation (PostNet 9). NatPN (d, fi)), and using OOD data during training (RPrionNet (b, f)), play EDL methods' UQ performance. Without auxiliary techniques, BM (a, e) cannot distinguish in-distribution and OOD regions. Similar behavior holds for total uncertainty quantification (see Fig. 8).
significant role
There could be many other behavior. For instance, we can consider a simple variant of MSE loss as follows:
objectives that capture similar
min (p)()))) |Il (ogg (x)) log X x-
(18)
The global minimum of optimizing this heuristic loss function can be characterized by log(a.- (z)) = Epiyla) [log(as + --ee), which implies ag (x:) R Ok) + ( under the assumption of a sharp true label distribution p(.)x). Independent of the training procedure, we can artificially define epistemic uncertainty metric, while the entropy of normalized model output
log (T)(()) (x) (x)) can represent total uncertainty.
We conduct an ablation study by replacing the Leverse-KL. objective with the MSE objective in Eq. (18) for model training. First, we provide visualization of epistemic uncertainty, aleatoric uncertainty, and total uncertainty 2D Gaussian data quantified by different EDL methods in Fig. 6, Fig.
23

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Posther
KarPP
822
622
(a) r-KL
(b) r-KL OOD
rc) multiple flow
(d) r-KL single flow
2N
#Ptofie:
Pouthee
972
72
(e) MSE
(f) MSE + OOD
(g) MSE multiple flow
(h) MSF single flow
Figure 7: Visualization Alcatoric Uncertainty on Gaussian Data. The row (a d) corresponds to the aleatoric uncertainty quantified by EDL models trained using their original proposed training objective (reverse-Ki) KI., and the second row (e h) corresponds aleatoric uncertainty quantified by the same EDL methods ablated with MSE objective in the equation Eq. (18). Based these results. We can draw similar conclusion Fig. 6: specific training objective actual impact on the methods' ability quantify aleatoric uncertainty either.
(a) r-KL
(b) r-KL + OOD
rc) + multiple flow
(d) r-KL single flow
Poother
KwPP
120
3.8
520
3.
55
655
0.4
34
654
554
(e) MSE
( MSE + OOD
(g) MSE multiple flow
(h) MSE single flow
Figure 8: Visualization of Total Uncertainty on Gaussian Data. Total uncertainty is measured using entropy. We can draw a similar conclusion as Fig. and Fig.
and Fig. 8, respectively. Next, extend the ablation study to OOD detection (using epistemic (using aleatoric uncertainty) tasks using real data in Fig.
uncertainty) and ambiguous data detection and Fig. 10, respectively. The results indicate that EDL methods exhibit similar behavior by using reverse KL or MSE objectives.
H.3 Ablation Study in Sec. 5.3
Empirical Finding 1. DOD-Data-Dependert Methods are Sensitive to Model Architectures
Firstly, we remark on the conflicting empirical findings regarding the Prior Network [7, 8] as reported in current EDL literature. Baseline results of [8] on the OOD detection task, reproduced by [15, 11]. deviate lot from the original paper's results. Upon closer investigation, we can attribute these conflicting results nuances in different model architectures, i.e., the use of BatchNorm has significant impact when incorporating OOD data during training. Specifically, we conduct an ablation study the model architectures of RPriorNet, using VGG-16 and ResNet backbone models, and

---

CIFAR10
CIFAR100
1.0-
Reverse
Reverse-K
Loss
Averbge AUROC Score
Average AUROC Score
0.2
0.2
0.0
PostNet
NatPN
PostNet
NatPN
RPriorNet
EDL Methods
EDL Methods
Figure 9: Comparison of OOD Detection Performance of EDL methods with Original Objective MSE Ohjective. The EDI. models trained using MSF objective in Eq. (18) demonstrate comparable performance. These results further justify that the specific training ohjective has no actual impact on the downstream task performance.
CIFAR10
CIFAR100
Reverse
MSE Loss
MSE Loss
0.5
AUROC Score
AUROC Score
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
PostNet
NatPN
RPriorNet
BM
PostNet
NaiPN
RPriorNet
EDL Methads
EDL Methads
Figure 10: Comparison of Ambiguous Data Detection Performance of EDL methods with Original Objective v.s. MSE Objective. We can draw similar conclusion as Fig. 9.
compare their performance with and without BatchNorm layers removed. As illustrated in Figures 11, the OOD detection performance of RPriorNet shows sensitivity to the model architecture, with the removal of BatchNorm layers leading performance improvement. This observation further reflects the limitations of utilizing OOD data during training, particularly for large-scale models in which BatchNorm is usually employed.
Empirical Finding 2. Density Models May Not Perform Well with Moderate Dimensionality
Charpentier et al. 20] claim that utilizing density parametrization can effectively address the OOD detection task without the need observe OOD data during training. This seems to be intuitive, since a perfect density estimator should be capable of identifying OOD data, as OOD data are typically from low-density regions. However, density estimation itself may encounter challenges, particularly in high-dimensional spaces. To validate this conjecture, we train PostNet [15] with a latent feature dimension of 2. As illustrated in Fig. 12, we visualize the epistemic uncertainty (measured by differential entropy) in the latent feature space and plot the features both in-distribution and OOD data. Despite the density model effectively generating higher uncertainty for OOD regions, the OOD data are mapped to the same region as the in-distribution data. This phenomenon, known as model collapse [45], limits the model's capability distinguish between in-distribution and OOD data.
H.4 Bootstrap Distillation Method Faithfully quantify Epistemic Uncertainty
The comparison of epistemic uncertainty quantified by existing EDL. methods and our proposed Bootstrap Distillation method is shown in Fig. 13. Compared to existing EDL methods, our proposed
25

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SVHN
FMNIST
TimageNet
Corrupted
2.0
2.0
Scree
ALIOCC StOR0
0.6
0.5
0.4
0.44
VOA
und
Festre--0 Batcwoom
Barkeock RS Wac
Seatuo w/s baichior
Seatto- Eecctio
Figure 11: OOD Detection Performance of RPriorNet with Different Model Architecture. RPriorNet trained using CIFARIO as in-distribution data and CIFAR100 as OOD data. The OOD detection performance (showing four OOD datasets) is sensitive to the choice of model architecture.
46
(a) SVHN
(b) FMNIST
(c) TlmageNet
(d) Corrupted
Figure 12: Visualization of Epistemic Uncertainty in PostNet's 2D Latent Feature Space. PostNet leverages
a flow-based density estimation model, which outputs low (high) uncertainty in-distribution (OOD) regions as expected. However, given that input data is high-dimensional, PostNet suffers from feature collapse issue that mapping OOD data to the same region as ID data in the latent space, making them indistinguishable.
Bootstrap Distillation method demonstrates monotonically decreasing and eventually vanishing uncertainty, consistent with the dictionary definition of epistemic uncertainty.
BM
RPriorNet
PostNet
NatPN
Bootstrap
epistemic uncertainty
10-1
Test acc.
0.8
0.6
10
0.4
10a
10a
10
10a
10*
10a
10a
10a
10*
sample size
sample size
sample size
sample size
sample size
Figure 13: Epistemic Uncertainty and Test Accuracy v.s. Number of Training Data. Compared other EDL methods, our proposed Bootstrap Distillation method can faithfully quantify epistemic uncertainty, i.e., the uncertainty is monotonically decreasing with increasing number of data.
H.S Bootstrap Distillation Method Benefits from More Samples
As discussed in Sec. 6, the Bootstrap Distillation method needs the sampling multiple bootstrap samples, with each requiring the training of a model using randomly sampled datasets. To understand the impact of varying the number of samples 5, 10, 20, 50, 100, we conduct an empirical investigation on the OOD detection performance of Bootstrap Distillation. The results shown in Fig. 14 reveal that Bootstrap Distillation indeed benefits from a larger number of samples, particularly in more complex OOD detection tasks using CIFAR 100 as the in-distribution data.
26

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CIFARIO
CIFAR100
1.00
Score Score
Average AUROC Score
0.9
0.96
Average AUROC
0.8
0.7
0.88
0.6
10
100
Number of Bootstrap Samples
Number Bootstrap Samples
Figure Bootstrap Distillation's OOD Detection Performance v.s. Number of Bootstrap Samples. The æ-axis represents the number of bootstrap samples, and the y-axis represents the Average AUROC score of OOD detection tasks. Bootstrap Distillation's UQ performance will improve by obtaining more bootstrap samples, but more computational cost is the price needs pay.
Table 2: OOD detection AUROC score. MI and Dent stand for Mutual Information and Differential Entropy.
ID Data
Method
Metric
SVHN
FMNIST
TimageNet
Corrupted
CIFARIO
RPriorNet
MI
72.6
79.8+2.9
74.0
77.6.9.8
77.6+6.8
82.7+2.1
77.4+1.6
81.3+7.1
BM
MI
97. 97.540.8
93.640.8
87.4 87.4+12
98.8 98.8
Dent
97.0 97.0...5
93.321..
87.7
28.32
PostNet
94.411.0
89.811.6
86.310.3
96.810.8
Dent
95.5+1..
90.6+1.2
86.840.3
97.820.6
NatPN
17.842.0
21.0+2.7
26.343.2
14.9+2.1
Dent
67 67.016.7
60.6114.1
54.612.
69.7
96.2+11.1
90.3+0.6
86.3+0.4
98.7+0.5
Dent
96.540.9
90.3
86.9
98.8
His
91.310.3
86.91n.
86.110.5
95.111.0
Dent 93.711.3
88.1111
86.610.
97.110.7
END2
95.840.4
g2.7+0.7
89.040.4
95.7+0.5
Dent
96.940.2
93.440.7
87.140.
96.920.4
S2D
MI
80. 80.1
87.910.6
83.910.
90.313.5
93.311.5
90.010.4
87.010.
93.9 93.92o.2
Bontstrap Distill
97.240.1
94.440.6
89.640.3
98.2
Dent
97.8+0.1
94.3+0.8
87.6+0.4
98. 98.4+0.2
ID Data
Method
Metrie
SVHN
EMNIST
TimageNet
Corrupted
CTFAR100
RPriorNet
MI
74.8
73.1
61.642.5
72.9
74.3+1.8
73.744.5
63.1
2..4
BM
MI
71.519.8
74.616.8
71.415.
61.816.a
Dent
68.5
74.216.7
70.216.0
58.616.5
PostNet
75.948..6
70.8+2.4
64.441.6
57.9+11.6
Dent
79.4+s..
74.7+1.8
72.040.9
55.5+
NarPN
56.4+5.6
41.6+4.1
43.5+1.6
53.2+6.9
Dent
64.7+4.4
47.0+4.6
49.640.9
57.0+3.5
EDL
68.7400.4
74.8
73.740.5
50.5.6.5
Dent
68.213.4
74.412.5
73.510.5
49.916.3
Fisher: EDI.
71.342.6
71.043.2
74.840.6
54.847.5
Dent
71.542.0
69.643.5
75.4+0.3
63.6+6.3
END2
MI
78. 78.142.9
79.611.5
76.6404
68.442.8
82.012.4
85.010.6
83.210.
73.4
S2D
MI
62. 02.1
69.8
71.540.2
31.6
Dent
50.310.2
50.410.3
50.910.
50.3g0.3
Boutstrap Distill
MI
77.9+
83.3
78.240.5
53.44:3.0
85.6+1.0
86.8+1.2
84.1
6
H.6 Detailed Results UQ Downstream Tasks
The complete AUROC and AUPR results of OOD data detection are shown in Table 2 and Table 3, respectively. The complete results of selective classification are shown in Table 4.
27

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Table 3: OOD detection AUPR score. MI and Dent stand for Mutual Information and Differential Entropy.
ID Data
Method
Metric
SVHN
FMNIST
TTmageNet
Corrupted
CIFARIO
RPriorNet
MI
68.2-7.5
15.6+33.0
69 09.3442.9
73.6
Dent
73.5.6.3
79.941.a
72.911.9
75.428.9
BM
MI
96.920.8
92.310.a
85.311.3
98.220.6
Dent
95.0=1.7
90.5+1.2
84.2+1.6
96.4+1.2
Postee
MI
90.9-1.6
86 10.64+2.3
82.3++.a
93.3++2.2
93.5
88.5411.3
83.7.20.4
96.121.2
NaPPN
MI
43.9
40.412.0
38.711.2
44.919.6
Dent
70.2=
65. 15.22
59.2+1..
73.6.3.9
EDL
MI
93.7221.9
87.610.8
82.610.5
97.620.6
Dent
94.421.6
87.710.8
83.410.4
98.020.4
Fis
MI
82.2-113
80.5+1.11
80.9+0.0
89.0+2.7
Dent
88 88.2=2.7
84.7441.5
82.5+1.1
94.0=1.7
END2
93.320.2
90.9.10.
84.3.0.7
91.721.0
Dent
95.120.2
91.810.7
83.710.8
93.820.9
S2D
MI
83.02a.9
82.640.9
77.740.7
83.0g5.2
Dent
89.0
86.410.7
82.410.4
87.5g3.6
Bookstrap Distill
94.6.0.4
92.522.7
85.622.5
25.2
Dent
95.720-3
92.510
84.520.6
96.020.4
Data
Method
Metric
SVHN
FMNIST
TImageNet
Corrupted
CIFARI00
RPriorNet
MI
20.3
69 4.9
58.712.1
71.6
Dent
69.1
69 69.244.6
59.221.6
68.5
BM
MI
67.3-7.8
68 68.9+5.8
67.246.5
59.3
65.0=7.5
68.14s.1
66.44s.5
53.7+4.5
PestNer
MI
69.726.6
64.412..
57.511.4
69.71.0033
Dent
76.4+6.5
09.3+22.3
06.2+1.2
57.6+9.5
NatPN
MI
51..3
11.142.5
15.5
50.223.7
Dent
59.8 80.8
47 47.012.7
48.910.8
52.0
EDL
MI
63.9
69.742.7
68.9
48.5
Dent
63. 1-3.3
68.8+2.8
68.6+0.5
47.8+3.6
Fisher
64.0=3.1
66.643.1
69.6
53.026.1
Dent
65.02a.2
64.913.2
71.010.7
49.91
END2
MI
71.1=a.6
70.3++1.3
70.340.4
57.9+2.3
Dent
75. 75.14:2.7
77.010.4
78.210.3
63.512.4
S2D
MI
58.522.0
63.910.7
68.610.3
38.020.4
Dent
46.4
45.6+0.1
45.8+0.1
52.5.0.7
Bootstrap Distill
MI
68.5+1.a
73.6+2.5
71.9+..8
46.6+1.4
Dent
77.721.0
78.642.0
79.440.4
55.8
Table 4: Selective classification Test Accuracy, AUROC, and AUPR score. Ent and MaxP stand for Entropy and Max Probability, respectively.
Method
Metric
CIFARIO
CIFARI00
Ace
AUROC
AUPR
Ave
AUROC
AUPR
RPriorNer
Ent
85.44
86.0+9.4
49.7+2.0
57.941.3
80.8+0.6
72. 02.2+0.9
MaxP
8..4
86.620.5
51.321.6
57.9
81.8
04.0=a.e
BM
Ent
88.7+0.1
90.1
51.5+0.7
54.3+1.2
81.8+0.1
76.0+
MaxP
88.7+.0.1
90.0+0.2
21.0
54.3+1..2
82.6+a.1
76.9+0.7
PostNet
Ent
88.410.1
89.210.2
50.420.4
58.01o.s
81.610.2
74.1
MaxP
88.4
89.210.2
50.7
58.010.s
82.610.2
75.4
NaPN
Ent
83.24.0.6 6
86.240.2
53.520.5
56.040.4
80.840.2
73.840.6
MoxP
83.2+...6
86.4+9.2
54.4+0.6
56.0 56.0
82.2+0.2
76.2+0.3
89.9..0.1
89.720.2
49.220.7
62.910.3
84.510.2
74.1
MaxP
89.9+9.11
89.7+0.2
48.8+0.7
62.940.5
84.9+0.2
74.9+
Fisher-EDL
Ent 88.2+..4
90.1g0.1
53.720.9
55.940.5
85.840.2
79.6
MaxP
88.210.4
90.410.1
53.810.8
55.910.s
86.110.2
80.1g0.3
END2
Ent 89.4
89.820.a
50.611.0
65.0
83.040.1
69.3
MaxP 89.4+...1
89.6+0.3
51.1+1.
65.040.2
85.3+0.1
72.640.2
S2D
Ent 89.1+a.1
89.4.0.1
50.5
61.0
83.44..1
72.7
MaxP
89.1
89.320.1
51.020.5
61.0.o.1
84.410.1
74.220.2
Bootstrap Distill
Ent 89.4
90.3g0.2
51.2g0.5
65.4
83.240.1
69.4g0.3
MaxP
89.440.1
90.2+0
52.1
65.440.1
85.740.1
73.240.3
28

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Reviews:
[{"review_id": "cueldQEOad", "reviewer": "Reviewer_Sxes", "summary": "In this paper, the authors propose a novel analysis of existing evidential learning approaches, which have recently gained significant attention in the domains of uncertainty estimation and probabilistic modeling. The two major contributions of this work are as follows: First, the authors provide a clearer understanding of the asymptotic behavior of a wide class of EDL methods by unifying various objective functions. Second, they reveal that EDL methods can be better interpreted as out-of-distribution detection algorithms based on energy-based models. Specifically, the first contribution explains why evidential learning might have poor uncertainty quantification capabilities (or probabilistic modeling quality), while the second explains why EDL methods empirically show good performance on a number of downstream tasks. The authors validate their insights through extensive experiments on several image classification tasks (CIFARs and TinyImageNet) and EDL baselines.", "questions": "* How do the proposed results translate to the regression case for evidential learning? Is there any way to extend the mentioned insights into this case? For example, it seems that the EBM intuition of OOD provided for classification potentially wouldn't work for the regression case.\n\n* The paper highlights a very interesting problem with EDL models, which have gained a lot of attention recently. In Section 6, the authors discuss the reason for these problems and the potential solution, which is stochastic modeling either through ensembling or dropout (plus the proposed bootstrapping). If stochasticity is a solution for EDL drawbacks, could we apply efficient ensembling alternatives [1, 2, 3, 4] to achieve the same results? I think it could be valuable to discuss this.\n\n[1] Wen, Yeming, Dustin Tran, and Jimmy Ba. \"Batchensemble: an alternative approach to efficient ensemble and lifelong learning.\" ICLR 2020\n\n[2] Durasov, Nikita, et al. \"Masksembles for uncertainty estimation.\" CVPR 2021\n\n[3] Laurent, Olivier, et al. \"Packed-ensembles for efficient uncertainty estimation.\" ICLR 2023\n\n[4] Turkoglu, Mehmet Ozgur, et al. \"Film-ensemble: Probabilistic deep learning via feature-wise linear modulation.\" NeurIPS 2022", "limitations": "No", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* The paper is clearly written and easy to follow. The idea is intuitive and easy to grasp. The related work section provides an adequate discussion of existing approaches to anchoring-based training. The analysis narrative, with the presented drawbacks of existing methods, is very clear and easy to understand.\n\n* The authors derive a novel way to unify existing evidential learning approaches within Unified EDL Objectives, which provides a new perspective on these types of models and facilitates further analysis. As an example, the authors demonstrate that a typical EDL approach suffers from asymptotic behavior when aleatoric uncertainty does not vanish with an increasing training set and remains constant for ID data.\n\n* The analysis also provides an interesting finding that the EDL models could be considered within the energy-based OOD framework, which partially explains their success in a number of downstream tasks.", "weaknesses": "* From the perspective of the experimental evaluation, I would be curious to see evidence that the behavior demonstrated in the paper would hold in other domains, such as texts, graphs, more complicated vision tasks (e.g. segmentation), not limiting to image classification task. Additionally, evidential learning also covers regression tasks, which would also be very beneficial for the paper to discuss.\n\n* The authors conducted their evaluation exclusively using ResNet18 and VGG16 models. Incorporating more recent models and state-of-the-art architectures would likely provide a more comprehensive and robust assessment of their approach, ensuring validity of the results across a wider range of scenarios. \n\nThe paper does not have any major flaws or weaknesses that would warrant rejection, and I tend to assess the paper positively. I think that the paper would be very interesting for the community since it provides several important insights about a popular family of uncertainty estimation approaches.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their thoughtful comments. We address them in detail below.\n\n**1. Whether EDL’s behavior demonstrated in the paper would hold in other modalities and domains.**\n\nWe totally agree with the reviewer that analyzing EDL’s behavior in other domains and modalities is worth exploring. Since the classification setting for the vision domain is standard in the UQ literature, however, we focus on this setup to clearly demonstrate the implication of our analyses. While we leave such extensions as future work, we refer the reviewer to Appendix F.2, where we provide some analysis for the regression case.\n\n**2. Incorporating more recent models and state-of-the-art architectures for evaluation.**\n\nThis is also a good suggestion. In fact, most EDL methods evaluate their performance on classical neural networks and standard benchmark datasets. Conducting a comprehensive empirical study to benchmark all existing EDL methods under a unified framework would also be an interesting future work. \n\n**3. How do the proposed results translate to the regression case? The EBM intuition of OOD wouldn't work for the regression case.**\n\nAs the reviewer noted, the EBM intuition would not hold for the regression case. However, a similar observation can be made: the existing EDL methods for regression only aim to fit to a “fixed” uncertainty target. This is consistent with the observation made by a recent paper (Meinert et al., 2023). We kindly refer the reviewer to Appendix F.2, where we elaborate further on this case.\n\n**4. If stochasticity is a solution for EDL drawbacks, could we apply efficient ensembling alternatives to achieve the same results?**\n\nIn this work, we argue that the EDL framework can be better used as a computational  tool to emulate the behavior of classical ensemble approaches, while reducing their computational burden. In this context, using more advanced ensembling techniques can indeed boost the UQ performance of distillation-based EDL methods. We will add the relevant discussion in the revision.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "YNgrvg7aaB", "reviewer": "Reviewer_LBKP", "summary": "The paper focuses on Evidential Deep Learning (EDL) models developed for uncertainty quantification in a computationally efficient manner. It identifies key limitations of the EDL methods: their inability to faithfully express both the epistemic and aleatoric uncertainties. The paper then proposes to integrate model uncertainty into the EDL-based models to address the limitations of existing EDL models with a Bootstrap-distill method. The bootstrap-distill methods train multiple models on a subset of training data. Each model trains on a disjoint subset of the dataset to capture the model uncertainty. Experiments are carried out over multiple benchmark datasets to demonstrate the weaknesses of EDL models and the effectiveness of their proposed bootstrap-distill model.", "questions": "→ Impact of hyperparameter lambda in EDL (Figure 2): I wonder how the OOD detection performance works for different evidential models for lambda = 0 (It seems as if using no regularization (\\lambda = 0) might lead to the best performance.)\n→ The proposed solution: Using model distribution p(\\psi|D) for EDL models instead of the point estimates would lead to three levels of uncertainty: Model distribution through p(\\psi|D), distributional uncertainty p(\\pi|x,\\psi) and aleatoric uncertainty through p(y|\\pi). I wonder if there is any benefit of such a three-level framework over standard Bayesian approaches (i.e. without any distributional uncertainty)? \n→ Will the proposed solution be theoretically better than the EDL methods i.e. will the method guarantee desired epistemic/aleatoric uncertainty behavior?\n--> Recent works have shown learning deficiencies in evidential deep learning models[1],[2]. I wonder if the proposed model will be robust to such learning deficiencies.\n[1] Learn to accumulate evidence from all training samples: theory and practice (ICML 2023)\n[2] Ucertainty regularized evidential regression (AAAI 2024)\n→ The claim that EDL methods perform strongly for OOD detection is not well justified. Do these EDL-based models outperform the SOTA OOD detection methods? Comparison with SOTA OOD detection methods will further strengthen the work.\n--> Uncertainty behavior of the developed model (bootstrap): Is the aleatoric/epistemic uncertainty of this model accurate on all datasets and settings? How does the aleatoric uncertainty behave with an increase in sample size for the baseline and the developed model?", "limitations": "The work is towards understanding the limitations of EDL-based models that is well presented. The limitations of their proposed model could be further discussed.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- The paper is generally well-written and easy to follow\n- The paper discusses the effectiveness of the evidential deep learning models and highlights their limitations regarding uncertainty quantification. \n- The relationship of EDL methods with EBM-based OOD detectors further helps better understand the EDL based methods.\n- Experiments are carried out with benchmark datasets to empirically validate their claims. The experiments justify the claims regarding the weaknesses of EDL models.", "weaknesses": "- The proposed solution of introducing model uncertainty p(\\psi|D) over the point estimate \\delta(\\psi - \\psi*)of the EDL approach to address the limitations seems to be a heuristic choice. Is the proposed solution theoretically guaranteed to faithfully express the epistemic/aleatoric uncertainties?\n- In my understanding, EDL approach presents a computationally efficient Uncertainty Quantification (UQ) approach that is alternative to Bayesian approaches as it only involves one forward pass. However, introducing the model uncertainty p(\\psi|D) would lead to the same computational expense compared to Bayesian approaches. It is not clear if there is any benefit of the p(\\psi|D)-based EDL framework in UQ over Bayesian approaches for UQ.\n- The proposed solution  (bootstrap-Distil)'s aleatoric/epistemic uncertainty behavior is not well justified.  Maybe, the uncertainty behavior of bootstrap could be added in Figure 1/4 to better justify the proposed model. Also Figure 10: Which dataset is used? How is the epistemic uncertainty quantified for the bootstrap method? How does it compare with standard UQ methods (eg. bayesian ensembling and dropout uncertainty). Also, the range of epistemic uncertainty seems to be small (0.1 - 0.0001). Is there any range of the aleatoric/epistemic uncertainty values? Are the uncertainty values relative, or is there any meaning to the values?\n- Comparison with SOTA OOD detection methods is missing. Only EDL-based models are considered where the proposed model is superior.\n- Experiments and results are shown on simple datasets such as Cifar10 and Cifar100. It would be interesting to see the developed model's uncertainty results along with baseline model's results for more challenging datasets such as imagenet.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their thoughtful comments. We address them in detail below.\n\n**1. Is the proposed solution theoretically guaranteed to faithfully express the epistemic/aleatoric uncertainties?**\n\nPlease refer to global response 1.\n\n**2. It is unclear if there is any benefit of the $p(\\psi|D)$-based EDL framework over Bayesian approaches.**\n\nThe main benefit of distillation-based EDL is its computational efficiency at inference time. Once the model is trained, it does not require multiple inferences like Bayesian or ensemble methods.\n\n**3. The proposed solution (bootstrap-Distil)'s aleatoric/epistemic uncertainty behavior is not well justified.**\n\nThe bootstrap-distill method does not have any hyper-parameters in its objective, so the aleatoric uncertainty is not model/hyper-parameter dependent. Regarding the behavior of epistemic uncertainty, we provide the figure of the epistemic uncertainty vs. sample size curve on the CIFAR10 dataset in Figure 10. The behavior of uncertainty quantified by bootstrap, ensemble, and Bayesian approaches is similar, as the epistemic uncertainty is induced by model uncertainty. All these methods are well-studied in literature, and they are usually considered as faithful uncertainty quantifiers.\n\n**4. Regarding the range of the aleatoric/epistemic uncertainty values. Are the uncertainty values relative, or is there any meaning to the values?**\n\nThe range of aleatoric/epistemic uncertainty depends on the uncertainty metric. For example, epistemic uncertainty measured using \"mutual information\" will be non-negative. For typical downstream tasks such as OOD data detection, only the relative uncertainty matters for identifying OOD samples. However, there might be practical scenarios where the absolute value of uncertainty also matters. For example, in active learning, we cannot rely on the EDL model’s epistemic uncertainty as a signal to indicate the model’s actual knowledge at a certain stage because it never vanishes. The $\\lambda$-dependent aleatoric uncertainty also may have issues in certain applications. For example, suppose an application relies on multiple pre-trained models to make predictions and identify ambiguous samples. If the learned aleatoric uncertainty of each model is trained with different $\\lambda$'s in an EDL framework, then they might not be in a comparable scale.\n\n**5. Missing Comparison with SOTA OOD detection methods.**\n\nFirst, the focus of this work is NOT to propose a novel algorithm that outperforms all SOTA OOD detectors. Instead, the main contribution is to offer an analysis to better understand how a specific type of UQ method (EDL) behaves and to provide insights on how to mitigate its limitations. The bootstrap-distill approach serves as one of many possible solutions to address the issues of existing EDL methods. Furthermore, while comparing EDL methods with algorithms specifically designed for OOD detection might not be fair, we include an additional baseline comparison with a classical cross-entropy-trained network (see global response 2.3).\n\n**6. how the OOD detection performance works for EDL methods with $\\lambda=0$?**\n\nWe experimented with the $\\lambda=0$ case, but observed that this simply encourages the model to output an extremely large value and leads to numerical errors during training. This is consistent with what Theorem 5.1 and Example 5.2 revealed, i.e., $\\lambda=0$ corresponds to that the target for ID data is $\\infty$.\n\n**7. What’s the benefits of three levels of uncertainty approach over Bayesian approaches?**\n\nFirst, our main claim in this paper is that distributional uncertainty alone while ignoring model uncertainty cannot capture uncertainty in a faithful manner. Second, we suggest that distillation-based methods could be a straightforward solution to remedy these issues. While distillation-based EDL methods have less computational complexity than Bayesian methods, we do not claim that distillation-based EDL methods are fundamentally superior, and further studies are warranted to compare the two approaches.\n\n**8. if the proposed method will be robust to learning deficiencies issue in EDL literature [1, 2]?**\n\nThe learning deficiencies issue is mainly related to the activation function in the EDL model architecture, which might be a tangential problem to this work.\n\n**9. How does the aleatoric uncertainty behave with an increase in sample size?**\n\nAleatoric uncertainty does not have asymptotic behavior. In practice, aleatoric uncertainty will converge to a constant when the model is trained with a sufficient number of samples.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "C3bTMWtTFj", "reviewer": "Reviewer_gUpP", "summary": "The presented paper offers a (further) critique on EDL. They show that a range of EDL objective functions are largely equivalent by presenting a unified objective function that subsumes many existing objective functions. With this unified objective function some problematic properties are shown that prove that these EDL methods cannot properly represent aleatoric and epistemic uncertainty. Following this, they show various failure modes of EDL. Additionally, they show that observed good performance on OOD-detection is not robust and may not work well with even moderate dimensionality. Lastly, they say that distillation-based methods are a better alternative and demonstrate an adaptation to a distillation-based method that outperforms existing methods.", "questions": "- The bootstrap-distillation method seems to have limited relation to the current work. Distillation methods in general seem relevant, but the newly proposed bootstrapping method compared to END^2 seems to be unnecessary and not completely discussed. Would the authors consider introducing the bootstrap-distillation method as a separate publication (with space for more theory and more experiments)? I think this would also allow some of the more important parts of the Appendix to be moved into the main body of the current work. \n- The demonstration in Fig 1.a is a bit confusing. How do you define epistemic uncertainty in the EDL? E.g. PriorNet discusses “distributional uncertainty”, but does not consider this to be equivalent to epistemic uncertainty. Can you cite any sources that claim that the distributional uncertainty from EDL should be equivalent to epistemic uncertainty? And ideally that this kind of behavior of being “reducible” with more data should follow?\n- Line 105 says “distributional uncertainty” is a kind of “epistemic uncertainty”. What is this claim based on? Is there a source for this? It seems that p(psi|D) would still be the epistemic uncertainty in this decomposition (which is reduced to a single point). \n- It is correct that lambda affecting the predicted aleatoric uncertainty is problematic. However, what practical problems will this cause (and will this not cause) when aleatoric uncertainty is considered for downstream tasks? Identifying difficult, ambiguous or mislabelled samples may still be perfectly functional.", "limitations": "EDL objective functions that are not subsumed by the unified objective function are discussed. \n\nThe authors give a thorough critique of the failures of EDL, with the implication that Bayesian NNs would be the better alternative. However, they are not explicitly compared. Adding the behavior of a BNN in Figures 1.a, 4.a and 10 would demonstrate whether EDL are also worse than BNN-based methods. This is a limitation and it is not discussed.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The authors expand upon an existing critique by showing that many seemingly different EDL methods are actually very similar, collapsing multiple different methods into a single conceptual understanding (originality). \n- Some of the practical weaknesses shown of EDL also appear to be novel, such as unchanged epistemic uncertainty under different dataset sizes (significance).\n- The claims surrounding the critique of EDL appear to be well supported and sound (quality)\n- The observation that EDL methods for OOD detection can be sensitive to changes in architecture and that it may fail in moderately high dimensionality give suggestions on how OOD detection using density estimation generally may or may not work. This may have implications outside of EDL methods, and generally says something about how Neural Networks learn. (significance)", "weaknesses": "- The central point of critiquing EDL is not novel. The authors acknowledge that and do progress the critique, but I do think doubling down on the critique may have limited impact (originality)\n- The investigation into the newly proposed Bootstrap distillation is section 7 is lacking. Leave one-class-out OOD detection is not considered. The observation that the predicted epistemic uncertainty is monotonically increasing with accuracy is not sufficient to claim that it is “faithfully representing” epistemic uncertainty. (quality)\n- Some parts of the critique are about EDL’s failure in estimating epistemic uncertainty. However, the claim that EDL’s “distributional uncertainty” is “a kind of epistemic uncertainty” does not seem to have much support. Theoretically that claim is not sound (as epistemic uncertainty is the uncertainty in the model parameters), and the authors provide no sources that make this claim. Without a good source of why EDL is expected to represent epistemic uncertainty the critique on its epistemic uncertainty is meaningless. I was able to find some papers that make this claim: https://proceedings.neurips.cc/paper/2018/file/a981f2b708044d6fb4a71a1463242520-Paper.pdf, https://openreview.net/pdf?id=UI4K-I2ypG. Consider adding some discussion on the (mistaken) belief in the field that EDLs distributional uncertainty represents epistemic uncertainty. (clarity/quality)\n- The density of the paper (many experiments and findings for only 9 pages) means a large part of the methods and results are pushed to the appendix. As a result, the implementation details are not always clear. How distillation-based methods work is not sufficiently explained. (clarity)\n- The newly proposed bootstrap-distillation method is unlikely to have a meaningful impact considering it’s limited theoretical argumentation and the limitations of that method are minimally explored. (significance)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their thoughtful comments. We address them in detail below.\n\n**1. The central point of critiquing EDL is not novel.**\n\nWhile novelty is in the eye of the beholder, and critiquing EDL is indeed a primary focus of this work, our contribution is non-trivial compared to existing papers. First, we offer a more advanced understanding of EDL methods compared to existing critiques [4, 5, 6] (see Appendix A). Second, beyond the critique, this work is the first to attempt to remedy the limitations of EDL methods. Third, to the best of our knowledge, this work is also the first to unify multiple representative EDL methods into a single framework for theoretical analysis, supported by empirical justification with real data experiments.\n\n**2. Regarding the support and the source of the claim “distributional uncertainty is a kind of epistemic uncertainty”.**\n\nWe respectfully point out that this is a common claim in the EDL literature. We provide a list of supporting evidences from the literature as follows:\n- Malinin et al. (NeurIPS 2018) [7] proposed the \"forward Prior Network\". In this work, they introduced the concept of \"distributional uncertainty,\" which is different from \"model uncertainty\" in Bayesian methods. However, similar to \"model uncertainty,\" \"distributional uncertainty\" also measures the model's lack of knowledge, so it is aligned with the definition of epistemic uncertainty. They mentioned, “Distributional uncertainty is an ’unknown-unknown’ - the model is unfamiliar with the test data and thus cannot confidently make predictions.” The follow-up work of the same group of authors Malinin et al. (NeurIPS 2019) [8] also noted that “Knowledge Uncertainty, also known as epistemic uncertainty or distributional uncertainty, arises due to the model’s lack of understanding or knowledge about the input.” \n- Malinin et al. (ICLR 2020) [9] proposed the distillation-based method “END^2”. Therein, the authors again remarked that “Knowledge uncertainty, also known as epistemic uncertainty, or distributional uncertainty, is uncertainty due to a lack of understanding or knowledge on the part of the model regarding the current input for which the model is making a prediction.”\n- Charpentier et al. (NeurIPS 2020) [15], where the“Posterior Network” was proposed, uses \"epistemic uncertainty\" to refer to the \"distributional uncertainty.\" The same usage can be found in its follow-up work (Charpentier et al., ICLR 2022) [20], in which “Natural Posterior Network” was proposed. There is no mention of “distributional uncertainty” in both works.\n- Recently, several critiques of the EDL methods such as Bengs et al. (NeurIPS 2022) [4], Bengs et al. (ICML 2022) [5], Jürgens et al. (ICML 2024) [6] have adopted the same convention. In these papers, they all use “epistemic uncertainty” as the terminology to refer to EDL's “distributional uncertainty” and criticize that EDL cannot faithfully quantify epistemic uncertainty.\n\nWe will include these sources in the manuscript to avoid any confusion.\n\n**3. Regarding paper presentation and clarity.**\n\nWe acknowledge the content is a bit cluttered in the submission due to the page limit. In the revised manuscript, we will do our best to improve clarity and elaborate on the details with the additional page.\n\n**4. Lacking theoretical justification and further investigation of proposed Bootstrap distillation.**\n\nPlease refer to global response 1.\n\n**5. Would the authors consider introducing the bootstrap-distillation method as a separate publication?**\n\nThe main point of this work is to advocate for distillation-based EDL methods in general, and the proposed bootstrap-distillation method serves as another alternative in the ensemble approach to justify this argument. We agree with the reviewer that conducting a comprehensive analysis (both theoretical and empirical) of distillation-based EDL methods can be an interesting future work.\n\n**6. What practical problems will $\\lambda$ dependent aleatoric uncertainty cause in downstream tasks?**\n\nIt is correct that identifying difficult, ambiguous, or mislabelled samples is still possible in practice, as long as the downstream task only relies on a relative scale of uncertainty values. However, there might be practical scenarios where the absolute value of uncertainty also matters. For example, suppose an application relies on multiple pre-trained models to make predictions and identify ambiguous samples. If the learned aleatoric uncertainty of each model is trained with different $\\lambda$'s in an EDL framework, then they might not be in a comparable scale, which prohibits any further comparison.\n\n\n**7. Missing the behavior of a Bayesian approach in Figures 1.a, 4.a and 10.**\n\nPlease refer to global response 2.4.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "xgHjKO4qIw", "reviewer": "Reviewer_cF7x", "summary": "This paper highlights problems with the Evidential Deep Learning (EDL) framework, analyzes these problems, and proposes possible solutions for some of them. The paper provides a new taxonomy and a unifying objective function for a wide range of EDL methods. The authors identify that aleatoric and epistemic uncertainty estimates of EDL methods do not conform to their textbook definitions and provide a bagging distillation technique that improves the learned epistemic uncertainty.", "questions": "- \"We use validation accuracy as the criterion to save the optimal model, as we observe using validation loss leads to unstable results.\" What does \"unstable results\" mean here? Additionally, have the authors tried to choose the optimal model based on their evaluation criteria (e.g., the selective classification performance)?\n- Does bootstrapping lead to a qualitatively different uncertainty quantification behavior compared to regular ensemble training? This is hard to judge, as Fig. 10 only features Bootstrap.\n- The extrapolated epistemic uncertainties of methods not utilizing OOD data are seemingly random based on Fig. 5. What might be the reason for the high AUROC score for most of these methods in Fig. 3? In particular, might it be the case that the OOD datasets are so different from their ID variants that the task becomes easy even with ill-posed epistemic uncertainties?", "limitations": "The authors do not adequately discuss their work's limitations, as they only acknowledge that their unified objective function does not cover all EDL methods and that their Bootstrap Distillation method is costly.\n- Several standard evaluation metrics are missing from the empirical evaluation.\n- The aleatoric uncertainty estimates are not evaluated quantitatively.\n- The paper does not present the performance of a simple baseline (e.g., a deterministic neural network's predictive entropy), making the difficulty of the tasks hard to estimate.\n- The authors do not formally argue why incorporating model uncertainty would alleviate the problems with aleatoric and epistemic uncertainty estimation outlined in Section 5.1.", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "- The range of EDL methods considered in theoretical arguments and empirical experiments is extensive which allows for convincing conclusions.\n- Proving properties of a novel unified objective function is a nice way to make general claims about multiple popular EDL methods.\n- The introduced taxonomy of EDL methods is easy to understand and extend.\n- The OOD datasets considered in the OOD detection benchmarks are diverse and lead to OOD tasks with varying characteristics. The average score describes OOD detection capabilities well.\n- The mathematical argumentations are substantial and clear.", "weaknesses": "- The paper has many typos and grammatical errors.\n- \"Second, the VI loss in Eq. (3) and UCE loss Eq. (4) turn out to be equivalent to the RPriorNet objective [...]\" While the RPriorNet method uses a held-out out-of-distribution (OOD) dataset and incorporates a positive $\\gamma_\\text{ood}$ scalar, the VI and regularized UCE objectives only utilize an in-distribution (ID)  dataset during training. This is an essential difference between the two method classes that does not allow such a conclusion. However, all three methods are, indeed, special cases of the unified objective function in Eq. (6), utilizing different parts of it.\n- The distinguishing features (2-4) discussed in L180-181 are already discussed in the referenced survey of Ulmer et al. (2023; TMLR) (e.g., in Tables 1-2 for the use of OOD data).\n- The fact that epistemic uncertainty does not vanish in EDL methods is not surprising, as, considering a constant regularization factor $\\lambda$, their explicit regularizer that drives them to a flat Dirichlet distribution trades off with the (deterministic) training signal from the labels. Therefore, the global minimizer will not be Bayes optimal even in the limit of infinite data. However, considering a regularization factor $\\lambda$ (or, equivalently, a scaling factor $\\nu = \\lambda^{-1}$) that depends on the dataset size readily alleviates this problem. It also fixes the misalignment with the \"dictionary definition\" of aleatoric uncertainty. Based on these observations, the proof shows that keeping the regularization strength constant is a bad idea, but does not identify problems with the EDL framework. This conclusion is well known, however: in Bayesian learning, the effect of the prior also vanishes as more data is observed, and the prior can be viewed as a regularizer.\n- Similarly, the fact that in the empirical results the aleatoric uncertainty is not constant is expected: estimates derived from the model are always inherently model- and objective-dependent. This holds not only for EDL methods but also for any learned uncertainty estimator.\n- For many practical tasks (such as OOD detection, a widely used proxy task for epistemic uncertainty, or correctness prediction, a predictive uncertainty benchmark), the absolute scale of the epistemic uncertainty value does not matter. In particular, an estimator can be performant without being constrained to vanish in the limit of infinite data, e.g., if it reaches a strictly positive minimum in this limit. The observation of non-vanishing epistemic uncertainty does not affect the practical use of these uncertainty estimates. The authors do not discuss such practical implications, even though they use the AUROC metric for OOD detection evaluation which is scale- and shift-variant by design.\n- L242 contradicts with L207. According to L207, the objective enforces $p_\\psi(y \\mid x)$ to fit $\\frac{\\mathbf{1}_C + \\nu\\boldsymbol{\\eta}(x)}{C + \\nu\\mathbf{1}_C^\\top\\boldsymbol{\\eta}(x)} \\ne \\boldsymbol{\\eta}(x)$. This, in turn, makes the correspondence between L250 and L207 weaker.\n- In the abstract, the following is stated: \"[we] reveal that the EDL methods can be better interpreted as an out-of-distribution detection algorithm based on energy-based-models\". This claim is unsubstantiated, as the parallel with energy-based models is only drawn for EDL approaches that utilize ID + OOD mixture datasets for training. Having access to different training signals for ID and OOD samples naturally makes the learned models better OOD detectors (c.f. Figure 5). However, as Charpentier et al. (2020; NeurIPS) point out, this assumption is unrealistic and fails to cover _all_ meaningful OOD regions. In particular, eleven out of the nineteen methods Ulmer et al. (2023; TMLR) consider do not require any form of OOD data.\n- The paragraph at L304 suggests that incorporating model uncertainty would result in the \"dictionary definition\" of distributional uncertainty (which remains to be provided). This claim is not backed theoretically and is only supported empirically via distillation. The success of distillation might be mostly explained by the use of model ensembles (or MC-Dropout) and not the framework of Eq. (6) used for theoretical reasoning by the authors. The bridge between the theoretical argumentations and the empirical results L304 foreshadows is missing.\n- The improvement upon ensembling by further bootstrapping the training datasets is marginal. Due to the lack of error bars, Figure 3 can only be used to judge the result of a single seed, but the error bars of ensembles and bootstrapped ensembles might overlap significantly. The novelty of the use of bootstrapping is only in the context of EDL: Lakshminarayanan et al. (2017; NIPS) already considered bagged ensembles (and found that non-bagged variants performed better).\n- In the empirical evaluation several standard evaluation metrics are missing, e.g., the ECE score and (strictly) proper scoring rules. While total uncertainty and predictive uncertainty are evaluated, the aleatoric uncertainty estimate is not benchmarked separately. This would be key to back the claim that EDL methods are (mostly) OOD detectors: perhaps they are just as aligned with aleatoric uncertainty as epistemic, which would refute this claim.\n- The authors do not benchmark against a baseline, e.g., a cross-entropy-trained deterministic network. This makes the connection of results with methods outside the EDL framework hard.\n\n**Minor issues:**\n- \"underlying distribution\" appears twice in L79, making the sentence hard to parse.\n- $D(p\\ \\Vert\\ q)$ is an ambiguous notation for the Kullback-Leibler divergence. For example, it is often used for general Bregman divergences. Consider using the notation $D_\\text{KL}(p\\ \\Vert\\ q)$.\n- $h(\\cdot)$ is also an unusual notation for entropy. $\\mathbb{H}(\\cdot)$ or $\\mathrm{H}(\\cdot)$ are the usually used notations.\n- The UCE loss was not proposed by Charpentier et al. (2020, NeurIPS). Their contribution was the Regularized UCE. UCE was proposed by Biloš et al. (2019, NeurIPS). Therefore, the notation $\\ell_\\text{UCE}$ is also questionable.\n- \"Other choices of prior were also proposed to promote some other desired property.\" Such sentences are too ambiguous for an academic text. Consider naming (at least some of) the desired properties before pointing to further resources.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their thoughtful comments and address them in detail below.\n\n**1. The difference between RPriorNet and VI, UCE cannot support the claim \" the VI loss in Eq. (3) and UCE loss Eq. (4) turn out to be equivalent to the RPriorNet objective [...]\"**\n\nWe acknowledge that this sentence could be misleading, and we will revise it accordingly. What we want to convey is the resemblance in the in-distribution (ID) part of the objectives. Specifically, although VI loss and UCE loss were originally derived from different motivations, they are equivalent to the ID part of the VI objective in disguise. To our knowledge, this observation has not been made before, and we believe it provides a unified way to reason about a class of EDL methods in the literature.\n\n**2. EDL’s non-vanishing epistemic uncertainty is not surprising, and a data size dependent $\\lambda$ can resolve this issue.**\n\nWe agree that epistemic uncertainty would vanish with a choice of data size dependent $\\lambda$’s. However, we would like to point out a few key considerations. First, the classical Bayesian approach does not require a data size-dependent prior. As long as the data size goes to infinity, the epistemic uncertainty will eventually vanish regardless of the choice of prior. Second, it remains unclear how to properly define the data size-dependent $\\lambda$, in particular, its decay rate. Third, while it is true that one can always propose heuristic tricks to make the uncertainty of any algorithm align with the dictionary definition, this does not imply that such an algorithm is a faithful uncertainty quantifier.\n\n**3. EDL’s non-constant aleatoric uncertainty is expected, this holds for any uncertainty estimator.**\n\nAlthough it might be correct to say that aleatoric uncertainty is implicitly dependent on the model's capacity, randomness in training, data sampling, and other artifacts, what we are trying to argue is that, even under ideal scenarios (infinite sample size, overparameterized model), aleatoric uncertainty quantified by EDL methods can still be dependent on hyper-parameters ($\\lambda$).\n\n**4. EDL’s non-vanishing epistemic uncertainty does not affect the practical use of this uncertainty in downstream tasks.**\n\nFirst, in this paper, we do not claim that EDL methods are ineffective for downstream tasks. Instead, we attempt to answer the following question: “Why and how do EDL methods work in practice?” As a first step, we characterize the global optimizer of several EDL methods and highlight several consequences, including the non-vanishing epistemic uncertainty. We believe that a clear understanding of the objectives used for \"uncertainty quantification\" is crucial to understanding how EDL methods actually behave, how to further improve these methods, and how to lead to a more reliable and grounded UQ approach.\n\nSecond, while we agree that non-vanishing epistemic uncertainty might not be directly related to common UQ downstream tasks such as OOD detection, there are practical applications that require accurate quantification of epistemic uncertainty. For example, in active learning, we cannot rely on the EDL model’s epistemic uncertainty as a signal to indicate the model’s actual knowledge at a certain stage because it never vanishes. The $\\lambda$-dependent aleatoric uncertainty also may have issues in certain applications. For example, suppose an application relies on multiple pre-trained models to make predictions and identify ambiguous samples. If the learned aleatoric uncertainty of each model is trained with different $\\lambda$'s in an EDL framework, then they might not be in a comparable scale.\n\n**5. L242 contradicts with L207.**\n\nIn the paragraph, we consider the standard regime where $\\nu$ is sufficiently large, so that $\\alpha_0+\\nu\\eta(x)$ is approximately $\\nu\\eta(x)$, and the correspondence approximately holds in that regime.\n\n**6. The claim “the EDL methods can be better interpreted as an out-of-distribution detection algorithm based on energy-based-models”' is unsubstantiated, as multiple EDL methods do not require any form of OOD data.**\n\nWe acknowledge that such resemblance applies to not all EDL methods, but to the methods covered by the unified objective (6). The main claim of this section is that the unified objective (6) closely resembles the EBM-based OOD detection objective. When no OOD data is available, the resemblance still remains, if we ignore the OOD regularization term. In that case, they simply promote very large outputs for ID data.\n\nThe empirical success of EDL methods that do not assume OOD data, such as PostNet, implies that OOD data may not be required for OOD detectors to distinguish OOD data for some reason. We conjecture that this success is an artifact due to a combination of some optimization phenomena and neural networks’ extrapolation ability. In fact, similar to the eleven out of the nineteen methods in Ulmer et al., there are also many effective OOD detection algorithms in the literature that do not require any OOD data during training. \nWe acknowledge that this point was not made clear in the submission, and we will revise the section accordingly, especially in the case when no OOD data is available.\n\n**7. The claim about the benefit of incorporating model uncertainty is not backed theoretically.**\n\nPlease refer to global response 1.\n\n**8. Missing standard evaluation metrics such as ECE.**\n\nPlease refer to global response 2.(1) and Figure 13.\n\n**9. Missing aleatoric uncertainty benchmark.**\n\nPlease refer to global response 2.(2) and Figure 14., 15.\n\n**10. Missing benchmark against a cross-entropy-trained deterministic network.**\n\nPlease refer to global response 2.(3).", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
# To ALL Reviewers:
We thank all the reviewers’ effort in reviewing our paper and providing thoughtful comments. We would like to take this opportunity to further clarify our contribution, and resolve some of the common concerns as follows:

**1. Theoretical justification of proposed Bootstrap Distillation**

As several reviewers asked about a further justification of the proposed bootstrap distillation in Section 6, we provide our response here.

- We first emphasize that the main contribution of this paper is to analyze the common limitation of EDL methods with empirical support, show that standard EDL methods simply fit the UQ model to a fixed target (Section 5.1), and reveal that the learned uncertainties bear no statistical meaning. Since this is due to the ignorance of model uncertainty in the framework, we show that incorporating model uncertainty in the EDL framework via distillation can alleviate the characterized issues at the cost of computational complexity. We propose the bootstrap procedure in this context as a new mechanism to induce model uncertainty. While we do not claim that bootstrap distillation (BD) is the best approach, our experiment shows that BD attains a good downstream task performance even compared to the existing distillation based approaches END2 and S2D; see Tables 2, 3, and 4 in Appendix.
- This suggests that a further study of the distillation-based EDL framework is warranted. We acknowledge that the BD lacks a rigorous justification, but conducting a comprehensive theoretical analysis of the method is rather nontrivial. As reviewer gUpP also suggested, such an analysis could be considered as a separate publication in the future.
- While it is relatively easy to argue why the epistemic uncertainty would vanish in the sample limit ($|\mathcal{D}|\to\infty$) with the bootstrap from the intuition (Section 6), we believe that a more sophisticated asymptotic analysis for vanishing epistemic uncertainty can be carried out with overparameterized neural networks, adopting a similar setting in [A]. Succinctly speaking, if the trained model's prediction can be shown to be asymptotically normal in the limit of the sample size, we can argue that the uncertainty captured by bootstrap behaves as Gaussian of vanishing variance in the sample limit. This implies a naturally vanishing (epistemic) uncertainty. We will carry out the analysis and discuss this in more detail in the revision.

[A] Huang et al., Efficient Uncertainty Quantification and Reduction for Over-Parameterized Neural Networks, NeurIPS 2023.

**2. New experiment Results**

As some reviewers suggested, we provide additional experimental results to further strengthen our work. We summarize the results below and the accompanying figures can be found in the attached pdf:

- (1) **Calibration Performance.** We measure and compare the Expected Calibration Error (ECE) score of different EDL methods in Figure 13. Calibration measures how well the model's confidence aligns with its prediction accuracy. Good calibration performance also indicates accurate “total uncertainty” quantification. From Figure 13, we observe a behavior similar to the selective classification task in Figure 12. Distillation-based methods achieve lower ECE scores on calibration and better AUROC scores on the selective classification task, which relies on total uncertainty. We also observe that Fisher-EDL achieves the lowest calibration error, aligning with its promising performance on the selective classification task in Figure 12.
- (2) **Aleatoric Uncertainty Benchmark.** In Section 5.2 and Section H.2 of Appendix, we argue that using a Dirichlet-framework-independent objective that promotes the same behavior as the EBM-based OOD detection algorithm suffices for downstream task performance. We justify this through the epistemic uncertainty benchmark. We further examine the behavior of aleatoric uncertainty quantified by EDL methods. First, we provide a visualization of aleatoric uncertainty on 2D Gaussian data in Figure 14. Second, we evaluate EDL methods’ capability to detect ambiguous data (linear interpolation between two test images) using aleatoric uncertainty in Figure 15. The results in Figures 14 and 15, together with Section 5.2, further support that EDL's UQ capability (on both epistemic and aleatoric uncertainty) is not due to the Dirichlet framework but rather due to other auxiliary techniques.
- (3) **Comparison with a cross-entropy-trained deterministic network.** Due to space limitations, we summarize the results in words as follows. On the OOD detection task, the cross-entropy-trained network achieves an average AUROC score of 91.6 (66.8) on CIFAR10 (CIFAR100). On the selective classification task, the cross-entropy-trained network achieves AUROC scores of 90.2 (84.1) on CIFAR10 (CIFAR100). The performance of this baseline is suboptimal compared to most EDL methods, especially on the OOD detection task, as expected. We will add the concrete results (using tables and figures) in the revision.
- (4) **Behavior of deep ensemble’s epistemic uncertainty.** Although the behavior of classical Bayesian approaches and ensemble approaches has been well studied in the literature, we provide the epistemic uncertainty vs. sample size curve of the deep ensemble method in Figure 16. Unlike EDL methods, the epistemic uncertainty of the deep ensemble demonstrates a vanishing trend, as expected.

Meta-review: To all reviewers I would like to thank you for the time and effort you have put into the review process and for the in depth discussion with the authors. To the authors, thank you so much for engaging with the reviewers and for submitting your paper to NeurIPS.

After reading all of the reviews and all of the rebuttals, and reading the paper itself in depth, I would recommend this paper to be accepted.
I think this is a novel method to look at aspects of quantify Uncertainty in such models. This is heightened by the analytical treatment of the problem at hand and  specifically I think Lemma 5.1 can be something the community build upon and expand perhaps by testing with more empirical experiments. I think this is enough to compensate for the small scale fo experiments.

I would like to highly encourage the Authors to take into account all of the feedback  given and incorporate the remarks for the camera ready copy. 

Best wishes,

    AC
Decision: Accept (poster)