Credal Wrapper of Model Averaging for Uncertainty Estimation in Classification
Authors: Kaizheng Wang
Fabio Cuzzolin
Keivan Shariatmadar
David Moens
Hans Hallez
Venue: ICLR 2025 Spotlight (2025)
Keywords: Uncertainty Estimation
Model Averaging
Credal Stes
Probability Intervals
Out-of-Distribution Detection
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

Abstract: This paper presents an innovative approach, called credal wrapper, to formulating a credal set representation of model averaging for Bayesian neural networks (BNNs) and deep ensembles (DEs), capable of improving uncertainty estimation in classification tasks. Given a finite collection of single predictive distributions derived from BNNs or DEs, the proposed credal wrapper approach extracts an upper and a lower probability bound per class, acknowledging the epistemic uncertainty due to the availability of a limited amount of distributions. Such probability intervals over classes can be mapped on a convex set of probabilities (a credal set) from which, in turn, a unique prediction can be obtained using a transformation called intersection probability transformation. In this article, we conduct extensive experiments on several out-of-distribution (OOD) detection benchmarks, encompassing various dataset pairs (CIFAR10/100 vs SVHN/Tiny-ImageNet, CIFAR10 vs CIFAR10-C, CIFAR100 vs CIFAR100-C and ImageNet vs ImageNet-O) and using different network architectures (such as VGG16, ResNet-18/50, EfficientNet B2, and ViT Base). Compared to the BNN and DE baselines, the proposed credal wrapper method exhibits superior performance in uncertainty estimation and achieves a lower expected calibration error on corrupted data.

Published as a conference paper at ICLR 2025
CREDAL WRAPPER OF MODEL AVERAGING FOR UN- CERTAINTY ESTIMATION IN CLASSIFICATION
Kaizheng Wang -,55,- W4,5 Fabio Cuzzolin Keivan Shariatmadar David Moens 2,5 Hans Hallez!
DistriNett Department of Computer Science, KU Leuven, Belgium
2LMSD, Department of Mechanical Engineering, KU Leuven, Belgium
School of Engineering, Computing and Mathematics, Oxford Brookes University, UK *M-Group, KU Leuven, Belgium "Flanders Make KU Leuven, Belgium ka zheng. wang@kuleuver be fabic. cuzzolin@brookes. .ac. uk
(ke (keivan aivaan.hha sharisaraaae aataddar,, david.
hans. ha lez) @ku luven. be
ABSTRACT
This paper presents an innovative approach, called credal wrapper, to formulat- ing a credal set representation of model averaging for Bayesian neural networks (BNNs) and deep ensembles (DEs), capable of improving uncertainty estimation in classification tasks. Given a finite collection of single predictive distributions derived from BNNs DEs, the proposed credal wrapper approach extracts up- per and a lower probability bound per class, acknowledging the epistemic uncer- tainty due to the availability of limited amount of distributions. Such probability intervals over classes can be mapped on a convex set of probabilities (a credal set) from which, in turn, unique prediction can be obtained using a transforma- tion called intersection probability transformation. In this article, we conduct ex- tensive experiments on several
(OOD) detection benchmarks. encompassing various dataset pairs (CIFARI0/ 10/100 vs HrIN/Tiiy-mmageNe CI- FARI0 vs CIFARIO-C, CIFARI00 vs SIFAR100-4 and ImageNet vs ImageNet- O) and using different network architectures (such as VGG16, ResNet-18/500 Ef- ficientNet B2, and VIT Base). Compared the BNN and DE baselines, the pro- posed credal wrapper method exhibits superior performance in uncertainty esti- mation and achieves a lower expected calibration error on corrupted data.
INTRODUCTION
Despite their success in various scientific and industrial areas, deep neural networks often generate inaccurate and overconfident predictions when faced with uncertainties induced by, e.g. out-of- distribution (OOD) samples, natural fluctuations, or adversarial disruptions (Papernot et al.] 20166] Zhang et al., 202T; Hüllermeier & Waegeman] [02T Hendrycks & Gimpel, 2016)) Properly es- timating the uncertainty associated with their predictions is key to improving the reliability and robustness neural networks (Senge et al. 2014, Kendall & Gal [017)] Sale et al. 2023b). Re- searchers commonly distinguish two types of uncertainty in neural networks (and machine learning models in general). Aleatoric uncertainty (AU), also known as data uncertainty, arises from inherent randomness such as data noise, and is irreducible. Epistemic uncertainty (EU) stems from the lack of knowledge of the data generation process and can be reduced with increased availability of train- ing data (Hüllermeier Waegeman 202T). The distinction between AU and BU can be beneficial in applications such as active learning or OOD detection, particularly in safety-critich and practical fields such as autonomous driving (Zhou al. 2012) and medical diagnosis (Lambrou al.) 2010). For example, in active learning. the objective is to avoid inputs that exhibit high AU but low EU. Similarly, effective EU estimation can prevent the misclassification of ambiguous in-distribution (ID) examples OOD instances (Mukhoti et al. 2023). for the inherent ambiguity of the former does not arise from more epistemically uncertain regions of the ID distribution.
To represent and estimate both AU and EU, researchers have recently proposed neural networks able to output a "second-order" representation in the target space, to express their uncertainty about a pre-
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diction's 's uncertainty itself (Hüllermeier al., 2022 Sale et al. 2023b Cuzzolin 2024). Sampling- based approaches, among which Bayesian neural networks (BNNs) and deep ensembles (DEs), have emerged powerful methods in this domain.
The BNN framework (Blundell et al. 2015 Gal & Ghahramani 2016 Krueger ct al. 2017 Mobiny et al. 202T), in particular, quantifies uncertainty by learning posterior distribution over net- work's weights; this, in turn, is used to predict a "second-order" distribution (i.e., a distribution of distributions (Hüllermeier & Waegeman 202I)) over the target space. While BNNs are straight- forward to construct, their training presents a significant challenge due to inherent computational complexity. Consequently, several approximation algorithms have been developed, such as sam- pling methods (Neal et al., [011;; Hoffman et al. 2014:; Hobbhahn et al., [222]] Daxberger et al., 2021 (Tan et al., 2020; Rudner et al. 2022). variational inference approaches (Blundell et al. 2015, Gal & Chahramani 2016), and Laplace approximation methodologies (Jospin et al. As com- puting the exact 'second-order' distribution generated by full Bayesian inference at prediction time is of prohibitive complexity, Bayesian model averaging (BMA) is often applied in practice (Gal & Ghahramani 2016). The latter entails sampling a finite set of deterministic weight values from the posterior (obtained after training) to generate collection of single (softmax) distributions. How- ever, in practice, BNNs are generally challenging to scale to large datasets and architectures due to the high computational complexity (Mukhoti et al.) 2023), and limited number of samples is often used for B.M to reduce the inference complexity.
DEs (Lakshminarayanan et al. 2017). established alternative approach for quantifying uncer- tainty, have been recently serving as a strong baseline in deep learning (Ovadia et al. 2019 Custafs- son et 2020; Abe et al. 2022 Mucsányi al. 2024). DEs marginalize multiple standard neural network models to obtain a predictive distribution instead of explicitly inferring a distribution over parameters such as BNNs (Lakshminarayanan et al., 2017 Band et al. 202T). At prediction time, DEs average finite set of single probability distributions for classification. Despite their successes in uncertainty quantification, a recent study has shown that DEs may demonstrate relatively low quality of EU estimation (Abe et al.) (022). They have also been criticized because of their substan- tial demand for memory and computational resources (Liu et al. 2020 He et al. 2020).
As previously discussed, practical challenge in both BNNs and DEs at prediction time is the limited number of individual distributions (e.g., most papers use up to five) used to approximate the true predictive distribution and quantify uncertainty. Therefore, an intriguing research question arises here: Can the uncertainty quantification performance of the campling-based approaches be enhanced given the constrained number of predictive distributions?
Novelty and Contribution In response, this paper presents an innovative method for formulating a credal set representation for both BNNs with BMA and DEs, capable of improving their uncertainty estimation in classification tasks, which we term credal wrapper. Given a limited collection of single distributions derived, at prediction time, from either BNNs or DEs, the proposed approach computes a lower and an upper probability bound per class, i.e., a collection of probability intervals [Tessem] 1992 De Campos et al.) 1994). acknowledge the epistemic uncertainty about the sought exact predictive distribution induced by the limited information actually available.
Such lower and upper bounds on each class' probability induce, in turn, a convex set of probabilities, known as credal set (Levi 1980, De Campos et al., 1994). From such a credal set (the output of our credal wrapper), an intersection probability can be derived, in whose favor theoretical arguments exist as the natural way of mapping credal set to single probability (Cuzzolin) 2009 2022).
Extensive experiments are performed on several OOD detection benchmarks including different CIFAR10 vs CIFAR10-C, CIFAR 100 vs
dataset pairs (CIFAR 10/100 vs SVHN/Tnn- CIFAR100-C. and ImageNet vs ImageNet-O) and utilizing various network architectures (VGG16, ResNet- 18/50, EfficientNet B2 and VIT Base). Compared to the BNN and DE baselines, the pro- posed credal wrapper demonstrates improved uncertainty estimation and a lower expected calibra- tion error on the corrupted test instances when using the intersection probability for class prediction.
Other Related Work Numerous mathematical theories, such as subjective probability (De Finetti 2017), possibility theory (Zadeh] 1978 Dubois & Prade 1990). credal sets (Kyburg Jr 1987] Levi
1980 probability intervals, random sets (Nguyen, 1978). and imprecise probability theory (Wal-
ley 1991), have been devised to estimate the (epistemic) uncertainty in response to the challenge posed by the limited availability of information (in our case, a limited number of sampled single
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distributions). These theories state that the exact probability distribution is inaccessible and that the available evidence should instead be used to impose specific constraints on the unknown distribution.
Among them, probability intervals represent one of the simplest approaches and have attracted sig- nificant interest among researchers (Yager & Kreinovich] 1999; Guo & Tanaka 2010 Wei et all 2021 Cuzzolin 2022). Instead of providing point-valued probabilities, probability intervals assume that the probability values p(y) of the elements (e.g. classes) of the target space (e.g., the set of classes) belong to an interval Pr. (v) Sp(y) S (r((), delimited by a lower probability bound pt. (v) and an upper probability bound pu (y). Researchers have claimed that probability intervals may express uncertainty more appropriately than single probabilities (De Campos et al.) [1944] [ano & Moral 2002; Guo & Tanaka 2010), particularly in situations where: (i) limited information is avail- able to estimate unknown and exact probabilities; (ii) individual pieces of information are in conflict: e.g., when three predictors for the weather condition (rainy, sunny, or cloudy) predict probability vectors (0.2, 0.6, 0.2), (0.1, 0.2, 0.7), and (0.7, 0.1, 0.2), respectively. Such undesirable scenarios can arise when BNNs DEs use a limited number of predictive samples, we discuss here.
Credal sets have recently attracted increasing attention within the broader field of machine learning for uncertainty quantification (Zaffalon 2002; Corani & Zaffalon, 2008 Corani et al. 2012 Mauá
ct al. 2017; Hüllermeier & Waegeman 2021; Shaker & Hüllermeier| 2021 Sale et al. 2023a). The
advantage of adopting credal sets is the integration of notions of the set and probability distribution in a single and unified framework. Using credal sets, rather than individual distributions, arguably allows models to more naturally express epistemic uncertainty (Corani et al., [012 HHällermeier & Waegeman 2021). Concerning deep neural networks, 'imprecise" BNNs (Caprio et al.) 2024) mod- cling the network weights and predictions as credal sets have been proposed. Despite demonstrating robustness, the computational complexity of imprecise BNNs is comparable to that of the ensemble of BNNs, posing significant challenges for their widespread application. In addition, Wang et al. (225) have introduced credal-set interval neural networks, which predict credal sets from probabil- ity intervals based on the deterministic interval output of interval neural networks. While effective in uncertainty quantification for small datasets, their scalability is limited by high computational costs.
In addition to BNNs and DEs, evidential deep learning models (Malinin Gales) 2018: 2019
Malinin et al.) 2019; Charpentier et al.) 2020) have also been developed that generate a Dirichlet distribution (another type of 'second-order' representation) in the target space. A drawback of these methods is the fact that suitable Dirichlet distribution labels are not available at training time. Fur- ther, the behaviour of these models often deviates from theoretical EU assumptions (Ulmer et al.) 2023). Recent research (Juergens al., 2024) has shown that evidential methods often fail faith- fully represent EU, and that the resulting EU measures lack a meaningful quantitative interpretation.
The rest of the paper is structured follows. Sec. presents how uncertainty estimation works in different model classes. Sec. 3 inttroduces our credal wrapper in full detail. Sec. describes the experimental validations. Sec. summarizes our conclusion and future work. Appendices report additional experiments in and implementation details in $ respectively.
Paper Outline
2 UNCERTAINTY ESTIMATION IN DIFFERENT MODEL CLASSES
Bayesian Neural Networks BNNs model network parameters, i.e., weights and biases, as proba- bility distributions. The resulting predictive distributions can thus be seen as 'second-order' distri- butions, i.e., probability distributions of distributions (Hüllermeier & Waegeman, 202T). As men- tioned, for computational reasons, BMA is often applied for BNN inference (Gal & Ghahramani 2016). Namely, in a classification context, we obtain
p- Σ N₂ p. N₂ ii A (x),
where P; is the sampled prediction from the deterministic model (h parametrized by w. Np is the number of samples, and p is the averaged probability. The parameter vector w; obtained from the i-th sampling instance of the parameter posterior distribution of BNNs (Jospin et al. 2022).
Employing Shannon entropy as the uncertainty measure, the total uncertainty (TU) and AU in BNNs can be approximately quantified by calculating the entropy of the averaged prediction and averaging the entropy of each sampled prediction (Hillermeier & Waegeman] 2021), respectively, as follows:
TU: H(p) = -E AU:=H(p) = * = ,, E
(2)
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where 9k and Piz the k-th elements of the probability vectors p and ps across the C' classes, respectively. An estimate of epistemic uncertainty can then be derived EU H()) (p) (Depeweg et al.] 2018), which can be interpreted as an approximation of 'mutual information Hüllermeier et al. 2022 Hüllermeier & Waegeman 202I).
Deep Ensembles DEs generate an averaged prediction from a set of M individuullly-taanned standard neural networks (SNNs), follows:
p M Em M Am (x)
31 Σm M
Pm:
where Pm denotes the single probability vector provided by the m-th SNN. Considered by some to
be an approximation of BMA (Wilson & Izmailov 2020: Abe et al. 2022). DEs also quantify TU, AU, and EU as in eq. 2 where the sampled single probability vector p: is replaced by Pm-
Credal Sets Uncertainty quantification for credal sets is an active research subject (Hüllermeier &
Waegeman 2021 Sale et al. 2023a). To that extent, researchers have developed the concepts of
generalized entropy (Abellán et al. (006) and generalized Hartley (GH) measure (Abellán & Moral, 2000 Hüllermeier et al. 2022). However, applying the GH measure practice is challenging due to the high computational cost of solving constrained optimization problems involving 2 subsets Hüülermeier& & Waegeman 2021 Hüllermeeer et al. 2022), particularly when the number classes C' is large (e.g.. .... C' = 100). The width of the probability interval has also been proposed as EU measure (Hllermeiir et al.) 2022), but is only applicable to binary cases. As a result, we opt for generalized entropy. Given credal set prediction, denoted by IP, its upper and lower entropy, (P)) and (P)), can be calculated as a generalization of the classical Shannon entropy, allowing us to estimate the TU and AU for credal prediction (Abellán et al. 2006), as follows:
TU: H (IP)= = max/ (p), AU:= (IP) minH(()).........
(4)
pEP
pEP
Such measures capture the maximal and the minimal Shannon entropy within the credal set, respec- tively. The level of EU can then be measured by their difference, namely EU (P))H((P).
3 METHODOL.OGY
Inspired by the use of probability intervals for decision-making YYgeer& & Kreinovich 1999 Guo & Tanaka 2010), we propose to build probability intervals by extracting the upper and lower bound per class from the given set of limited (categorical) probability distributions, validating this choice via extensive experiments in Sec. E.g., consider again the task of predicting weather conditions (rainy, sunny, cloudy). When receiving three probability values for the rainy condition, e.g.. 0.2, 0.1, and 0.7, using probability intervals we model the uncertainty on the probability of the rainy condition as 00.10.77, Each probability interval system can determine convex set of probabilities of classes, i.e., a credal set. Such a credal set is a more natural model than individual distributions for representing the epistemic uncertainty encoded by the prediction, as it amounts constraints on the unknown exact distribution (Hüllermeier & Waegeman, 2021] Shaker & Hüllermeier] [021] Sale al., (023a). Nevertheless, a single predictive distribution, termed intersection probabiliry, can still be derived from a credal set to generate unique class prediction for classification purposes.
the set
Our credal wrapper framework is depicted in Figure [ The remainder of this section discusses the credal wrapper generation, a method for computational complexity reduction for uncertainty estimation, and the intersection probability, in this order.
Credal Wrapper Generation Given a set of N individual distributions from BNNs DEs, an upper and a lower probability bound for k-th class element, denoted as pu, and pr. respectively, can be obtained from
PU2 =
max Pr.k: 12 I..I..
PL. =
min Pn.k-
(5)
n=I...,N
where Po,k denotes the k-th element of the n-th single probability vector Pn. Such probability intervals over C classes determine a non-empty credal set IP, as follows (Moral-Gareia & Abellán De Campos et al. 1994):
2021
P- e C-1 and E (pL,, Pu. Ah: 2,.,.., C} s.t. ΣE PL. < SIS PU., (6)
where p denotes the valid (normalized) probability vector and A C-1 represents the probability simplex on the set of C classes. The condition in eq. 66) guarantees that individual probabilities in
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Single prediction F
Extract Upper Lower Bound Per Class
Intersection Probability Transform
Uncertainty
Estimation TU: H(?) EU: (P) M
Multiple Sampled Single Probability Distributions
Credal Set (P) Determined
Intersection Probability (p') in Probability Interval System
Probability py Intervals
Figure 1: Credal wrapper framework for a three-class (A, B. D) classification task. Given set of individual probability distributions (denoted as single dots) in the simplex (triangle) of probability distributions of the classes, probability intervals (parallel lines) are derived by extracting the upper and lower probability bounds per class, using eq. ( Such lower and upper probability intervals induce a credal set on {A, B, D} (I, light blue convex hull in the triangle). A single intersection probability (the red dot) is computed from the credal set using the transform in eq. (5) Uncertainty is estimated in the mathematical framework of credal sets eq.
the credal set satisfy the normalization condition, and their probability value per class is constrained to the given probability interval. It can be readily proven that the probability intervals generated in eq. (5) meet the above condition, as follows:
ΣS = Ee min Pr.kk S P - A E max Pa.k m=
P
n= N
where n² is any index in 1,..., N. Computing (T)) in eq.
(4) for uncertainty estimation, on the
other hand, requires solving the following optimization problems:
HP)) = maximize Ee -x-Ph Pt -loge Pk s.t. Eer==
p =1 and pa E(pt.. PU,] Vk=1,..,,C;
which can be addressed by using standard solver, such as the SciPy optimization package (Virtanen cct al. 2020). To calculate (P) in eq. maximize is simply replaced by minimize. We further discuss an alternative approach for generating credal sets in Appendix C
The convex optimization problem in eq. (8) may present computational challenge for large value of C (c.g., C= 1000). To mitigate this issue, we propose an approach, termed Probability Interval Approximation (PIA), in Algorithm [ The PIA method initially identifies the top J relevant classes by sorting the probability values of the intersection probability p" (detailed in eq. (9)) below) in descending order. Then, the remaining elements are merged into a single class whose upper and lower probability are computed. As result, the dimension of the approximate probability interval is reduced from C' to J.
Computational Complexity Reduction Method
Algorithm Probability Interval Approximation
Input: P for 1= Intersection probability p¹; Chosen number of classes J Output: Approximated probability intervals [rT,, rU,l for 1=
1: Compute the index vector for sorting p" in descending order as
m:= (==m,, m2, .... mc) for Pm, 2 P'n, 2 ... 2 Pimc
2: Calculate the upper and lower probability per selected class
TLy ;- P=PLmy TU, := PU,,, for j= J-1
3: Calculate upper and lower probability for deselected classes as
r1 = max -imm, PU. )1,
= =min(1 (1- Limm, PLe ))1))
Intersection Probability In classification tasks. it is desirable to eventually map credal set to single probability prediction to make a unique class prediction. In our credal wrapper, from the probability interval predictions defined in eq. (5), we derive a single intersection probability vector for class prediction (Cuzzolin) 2022) the unique probability distribution which (i) ensures that the derived single probability is normalized within the probability interval system, and (ii) assumes equal trust in the probability intervals for each class, since there is no justification for treating them differently (Cuzzolin 2022). Mathematically, the intersection probability, p", must satisfy
Pt PL. + (PUs PL.)) Vk 1,...,C
(9)
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where pt is the k-th element of p" under constant a E [0. 1]. This is convex combination of PL. and pú., so that Pi E [PLL 1PLu-PG,] p.;) Due to the normalization constraint, > P =, it follows that:
[p +alpu, - PL.)) =1 = Σx (pp,)++ Σx (pU, PL, =1.
(10)
As a result, the unique a can be computed as follows (uuzzolin 2022
a= PL.)/(E$ C (pu, -PL.))
(11)
The intersection probability is thus given by eq. (9) with a as given in eq. ITI.. The intersection probability transform follows clear rationale prineiples and does not suffer from the drawbacks of other potential transforms. Let us see this in detail. (i) Taking the average probability value of the original sampled distributions produces a valid probability vector that falls within the probability interval system. However, such transform does not leverage the probability bounds, thus violating the semantics of probability intervals (Guo & Tanaka ing inferior performance in the experiments). (ii) Selecting the midpoint of each probability interval [PL. p. instead, does not generally result in a valid normalized probability vector. (iii) Normalizing the lower or the upper probability vectors, i.e., taking PU, = pu. Σx PU, or PL. PLe/ EE PL. yields a probability that not guaranteed to be consistent with the interval system (Cuzzolin (2020). In Appendix $D we further discuss the limit behavior of the intersection probability when the num- ber of sampled probability distributions goes to infinity.
& Moral (002) (and also exhibit-
4 EXPERIMENTAL VALIDATION
In this section, we apply OOD detection benchmarks for EU quantification assessment OOD (ukkhoti et al. 2023 Mucsányi et al., 2024). Regarding OOD detection, model is expected to exhibit high EU estimates on OOD samples in comparison to in-domain (ID) instances. Conse- quently, superior OOD detection performance provides evidence of the enhanced uncertainty esti- mation quality. To assess OOD detection performance, we use the AUROC (Area Under the Re- ceiver Operating Characteristic curve) and AUPRC (Area Under the Precision-Recal curve) scores. Greater scores indicate a higher quality of uncertainty estimation. In addition to OOD detection performance, expected calibration error (ECE) (Guo et al. [0177] Nixon et al. 2019) is also applied to measure the model uncertainty and calibration performance. A lower ECE value signifies a closer alignment between the model's 's confidence scores and the true probabilities of the events.
Evaluation using Small-scale Datasets In this experiment, we use dataset pairs of the type (ID samples vs OOD data), including CIFARIO (Krizhevsky et al.) 2009) vs SVHN (Hendrycks) et al. TiyTy/TiyylmmagNee (Le & Yang 2015) and CIFARIO vs CIFARIO-C (Hendrycks & Di- etterich (019)) As for BNN baselines, we choose two standardized variational BNÑs: BNNR (Auto-Encoding variational Bayes (Kingma & Welling 2013) with the local reparameterization trick (Molchanov et al.] (017 and BNNF (Flipout gradient estimator with negative evidence of lower bound loss (Wen al. 2018)). We do not consider BNNs using sampling approaches because of their generally high computational resource requirements (Gawlikowski et al. 2021 Jospin et al.) 2022). As for DEs, we aggregate five standard neural networks, trained using distinct random seeds. All models are implemented on the established VGG16 (Simonyan & Zisserman (015)) and ResNet- 18 (He al., 2016) architectures using the CIFARIO dataset. The number of sampled predictions for BNNs is set to N = 5. While the main aim of this work is to improve the uncertainty estimation performance of BNNs and DEs through the credal wrapper, is also intriguing to compare our ap- proach with an alternative method, Ensemble Distribution Distillation (EDD) (Malinin et al.) 2019), within the evidential deep learning family. EDD aims to approximate DE predictions on the simplex by leveraging single Dirichlet distribution. Thus, here we train EDD from the DEs under two dis- tinct learning rate scheduler configurations: (1) EDD-Fair, which uses the same basic scheduler as BNNs and SNNs to ensure a fair comparison; and (2) EDD, which employs the cycle learning rate policy described in the original EDD paper. More implementation details are given in Appendix SB)
Table I reports test accuracy (ACC) and ECE values on the CIFARIO test set and the corrupted CIFARTO (CIFARIO-C) data as well as the OOD detection performance comparison. Regarding the CIFAR10-4 10-C dataset, which applies 15 corruption modes to the CIFARIO instances. with 5 inten- sities per corruption type, Figures 2 and 3 show the OOD detection results and the ECE values of CIFAR10 vs CIFAR 10-C against increased corruption intensity, respectively. The results over the 15

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Table 1: Performance comparison between the classical and credal wrapper version of BNN and DE, as well as EDD models. All models are implemented on VGG16/ResNett 18 backbones and trained using CIFARIO data ID samples. The results are from 15 runs. The best scores per metric are in bold. The results on corrupted data are averaged over all corruption types and intensities.
Prediction Performance
OOD Detection Performance (5) using EU
ID Test Data
Corrupted Data
SVHN (OOD)
Tiny-ImageNet (OOD)
ACC (98)
ECE
ACC (%)
ECE
AUROC
AUPRC
AUROC
AUPRC
Bascline
90,98s0.11
0.04849.000
21.8611.88
0.47020000
86.65.1.26
90.6140.85
64.6220.28
80.0620.40
BNNR
Ours
90.98200.15
0.03420.003
21.9621.83
0.416:00000
87,85=1.45
92.3221105
85,55=0.30
82.7620.46
Baseline
90,8820.19
0.05520.000
23.00
0.498
86. 86.79.0.47
90.76.0.55
84.54.0 20
29.91
VGGL6
BNNF
Ours
90.84:0.20
0.04340.003
23.08:1.95
0.452:9.068
87.6620.50
92.26210.51
85,266:0.20
82.33
Baseline
93.5720.09
0.015:0.001
24.81=0.79
0.314
89.74.1.31
93 38,0.97
88, 88.49.0011 17
85. 35
DE
Ours
93.47=0.10
0.03340.002
24.42+081
0.200le0022
91.96a1.33
95.2440.79
89.80=8008
87.99:0.16
EDD
91.05=0.18
0,038+0..03
19.37+2.24
0.122=0.022
91.21+0095
93.62+1.70
87,04e0.75
82,53e2.18
BNNR
Baseline
91,87=0.17
0.05840.002
22.38
0.49820.030
88.32s1.22
93.03+0.74
85,83,0
81.43
Ours
91.8660.18 18
0.055+0.002
22.45e1.30
0.4601=0.030
88.40:1.25
93. 93.1699
83.99-0.20
=0.51
ResNet- 18
Baseline
91.91a0.19
0.057+0.002
22.38 1.30
0.498:0.024
88 $8.62.00.0
93. 22.2666..55 55
85, 35,9440 30
81 64:0 34
BNNF
Ours
91.90e0.18
0.055e0.002
22.43+1.30
0.464:0.024
88.71=1.81
93.41=0051
86.06ie0.31
82.69
Baseline
93.36s0.11
0.018s0.001
26.62+0.69
0.304+0018
67.40+0.68
81.8220.51
87,94+0 15
84 86.00
DE
Ours
93.35.0.13
0.02510.002
26,4110,73
0.224
88.6320.80
92.7610.64
89.1620015
87.41
EDD
92.9620.10
0.02740.001
27.00=1.20
0.27620:032
87.2621.57
91.9141.01
88,05
83.66.0.41
corruption types are averaged. Table[aand Figure demonstrate demunrt3emonnstrate that our intersection probability ex- hibits comparable test ACC and ECE on II) test data, compared to the classical averaged probability. Moreover, the calibration performance on corrupted samples is consistently better, as evidenced by the lower ECE values. Table T and Figure also demonstrate that our credal wrapper markedly and persistently enhances the EU estimation of baselines. This is evidenced by the augmented OOD de- tection performance applies a cycle learning rate policy, our credal wrapper still consistently achieves superior perfor- mance, as shown in Table [ and Figure _ Concerning EDD-Fair, we observe that fairly applying the simple learning rate scheduler leads to low prediction performance of EDD-Fair (considerably below the baselines) and poor EU estimation (evidenced by the lowest OOD detection values), as shown in Table i9 the Appendix.
various data pairs and across disparate backbones. Compared to EDD, which
0.852
Mergaa AUROC
0.920
0.800
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0.930
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500
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Overa Comuation Intensity
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-0- DNN:: (Dur) -d- DWNB: (Bauckire)
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-8
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0.955
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Overil Compution Intensiyy
Overal Compation interaity
Overall Comuption merrrlty
Ovaral Corrattinn momulty
---8RR: (Durs)
S0BR: (Basokine)
BNN- (Qus))
- BIAe (Bage
DE: (Qurs)
DE (Basclino)
800
Backbone: ResNet-18
Figure 2: OOD detection using EU as the metric on CIFARIO vs CIFAR10-C of the classical and credal wrapper version of BNNs and DE, and EDD against increased corruption intensity, using VGG16 and R8 as backbones.
Evaluation using Large-scale Datasets In this experiment, we exclusively employ DEs (combin- ing five SNNs) to assess the quality of EU estimation of our credal wrapper on large-scale datasets and network architectures. This is because, in practice, BNNs are typically unable to scale to large datasets and model architectures due to the high computational complexity (Mukhoti et al.] 2023
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04-
24
8
202
12
Aweraged Prb.
Aerraged Inlosettion Prbhs
Aperaged Frtt Riesetin
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Overa Gomarian trtennty
Overaa Caman reeesit DE (ReshetL8)
ENNR (VGG16)
DE (VGG16)
BNNF (ResNer18)
Figure 3: ECE values of BNNR, BNNF, and DE on CIFARIO-C against increased corruption in- lensity, using the averaged probability (Prob.) and our proposed intersection probability (Prob.). VGG16 and ResNet- are backbones. Results are from 15 runs.
Table 2:
OOD detection AUROC and AUPRC performance (%) of both the classical and credal wrapper version DEs using EU as the metric. The results are from 15 runs, based on the ResNet- 50 backbone. Best scores are in bold.
CIFARIO (ID)
CIFARI0O (ID)
ImageNet (ID)
SVHN (OOD)
Tiny-ImageNet (OOD)
SVHN (OOD)
Tin--mageNett (OOD)
ImageNet-O (OOD)
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
Baseline
89, 89.5810993
92.2921.00 86.8720.20
83,0280.16
73,8321.9
84.96&1.25
78 80
65, 65.702041 74.68+0.27
63.2020.35
Ours
93.77#0.60 96.06620.46 88.78a0.15 86.8360.23
80.22
89.40=1.03 81.0020.16
77.1648.23
66.20H0.38
63.23+9.34
For instance, training ResNet-50-based BNN on CIFAR-10 (resized to (224, 224, 3)) failed in our experiment due to exceeding the memory capacity of a single Nvidia A100 GPU. The dataset pairs (ID vs OOD) considered include CIFAR IO/CIIAAAARRIIIIIR 100 (Krizhessyy) 2012) vs SVHN/Tiny- ImageNet, ImageNet (Deng et al., 2009) ImageNet-O (Hendrycks
al. (021). CIFARIO vs CIFARIO-C, and CIFARIOO vs CIFAR00--C (Hendrycks & Dietterich] 2019). DEs are imple- mented the well-established ResNet-50 (He et al.] (016). All input data have shape of (224, 224, 3). More training details are given in Appendix S The PIA algorithm (Algorithm) T is ap- plied using the settings J 20 and J = 50 to calculate the generalized entropy (TP and H(P) on dataset pairs involving CIFARI00 and ImageNet, respectively. Compared to classical DEs, our credal wrapper demonstrates the enhanced OOD detection across spectrum of data pairs, as shown in Table suggesting that our proposed method can consistently improve EU estimation.
Ablation Study on Network Architectures We also perform an ablation study using Efficient- NetB2 (EffB2) (Tan & Le (021) and Vision Transformer Base (ViT-B) (Dosovitskiy et al., 2020) as architecture backbones, involving CIFAR10/100 vs SVHN and Tiny-ImageNet, CIFARTO T0 vs CIFARIO-C, and CIFAR100 vs CIFARI00-C. The key findings, presented in Table 3 and Figure demonstrate that our credal wrapper approach significantly enhances the EU quality of DEs, and robust against both the dataset pairs and the architecture backbones. Furthermore, 5 shows the ECE values exhibited by DEs on both CIFAR 10-C and CIFARI00-C datasets, using ResNet-50, EffB2, and ViT-B backbones. The results indicate that adopting the intersection probability as the final prediction, with its strong rationality foundations, can indeed empirically improve calibration performance (lower ECE) on the corrupted samples by using the classical averaged probability.
Table 3: OOD detection AUROC and AUPRC performance (%) of the classical and credal wrap- per version of DEs using EU as the metric. Results are from 15 runs, based on EffB2 and ViT-B backbones. Best scores are in bold.
CIFAR 10 (TD)
CIFAR 100 (TD)
SVHN (OOD)
Iny--ImgeeNet (OOD)
SVHN (OOD)
Tiny-ImageNet (OOD)
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
EAR2
Baseline Ours
95.76en.59
97.43+B447
92.3240.14
90.72=0.22
87.5241.52
93,81=080
85.29=0
82,9840.24
97.60s0.25
98.74s0.14
93.5840.15
93.17a0.16
89.20x1.19
94.58.0.56
86.23a0.10
83.69.0.11
Baseline
77.71+1.7
85. 1.41
82,2740.79
78.85=0.81
81.41+1
89.00=0.%
81. 18
77. 47.4640.49
VIT-B
Ours
80.71
88.43&1.26
84.28.0.76
82.82.0.85
84.87
91.40=0.67
82.53.0.32
79.04.0.39
Ablation Study on Hyperparameter of PIA Algorithm In this experiment, we investigate the effect of the hyperparameter J of the PIA algorithm [ The OOD detection tasks include CIFAR100 (ID) SVHN (OOD) and -mny-ImaeeNet (OOD) and deep ensembles are implemented on the ResNet-50 backbone. Table shows OOD detection performance and the time cost for uncertainty quantification per sample, using different settings of J for the PIA algorithm.
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2.980
0.520
0.825
gosgon A.ROC
3,440
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-220
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600
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880
0.800
Overal Compaption intensityy
Overa Compitioo Intenchy
Overs Comuption intensity
Overal Comuption mendty
EIBE :(.urs) -de-
VIT-B: (Cuus)
CIFARIO VS CIFARIO-C
CIFAR1DO vs CIFARIOO-C
Figure 4: OOD detection performance of the classical and credal wrapper version of DEs using EU as the metric on CIFAR vs ResNet-50, EffB2, and ViT-B as backbones.
against increased corruption intensity, using
fnnraged 200.. Massecile
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12
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Doeri VIT-B (CIFARLO)
ResNet-50 50 (CIFAR101
EMB2 (CIFAR30)
Restiet (CIFAR100)
EMSZICIFARRO0
VIT-B (CIFARI0O)
Figure 5: ECE values of DEs on CIFARIO-C and CIFAR100-C against increased corruption in- tensity, using the averaged probability (Prob.) and our proposed intersection probability (Prob.). ResNet-50, EffB2, and ViT-B are backbones. Results are from 15 runs.
The hyperparameter J is design choice intended to mitigate the computational cost associated with uncertainty quantification, particularly in situations with limited computational resources. The evaluation indicates that the proposed PIA algorithm can significantly reduce the computational complexity of estimating uncertainty in cases high-class classification while maintaining superior uncertainty performance
OOD detection benchmarks. We can also observe that increasing the value of J improves the OOD detection performance, but can lead to an increase execution time. This is because solving the constrained optimization problem in eq. (8) involves more variables and constraints. Furthermore, we present the uncertainty estimates for ID and OOD samples at different values of J in Figure i6 Appendix $A.10 The result shows that using small value of J leads to an underestimation of uncertainty values, particularly for OOD samples. This phenomenon explains why higher value of J can achieve better OOD detection performance.
Table 4: OOD detection AUROC and AUPRC performance (%) of credal wrapper of DEs using EU as uncertainty metrics and the time cost, using different setting of J of PIA algorithm. The OOD detection involves CIFAR 100 (ID) SVHN (OOD) and Tmag-ema (OOD). The results from 15 runs, based on the ResNet-50 backbone.
EU as Metric
Time Cost per Sample (ms)
SVIIN
Tiny ImageNet
AUROC
AUPRC
AUROC
AUPRC
78.81
87.9621.0 .9
80.1720.19
75.25a0.26
7.182#0.087
20
80.22+1.00 96
89 49.401100
81.00=0.16
77. 164163
11. 03334640.08
80.55A1.99
89.68e104
81.1620.16
77.76a0.1
109.469a1.330
Computational Cost The reported time in Table 4 cost is measured on single Intel Xeon Gold 8358 CPU@ 2.6 GHz, without optimization in the calculation process. As reference, the time cost of the classical method for uncertainty quantification per sample is 4.007s.. time cost for CIFAR 10 without reduction can also be seen as 7. 8ms from Table While bit high, remains practical without limiting computational resources. We also believe a more efficient code implementation of our approach could significantly mitigate the computational cost.
addition, the
Moreover, our method can achieve superior uncertainty quantification performance with fewer pre- dictive samples, as demonstrated in 5and the Appendix. Additionally, the computational cost analysis in Appendix $A.9 indicates that the overhead introduced by our method for entropy calculation is negligible, particularly when accounting for the significant reduction in overall in- ference cost. From this perspective, the proposed credal wrapper not only reduces computational complexity but also enhances practical applicability.

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Additional Experiments Appendix $A.1 presents visualizations of the credal set predictions and intersection probability, offering an intuitive graphical explanation of why our credal wrapper out- performs the baseline methods. Appendix $A.2 and AA.3 present ablation studies the number of predictive samples in DEs and BNNs, respectively. The results consistently demonstrate the superior performance of our credal wrapper method. Appendix $A.4 the EU quantification perfor- mance of our credal wrapper using the generalized Hartley measure (Abellán & Moral 2000). as an alternative to the difference between upper and lower entropy in eq. (4) The results demonstrate that our credal wrapper consistently enhances the EU estimation performance while being agnostic to the specific EU measure for credal sets employed. Appendix $A.5 assesses the TU estimation quality (as opposed to BU) OOD detection benchmarks, suggesting that our credal wrapper con- sistently improves TU estimation as well, across the different dataset pairs, network architectures, and test settings. Appendix in scenarios where the ensemble member (SNN) generates overconfident predictions. The findings also show the improved uncertainty quantification of our credal wrapper. Appendix $A.7 presents uncertainty estimate plots for both ID and OOD samples across various settings, providing further qualitative evidence that our credal wrapper consistently enhances baseline uncertainty quantifi- cation. Appendix &A.8 assesses intersection probability on corrupted samples using the negative log-likelihodd (NLL) metric, showing consistently improved performance with lower values across extensive test scenarios.
and
an examination of the performance of our credal wrapper
5 CONCLUSION AND FUTURE WORK
Conclusion This paper presents innovative approach, called credal wrapper, to formulating a credal set prediction for model averaging in BNNs and DEs, capable of improving their EU and TU estimation in classification tasks. Given limited collection of single distributions derived prediction time from either BNNs or DEs, the proposed approach computes lower and an upper probability bound per class, to acknowledge the epistemic uncertainty about the sought exact pre- dictive distribution due to limited information availability. Such lower and upper probability bounds on each class induce credal set. From such credal prediction (the output of our wrapper), an intersection probability distribution can be derived, for which there are theoretical arguments that it the natural way to map a credal set to a single probability.
Extensive experiments performed several OOD detection benchmarks, including CIFAR10/100 vs Tell//iinyIIageNNC CIFAR 00 vs CIFAR10/1000 10/100-C, and ImageNet vs ImageNet-O. and utilizing various network architectures 16 ResNct-18/50, EfficientNet B2 and ViT Base), demonstrate that, compared to BNN and DE baselines, the proposed credal wrapper approach ex- hibits better uncertainty estimation and lower ECE on corrupted data.
Limitation and Future Work Our credal wrapper functions as a plug-and- aage-and--lyy -lay method for enhanc- ing the uncertainty quantification of BNNs and DEs without requiring any additional training. How- ever, uncertainty quantification in this setting demands a higher computational investment. Thus, memory usage or computational resources are strictly limited, the proposed approach may not be the optimal solution. A primary objective of our forthcoming research is to apply and benchmark our approach on safety-critical and real-world applications, such medical image analysis.
ACKNOWLEDGMENT
We thank the anonymous reviewers for their valuable feedback. This work has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement No. 964505 (E-pi).
REPRODUCIBILITY STATEMENT
Detailed implementation description is provided in B The experiment codes provided here
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TABLE OF APPENDIX CONTENT
Additional Experiments
16
A.1
Visualization of Credal Set Predictions and Intersection Probability
16
A.2
Ablation Study on Numbers of Predictive Samples in DEs
18
A.3
Ablation Study Numbers of Predictive Samples in BNNs
19
A.4
Generalized Hartley Measure for RU Estimation of Credal Wrapper
19
A.5
Total Uncertainty Estimation Evaluation on OOD Detection Benchmarks
20
A.6
Ablation Study on Overconfidence Regime
20
A.7
Qualitative Evaluation of Uncertainty Estimation of Credal Wrapper
23
A.8
Evaluation of Intersection Probability on Corrupted Samples using NLL Metrie
24
A.9
Additional Results on Computational Cost
25
A.
10 Additional Results on Different Hyperparameterss of PIA Algorithm
25
Experimental Implementation Details
26
Discussion Generating Credal Sets from Individual Probability Distributions
27
D Discussion on Credal Wrapper in Cases of Infinite Samples
28
A ADDITIONAL EXPERIMENTS
A.1 VISUALIZATION CREDAL SET PREDICTIONS AND INTERSECTION PROBABILITY
Visualizing credal sets in high-ding -igeensiiiii (C) classification tasks is challenging, as it requires (C dimeessioaal probability simplex. To demonstrate our credal wrapper method, we con- sider three-class classification scenario derived from the CIFAR10 classification tasks in the main body, focusing on the classes 'airplane', 'automobile', and 'others'. The class 'others' includes the remaining class indices in the original CIFAR10 datasets. We visualize our credal wrapper of ResNet50-based DEs and VGG16-based BNNRs using different sample sizes on the 2D simplex in Figures 6 and Three cases considered: the correctly classified in-distribution (ID) sample (left), the incorrectly classified sample (middle), and the OOD samples (right).
From the visualizations, observe that the differences between individual predictive probability samples for correctly classified ID data and the resulting discrepancy between the intersection prob- ability and the averaged probability differences become more apparent. Since the intersection probability is derived from probability intervals, it tends exhibit less overconfidence in incorrect predictions. A similar observation can be seen in Figure g where the individual probabilities, the corresponding intersection probability, and the averaged probability for the CIFARIO and CIFARIO-C (corrupted) samples are visualized. Furthermore, as the ensemble size increases, this reduction in overconfidence becomes even more apparent (Figures(olandd 7. These observations explain the comparable performance of the intersec- tion probability and the averaged probability in terms of test accuracy and expected calibration error (ECH) on ID data but improved performance on corrupted data.
less visible. However, for incorrectly classified samples, the
Figures 6 and 7 also illustrate that the credal sets for incorrectly classified and OOD samples are notably larger than those for correctly classified ID samples. This behavior aligns with the construc- tion of our credal wrapper, which is formulated by extracting the upper and lower probability limits per class from individual probability samples. also shows that the credal sets naturally expand when newly sampled probabilities fall outside the original bounds (for example, the OOD cases in Figures/6Jand reflecting a conservation property that reasonably incorporates new evidence rather than disregarding it. This phenomenon also explain why the OOD detection performance can
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be further enhanced with an increased sample size, observed in Tables and 6 Furthermore, the theoretical foundation for uncertainty quantification using credal sets and probability interval systems is robust, detailed in Appendix A.4 and Appendix S
Correctlyy Classified Sample automobile
Incorrectly Classified Sample automobile
Out-of-distribution (OOD) Sample
automobile
automobile
airplane
airplane
airplane
label: airplane 5) automobile
label: airplane 5) automobile
000: SVHN
0OD: Tiny-lmageNet
automobile
automobile
airplane
airplane
airlaan
airplane
label: sirplane
label: airplane
0DD: SVHN
DOD: Ting-ImageNee
(N 25)
[N = 25)
= 25)
d 25)
individual probebility samples intersection probability
averaged probebility
credal set
Figure 6: Visualization of our credal wrapper of ResNet50-basse DEs on different ensemble sizes (N
and N 25) in three cases: correctly classified in-distribution (ID) sample (left), incor- rectly classified sample (middle), and OOD samples (right).
Correctly Classified Sample automobile
Incorrectly Classified Sample
Out-of-distriutiion (O0D) Sample
automobile
automobile
automobile
birplane
others
airpiane
others
airplane
others
airplane
others
airplane
g00: ImageNtt d (N 10) automobile
aNN - 10)
[ 10) automobile
(N - 10)
automobile
automablle
airplane
others
airplane
airplane
others
airplane
automobile IN = 50)
airplane
0OD: Tinp-mtattm
b IW = 50)
IN = 50)
dW = 50)
i individual probability samples
intersection probability
averaped probabiltyy
credal set
Figure 7: Visualization of our credal wrapper of 6 baseed BNNRs on different sample sizes (N 10 and N 50) in three cases: correctly classified in in-distribution (ID) sample (left), incor- rectly classified sample (middle), and OOD samples (right).
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1.0
1.0
0.8
0.8
probability value
probability value
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
airplane
automobile
bird
Cat
deer
dog
frog
horse
ship
truck
airplane
automobile
bird
cat
deer
dog
frog
horse
ship
truck
label:
label: ship
[contrast corruption severity 5]
individual probability samples
intersection probability
averaged probability
Figure 8: Visualization of the individual probabilities, the intersection, and the averaged probability for one CIFARIC 10 sample and its corrupted version sample. For the corrupted data, the intersection probability can remain the correct prediction while the averaged probability leads wrong pre- diction.
A.2 ABLATION STUDY ON NUMBERS OF PREDICTIVE SAMPLES IN DES
In this experiment, we train 30 ResNet-50-bbsed SNNs individually using the CIFAR10 dataset and under different random seeds. Then, we construct DEs using different numbers of ensemble members, namely N 3, 5, 10, 15, 20 and 25. Each type of DEs includes 15 instances using distinct seed combinations.
Table reports OOD detection performance comparisons under different numbers of samples. The findings suggest that (i) Increasing the number of samples can indeed improve the EU estimation of our proposed credal wrapper. (ii) Compared to classical DEs, our method consistently shows superior performance on EU quantification, robustly against the number of samples. Figure shows the ECE performance evaluation using different settings of N on the CIFARIO-C samples. We can observe that: (i) Increasing the number of samples overall can lead to a lower ECE value; (ii) Compared to the naive averaging DE predictions, intersection probability consistently achieves lower ECE values on corrupted instances.
Table 5: OOD detection performance (%) comparison in DEs.
SVHN
Tiny ImageNet
AUROC
AUPRC
AUROC
AUPRC
Buseline
Ours
Baseline
Ours
Baseline
Ours
Baseline
Qurs
89. 64..1.16 16
91.97a1.38
92.13a1.12
94.91=1.08
86.3 26.3120.24
87.57a0.25
81. $1.76a50.40
85.0220.32
89.
93.50)s9.79
91.9640.97
95.79.0.70
86.83s0.25
88.65en.25
10
90.54+0 72
95.033+9.40
93.22+0.70
96.50:00.35
87.75+0.15
89.62=n.14
84 99
S8.04=0.20
90.7840.68 68
95.31:9.47
93.23:0.62
96.91
88. 10:0.00
89.94:0.10
85 55.500000
88.44:0.13
09.00..550
95.6929..4
93.4920.48
97.1520.30
33.3380008
90.11:0.05
85. 85.9610011
88.5910.10
91.31
95.8429.15
93.6240.20
97.2420.12
88.4620.04
90.2220.05
86. 86.1720000
88.7220077
03
Intervctionn
10t morton
Intertativn
Inivmaation
Intersctionn Frn
00
Overl Compgtinn stanoity
Compptin Irterdty
Drandl Comrgtion Inieriity
Ovena wow.lionttttnnn Intenity
Comigiion
Ovans Camupt
Figure 9: ECE values of DEs with various N on CIFARIO-C against increased corruption intensity.
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A.3 ABLATION STUDY ON NUMBERS OF PREDICTIVE SAMPLES IN BNNS
In this experiment, we first increase the sampling size of BNNs at prediction time, namely N = 10 and N 50. Table 6 reports the OOD detection performance of BNNR and BNNF involving CIFARIO (ID) vs SVHN and
(OODs), based on the VGGI6 backbone. It shows that the credal wrapper consistently produces better EU estimates, as evidenced by enhanced OOD detection performance. Further, Table T3 in the Appendix reports the same comparison based on the ResNet-18 architecture, confirming those results.
Table 6: OOD detection comparisons using EU (%) of VGG16-bases BNNs.
N
Method
BNNK: CIFARIO (ID)
BNNF: CIFARIO (ID)
SVHN (OOD)
Tiny-ImageNet (OOD)
SVHN (OOD)
Imng-ImageNer (OOD)
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
Baseline
86.6011.28
90.5411.00
85.00s0.27
80.9010:42
86.6720.54
90.6410.71
84.7810.18
80.51.0338
Ours
88.38:1.53
92.61=1.12
86.2820.25
83.8820.32
88,0320.52
92.55:0.52
85.7940.18
83.29:0.28
Baseline
86 1.27
90.44:1.02
85.4120.21
81.7910.99
86.6320.60 60
90.5620.79
35.1010.11
81.2820.34
50
Ours
88 88.90=1.66
92.75x1.22
87.17H0.26
85.22.0.33
88.60+0.56
92.91=0.54
86.61s0.21
84.64.0.27
Tables 5 and G suggest that our credal wrapper provides better uncertainty estimation than Bayesian methods, even when the latter uses more samples. For instance, the OOD detection performance of traditional DEs with N 25 is significantly lower than that with N = 5 when using credal wrapper. Moreover, the OOD performance of our credal wrapper with = 10 consistently surpasses credal wrapper can reduce the
that of conventional BNNs with N 50. From this perspective, complexity as increasing the number of samples typically demands more computational resources and memory, such as the additional single models required for DEs during training and inference.
Table 7: Performance comparison when applying N = 1000 to the ResNet- 18 18-based BNNR.
ID Performance
CIFARIO SVHN |CIFARIO vs Tiny-ImageNet
ACC
ECE
AUROC
AUPRC
AUROC
AUPRC
Baseline |11 19 0.0557 27.24
93.0920.9 |3.9440022
81.1520.99
Ours
91.90.n.16 0.034.0.003] 89.36.1.25 93.88 (86.89.0.19
84.63.0.33
Additionally, we further increase the number of samples (N = 1000) for the ResNetl MoNNetl8-basse 8-based BNNR to explore an extreme case. Table 7 compares the test ACC and ECE values between two approaches, namely and p", well as the OOD detection performance. Although high sampling size is rarely employed in BNNs due to the inherent complexity, the results illustrate the stability of our methods and the consistent superiority of our credal wrapper compared the BNN baseline.
A.4 GENERALIZED HARTLEY MEASURE FOR EU ESTIMATION OF CREDAL WRAPPER
In this section, evaluate the EU quantification performance of our credal wrapper using the generalized Hartley (GH) measure (Abellán & Moral 2000), and briefly discuss the interpretation of the generalized entropy and the GH measure in uncertainty quantification of the credal set.
Definition and Implementation The generalized Hartley measure (Abellán & Moral 2000
Hüllermeier & Waegeman, 2021). GH(P), captures the non-specificity across the distributions in the credal set, and can be seen as a proxy for its volume (Hüllermeier & Waegeman 2021: Sale al., 2023a) Mathematically, GH(P) calculates the expectation of the Hartley measure over all possible subsets B on the target space Y. as follows (Abellán & Moral 2000):
GH(P) = Σ mr(B): log,(
(12)
BCY
in which my denotes the mass assignment function associated to a credal set P and [B] indicates the cardinality of B. m²(B) can be computed using the Möbius inverse of the capacity function Pp (Chateauneuf & Jaffray 1989) as follows:
ma (B)
(13)
ACB
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where B y/y B and y A and Vy describes the lower probability of all possible subsets S B. GH Measure is computationally complex to compute. However, in our case, the lower probability (A) associated with the predicted credal set can be readily computed as follows:
(A) max
Σ BAEA
Pi.. 1- - Σ PU;
(14)
MiRA
Mr
where Pr. and pa are the lower and upper probability values per class in the defined credal set.
Experimental Validation In this ablation study, we evaluate on EU estimation quality of our credal wrapper using GH(P) measure. Table 8 reports OOD detection performance tested on CIFARIO (ID) vs SVHN (OOD) and Tiny-ImageNet (OOD). All models are implemented on the VGGI6 backbones. The results demonstrate that our credal wrapper consistently enhances EU estimation performance and is agnostic in the sense that it can accommodate any EU measure for credal sets.
Table 8: OOD detection AUROC and AUPRC performance (%) comparison between classical and credal wrapper of BNNs and DEs using EU. The results are from 15 runs.
SVII (OOD)
Tiny ImageNet (OOD)
Model
EU
AUROC
AUPRC
AUROC
AUPRC
Baseline
H(p) (p))
86.65a126
90.61
84.6240.28
80.06a10.40
BNNR
Ours
GH(F)
87.24
91.5420.97
84,88:0.25
81.0520.37
BNNF
Baseline
H(p) ((())
79+9-47
90.76+0.55
84.54+0
79.91
VGGI6
Ours
GH(F)
87.2240-45
91,4920.99
84,72+0.20
80.72:0032
Baseline
( H((p)
74
88 18.499:.11 17
85.79+4033
DEs
Ours
GH(F)
90.37+1.33
94.21=0.89
88.98+0.13
86.88+1).26
Further Discussion on Different Uncertainty Measures Given credal set P. it was suggested to assume the following representation to express the uncertainty (Abellán Moral 2000 Hüllermeier
& Waagemann 2021
TU(P) EU(P) + AU(P).
(15)
As a result, given TU(P) and generalized measure of aleatoric uncertainty (conflict) AU(P), generalized measure of epistemic uncertainty (non-specificityy EU(P) can be derived via disag- gregation as BU(P)
TUPP) AU(P). The lower entropy (IP) applied in our work indicates the lower bound of the (aleatoric) uncertainty that remains even when all epistemic uncertainty is removed, i.e., when IP is reduced single distribution p P. From this perspective, (IP) cor- responds in fact to the natural measure of irreducible (aleatoric) uncertainty. Study (Abellán et al.) 2006) justifies the use of upper entropy as a global uncertainty measure for credal sets. As for the epistemic uncertainty, the difference H (IP) - (P) appears to be reasonable and refers to the size of the set of candidate entropies H (P), (P)). quantifying uncertainty about the aleatoric uncertainty H (p) of p. Different from (TP)) (TP) in EU estimation, the GH(P) measures the size of the set of candidate probabilities, and hence refers uncertainty or imprecision about the ground-truth probability. For further discussions on the different measures and their corresponding strengths and weaknesses measuring predictive uncertainty, we kindly refer to (Hillermeier et al. 2022).
A.5 TOTAL UNCERTAINTY ESTIMATION EVALUATION ON OOD DETECTION BENCHMARKS
In this section, we assess the total uncertainty (TU) estimation quality (as opposed to EU) of
credal wrapper on various OOD detection benchmarks. Tables 9 I0 I2 [3 and Figures I0 show that our credal wrapper can improve the quality of TU estimation as well, across different dataset pairs, network architectures, and test settings.
A.6 ABLATION STUDY ON OVERCONFIDENCE REGIME
In this section, we examine the performance our credal wrapper in scenarios where the ensemble member (SNN) generates overconfident predictions. In our credal wrapper framework, overconfi- dence in one of the predictions would stretch the resulting credal set. This is because the upper and lower probability bounds over classes are derived via max/min operation in eq. (5). For instance, in the case of 3 classes, if there are three distinct extreme probability vectors (three vertices of the simplex), our credal wrapper method will effectively convey complete uncertainty, with the resulting
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Table 9: OOD detection AUROC and AUPRC performance (%) comparison between the classical and credal wrapper version of BNNs and DEs, using TU as the uncertainty metric. All models are implemented on V backbones and tested on CIFARIO (ID) vs SVHN (OOD) and Tiny-ImageNet (OOD). The results are from 15 runs. The best scores are in bold.
ResNet- 18: CIFARIO (OOD)
VGGI6 CIFAR (OOD)
SVII (OOD)
Tiny ImageNet (OOD)
SVII (OOD)
ImageNat (OOD)
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
AUROC
Baseline
1841.30
92.68-0
56.40=0.24
83.164
88.57=1.47
93.26+1.00
85.50+0.31
82.600041
BNNR
Ours
88.3611.30
92.95:9.87
86.52:0.22
83.6620.38
88.79:1.57
93,4411.14
85.89
83,5340,43
Baseline
88.4320.83
92.90.0.51
86.45.0.33
83.2240.33
88.3210.50
93.0610.51
30.0.0
82.32,0.30
BNNF
Ours
88.6520.81
93.19
86.5829.32
83.75e0.31
88.47:9..2
93.27=00.22
85.62:0..0
83.2340025
Baseline
88.4720.80
92.82:0.4
88. 96.0 14
87.22.0.18
91.20.1.29
94.87a0.76
89 89.5520.08
87. 37.8920.14
DE
Ours
88.9820.86
93.1320.70
89.53.0.14
88.2350.17
92.12:1.32
95.4120.75
90.0019.07
88.4920.17
Table 10: OOD detection AUROC and AUPRC performance (%) of both the classical and credal
wrapper version of DEs using TU as the metric. The results are from 15 runs, based on the ResNet- 50 backbone. The hest scores are in bold.
CIFARIO (ID)
CIFARIO0 (ID)
ImageNet (ID)
SVHN (OOD)
Tiny ImugeNet (OOD)
SVIIN (OOD)
ITiny ImugeNet (OOD)
ImageNeit- (OOD)
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
Baseline
94,8023443
97. 27.2640.29
19
87.21
78.53s1.94
88.8341.01
80.7550.15
77.65
50.20
50.44
Ours
95.4420.37
97.5720.23
89.3020.17
87.9720.25
80.71#1.96
89.9720.99
81.4620.14
78.2920.17
54.8729.08
52.27
Table 11: OOD detection AUROC and AUPRC performance (%) of both the classical and credal
wrapper version of DEs using TU as the metric. Results are from 15 runs, based on EffB2 and ViT-B backbones. The best scores are bold.
CIFARIO (ID)
CIFARIOO (ID)
SVIIN (OOD)
Thy ImageNet (OOD)
SVIIN (OOD)
Thny ImageNet (OOD)
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
Baseline Ours
97.55.0.27
98.7810.16
93.4110.15
93.0110.17
88 4611...33
94,40.0.55 59
86.45a0.10
84.88.0.14
EMB2
97.91s0.1
98,9810.10
93.7410.15
93.5520.16
89.35s1.20
94,83=0.52
86.6110.09
84.8720.12
Baseline
79.80s1.75
87 17.97.1.17
83.81a0.81
81.6720.80 89
84.10a1.12
91.41:0.72
82.6 22.6440.28
79.90 43
VIT-B
Ours
81.88:182
88.4811.45
84.6220.75
82.8220.85
85.4021.05
92.0220.70
83.06:00.99
80.34:0.11
0.955
2.970
Aaggped..0 AURGC
Angiooug.nng JUPPIC
950
M0,950
3.02
805
2,960
00000
0,530
2.593
03445
0,000
Oxcol Compption intensiy
Overall Comupton Inteeslt
Overal interennnttttstty
Ovendl Comuption nicerity
BWW (saseline)
BRAF Haselinc
(ssselin) DE: (Dum)
DNVR: (Dun)
- Daar: 10um)
BIVE: (Duns)
Sha (Bun)
DE: (Oun)
ResNet-18 (TU)
VGG16 (TU)
Figure 10: OOD detection using TU as metric on CIFARIO vs CIFARIO-C of both the classical and credal wrapper version BNNs and DE against increased corruption intensity, using VGG16 and ResNet- 18 as backbones.
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0.800
0,629
Aegnorg Anggors AiRoc
Averane AiROC
0.992
520
2.302
2780-
0,740
0,875
9.720
6.680
3.850
0,700
0.700
Rechiet-5 S2: (0urs)
-
E7T2: (Durs)
-A-
VIT-B (Dur))
- Reeleet-3D: (Duns
CHB3: (Dun)
VIT- (Dun))
CIFARIO Vs CIFARIO-C (TU)
CIFAR100 CIFARIOD-C (TU)
Figure 11: OOD detection using TU as the metric on CIFAR10/100 vs the classical and credal wrapper version of DEs against increased corruption intensity, using ResNet- 50, EffB2, and ViT-B as backbones.
of both
Table 12: Ablation study on numbers of predictive samples in DEs: OOD detection AUROC and AUPRC performance (%) of both the classical and credal wrapper version of DEs using TU as uncertainty metrics, involving CIFARIO (ID) vs SVHN (OOD) and Tiny- ImageemaeNNet (OOD). The results are from 15 runs. The best scores are in bold.
CIFARIO SVHN
CIFARIO Tiny ImageNet
AUPRC
AUROC
AUPRC
2
Baseline
Baseline
Ours
Baseline
94.3020.99
94.881101
97.0340.62
97.21
28.2420.22
88.5420.23
86.4610 33
86.98...32
94.7844
95. 55.
97.19H0.38
97.49200.0
88.71=0.25
89.17
87.10e0.37
87.86::0.77
10
95.47700.4
96.18.0.31
97,6120.25
97.91=0.25
89.2720.13
89.8620.14
87 97.99000.19
88.63.0.19
95.55::0.32
96.2550.37
97.63+0.23
97.88
89.48=0038
90.11=0.10
88.87:0.12
20
95:72:0023
9646699.26
97.74+0.15
97.98=n.23
$9.60=0.07
90.24:n.08
88 34
88.95+0.11
95.80
96.53+9.12
97.78+0.08
98.01
89.6720.04
90.32:0.05
88.4 08.4220005
89.01=0007
Table 13: Ablation study on numbers of predictive samples in BNNs: OOD detection AUROC and AUPRC performance (%) of both the classical and credal wrapper version of BNNs with increased number of samples, involving CIFAR10 (ID) vs SVHN (OOD) and Tiny-ImageNet (OOD). The
results are from 15 runs and the best scores per uncertainty metric are in bold.
Model
Method
BU Measure Metrie
TU Measure Metric
SVHN (OOD)
Tiny- ImageNett (OOD)
(OOD)
Tiny- Imageen (OOD)
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
Raseline
86.60=1.28
90.54+1.0 100
85.00H6.27
80.90+0.42
88.63+1.4 1.9
93.26+1.10
85.77=0.27
82.9740.3 35
10
Ours 88.3821.3
92.61=1.12
86,28t0.25
83.88:9032
88.99#161
93.45+1.17
86.44H0.27
84,28:0.34
BNNR
50
Raseline
86.60=1.37
90.44+1
81.79+0.30
88.67+1.54
93.29
86.02+0.24
83.31 30
Ours 88.90=165
92.78+1.23
87.1750.26
85,22+0.33
89.17+169
93.24+1.25
87.22+0.26
85,35+0.33
VGG16
Raseline
86.67e0.54
90.64+0.71
84.7840.15
80.51
88.31
93.05+a.50
85.4440.19
82.5340.34 34
Ours 88.03=0.52
92.55en.52
85.7950.18
83.2950.22
88.58+0.53
93.31
85.9920.19
83.80-30.29
BNNF
Baseline
86.632n.00
90.5620.79
85. 10200.011
81.2810.34
88 34.043333
93.0720.51
85 85.6320.20
82.8011032
50
Ours 88.6020.56
86.6110.21
84.6422027
93.3420.54
86.6720.21
84.8240.27
Baseline
88.4621.23
93.1920.77
85.8520.22
81.5810.37
88.24:1.29
92.7110.5
86.4320024
83,2249.00
10
Ours 88.6121.26
93.355:083
16.16:20
82,9920.26
88.5361.19
93,0920088
86.6320.21
83,9120.35
BNNR
Baseline
88.4321.27
93.3320.77
85,5820.23
81.7020.0
88
92.7420.54
86.4520.24
83.25:0.00
ResNet-188
50
Ours 88.9321.24
93.612083
86.48190.21
83.80120.34
88.8421.36
93.3550036
86.8220022
84,3820.36
Bascline
88.7720.85
93.45220.5 56
85,9720.31
81.83a0,27
88.4920033
92.9320.5
86,4820.34
83.2620.35
Ours 88.962083
93.62e0.55
86.2849.33
83,2249.36
88.8420.11
93.36160.57
86.7020.33
84,0340.35
BNNF
Baseline
88.79=0.95
93.59+0.61
85.6740.35
81.88+0.35 35
88 94-5440.2
92.9740 50
86.49+0.35
83.30+4. 35
50
Ours 89,30=0.33
93.89en.55
86.59+40.3
83,99:0.35
89.18+0.81
93.64400.4
86.90=0032
84.5340.36
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credal set encompassing the entire simplex. This conservative nature can be sensible, expresses our full ignorance of the correct classification.
Furthermore, conduct additional ablation study explore the benefit of method given overconfident prediction samples. We manually and randomly transform one of the predictions of the same input image from five deep ensembles to one-hot probability and compare the OOD detection performance of our method and the classical approach. Table [4] also shows our credal wrapper can improve the uncertainty quantification even in overconfident scenarios.
Table 14: OOD detection using EU (left) and TU (right) uncertainty metrics in overconfident scenarios. The results are from 15 runs, based on the ResNet-50 backbone. The best scores per uncertainty metric are in bold.
Model
Method
Epistemiic Uncertainly Tiny-ImageNe (OOD)
as
Uncertainly
SVHN (OOD)
SVHN (OOD)
Tlay Tiay-ImageNet (OOD)
AUROC
AUPRC
AUROC
AUPRC
ALROC
AUPRC
AUROC
AUPRC
Baseline
90.4220.81
93. 33.3020.80
87.01
83.5340.28
94,3620.45
96.9810322
88.5 38.5420.19
86.8220.30
DE
88.9820.17
95.2440.38
A.7 QUALITATIVE EVALUATION OF UNCERTAINTY ESTIMATION OF CREDAL WRAPPER
In this section, we present the estimates of the EU and TU on the ID and OOD samples of the classical DE (using (()) H (p) and H (p). respectively) and our credal wrapper (using H(P) H(P) and (P), respectively) Figures 12 and 13 The results qualitatively demonstrate that our credal wrapper consistently enhances the uncertainty quantification of classical DEs. This is evidenced by the significantly larger uncertainty estimates for the OOD samples, which correspond the expected and desirable behavior.
8.0
3.0
8.0
6.0-
Frequency
2.0
Frequency
6.0
Frequency
2.0
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4.0
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1.5
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1.5
2,0
EU Estimates of assical DEs
EU Estimates of Credal Wrapper CIFAR10 (ID)
Estimates of Classical DES
TU Estimates Credal Wrapper
SVHN)
Tiny ImageNer (COD)
ResNet-50
5.0-
10.0
10.0
4.0
Prequency
Frequency
Trequency
7.3
3.0
7.5
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2.-
1.0
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0,0
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EU Estimates of Classical DEs
EU Eatimates of Credal Wrapper CIFAR10
TU Estimates of Classical DEs
TU Estimates of Credal Wrapper
SVHN1000)
Tiny-lmagtNer
EffB2
5.0
2.5
Trequency
4.0-
Frequency
2.0
Feequncy
40
requeene
2.0
3.0
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6.8
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EU Extimnates of Cassical DEs
EU Estimates of Credal Wi Wrapper
TU Estimates of Clessica Tiny-mageNer (COO)
Extimetes Credal Wrasper
CIFARTO (D)
SVHN1000)
VIT-B
Figure 12: EU and TU estimates of ID (CIFAR10) and OOD (SVHN and Imny-Ima samples of the classical and credal wrapper version of DEs, obtained using ResNet-50, EffB2, and ViT backbones. Results are from 15 runs.
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2,5
2.5
0.8
Frequeccy
6.6-
Fequuncy
2.0
Frequency
0.6
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2.5
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0
EU Estimates of Cassical DEs
EU Estimates of Credal Wrapper
TU Estimates of Classica DES
TU Estimates of Credal Wrapper
CIFAl110O (i))
100N)
Vimy-lmageNe (OOI)
ResNet-50
Frequency
Frequency
1.0
Franuancy
Fequency
2.0
0.8
0.5
1.0
10
0.2
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Estimatees of Cassical DEs
EU Estimates of Credal Wrapper
Estimates Classica
Estimaties Credal Wrasper
CIFAR100
SVHN
wmy-amageNtttccm
EffB2
3.0
3,0
0.8-
0.8
Frequency
2.0
Frequercy
0.6
Frecuency
2.0
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0.2
6.9
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Estimates Cassics DES
EU Estimates Credal Wrapper
Estimates Clessica
Estimates Credal Wrasper
GIIAR100 (I)
(000)
Timy-ImagnNer (000)
ViT-B
Figure 13: EU and TU estimates of ID (CIFARI00) and OOD (SVHN and Tiyy-mageNNet) samples of the classical and credal wrapper version of DEs, obtained using ResNet-50, EffB2, and ViT backbones. Results are from 15 runs.
A.8 EVALUATION OF INTERSECTION PROBABILITY ON CORRUPTED SAMPLES USING NLL METRIC
In this section, we further evaluate the intersection probability on corrupted samples using the neg- ative log-likelihood (NLL) metric. A smaller NLL indicates that the model is more confident and accurate predicting the correct class for each input (Dusenberry al,, 2020). Figures 14 and show the consistent superiority of the intersection probability on corrupted data in extensive test cases, as evidenced by smaller NLL values.
Aug.
ANE
Tool
ING PrLh
Aag PHLB
2v0 Pr/D
Iniemkyy (VGG16)
Comritionn Hiarsiy (V6616)
Uneu Dempptarn Enienalty
Cemust iltrrity (ReskettB)
Figure 14: NLL values of BNNR, BNNF, and DE on CIFARIO-C against increased corruption intensity, using the averaged probability (Avg. Prob.) and our proposed intersection probability (Int. Prob.). VGG16 and ResNet-18 18 are backbones. Results are from 15 runs.
Oved Compation Reshiet-S0 (CIFAR11))
Overl Compustin #1187 (CIFAR10)
Ovea Fommssttinn WINB (CIFARLO)
Over Cormigiinn Ruediiit (CIFAR100)
Overal Interalty EMFROCCIARROOO
ovendl rttreity WI-II (CFFA110)
Figure 15:
NLL values of DEs on CIFARIO-C and CIFARI00-C against increased corruption in- tensity, using the averaged probability (Avg, Prob.) and our proposed intersection probability (Int. Prob.). 50, EffB2, and ViT-B are backbones. Results are from 15 runs.
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A.9 ADDITIONAL RESULTS ON COMPUTATIONAL COST
In this section, we present the CPU time cost comparison in Table 15 The results demonstrate that, despite the higher time cost, our method remains practical, potentially even achieving real-time capability, particularly for low-dimensional classification problems without strict computational re- source constraints. addition, we are confident that more efficient code implementation of our approach could further reduce computational overhead.
Table 15: Time cost (ms) comparison for uncertainty (entropy) calculation of a single input instance. The isst measured by single Intel(R) XRon(R) Gold 6240 CPU @ 2.60GHz and is averaged from 20 runs.
C
20
100
Baseline
0.002
0.002
0.002
Ours
2.526
5.921
9.694
87,384
Furthermore, our method demonstrates superior uncertainty quantification performance while re- quiring fewer predictive samples (N), shown in Tables and 6 For instance, the credal set version of DEs-5 (N = 5) surpasses the classical DEs-25 in performance. Similarly, the credal set versions of BNNR and BNNs achieve higher performance using only 10 predictive samples com- pared to their classical counterparts, which require 50 samples. We report the inference complexity of these models across various values of N in Tables [6] and 17 The inference times measured using the same single CPU and an advanced Nvidia L100-SXM4-8000 that the additional overhead introduced by our method for entropy calculation is negligible, espe- cially when considering the substantial reduction in overall inference cost. In addition, our approach significantly reduces GPU memory consumption. For example, DEs-25 and BNNF (N 50) fail to make inferences on single Nvidia A A100-SXM4-40GB GPU due to memory constraints.
GPU. The results indicate
From the perspective delivering higher uncertainty quantification performance with fewer pre- dictive samples, the proposed credal wrapper reduces the computational complexity and enhances practical applicability.
Table 16: Inference complexity for single input instance from ResNet-50-based DEs with different ensemble sizes. The inference time is averaged from 20 runs. CPU: A single Intel(R) Xeon(R) Gold 6240 CPU @ 2.60GHz; GPU: A single Nvidia A100-SXM4-80GB GPU.
Ensemble Size
Model Size
Infereace CPU Cost
Infereace GPU Cost
N
(MB)
(in))
(ms)
5
500.05
9419 144.4 4495.3 # 120.8
300.4
Diss
25
2500.25
1429.7 110.0
Table 17: Inference complexity for a single input instance from VGG16-based BNNRs and BNNFs with sample ensemble size. The inference time is averaged from 20 runs. CPU: single Intel(R) Xeon(R) Gold 6240 CPU @ 2.60GHz; GPU: A single Nvidia A100-SXM4-800 -0GB GPU.
Sample Size
Inference CPU Cost (ms)
Inference GPU Cost
(ms)
BNNR
10
4951.4 a 91.8 23466.4 165.6
1982.9 a 204.3
BNNF
5474.9 a 92.7
2241.1 203.8 11046.2 + 214.6
BNNF
27498.8 478.1
A.10 ADDITIONAL RESULTS ON DIFFERENT HYPERPARAMETERS OF PIA ALGORITHM
The hyperparameter J of the PIA algorithm is design choice intended to mitigate the com- putational cost associated with uncertainty quantification, particularly in situations with limited computational resources. As demonstrated in the ablation study results in Table 4 the proposed PIA algorithm significantly reduces the computational complexity of uncertainty estimation high- class classification scenarios, while maintaining strong uncertainty performance OOD detection
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benchmarks. Increasing the value of J enhances the performance. Furthermore, present the EU and TU estimates for ID (CIFARI00) and OOD (SVHN and Iiny-ImageNet) samples at different values of J in Figure 4. The figure shows that using small value of J leads to underestimating the EU and TU, particularly for OOD samples. This phenomenon explains why a higher value of J can achieve better OOD detection performance.
=
10
= 10
/ = 10
0.8
20
20
/ = 20
100
100
/ 100
Frequency
0.4
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0.4-
0.2
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Estimates on CIFAR100
EU Estimates SVHN
EU Estimates Tiny-ImageNet
1.2
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10
/
= 20
1.0
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=
Frequency
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/ = 100
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0.6
0.4
0.2
0.2
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0.0
4
2
Estimates on CIFAR100
TV Estimates on SVHN
TU Estimates Imay-ImageNet
Figure 16: EU and TU estimates of ID (CIFARION and OOD (SVHN and Tiny-ImageNer samples of the credal wrapper version of DEs using different values of J, obtained using the ResNet-50 backbone. Results are from 15 runs.
Table 18: OOD detection AUROC and AUPRC performance (%) of credal wrapper of DEs using TU uncertainty metrics using different setting of J of PLA algorithm. The OOD detection involves CIFARI00 (ID) vs SVHN (OOD) and Tiny-ImageNet (OOD). The results are from 15 runs, based on the ResNet-50 backbone.
TU as Metric
SVHN
Tiay- Imageeet
AUROC
AUPRC
AUROC
AUPRC
80.0521.9
89.3011.97
81 243344
77.6010.18
80.7 96
89,9750.99
4680 14
78.29H0.17
80.90+1 98
90.07+1.00
81,5260.14
78.50H0.15
B EXPERIMENTAL IMPLEMENTATION DETAILS
Training Details In terms of the evaluation using the small-scale datasets, all models are imple- mented on the established VGG16 and ResNet-18 architectures using the CIFARIO dataset. The Adam optimizer is applied with learning rate scheduler, initialized 0.001, and subjected to 0.1 reduction at epochs 80 and 120. Standard neural networks (SNNs) and BNNs are trained for 100 and 150 epochs, respectively. Standard data augmentation uniformly implemented across all method- ologies to enhance the training performance quality of training data and training performance. The training batch size is set as 128. The device is single Tesla P100-SXM2-10 16GB GPU. The standard data split is applied. In terms of EDDs, for each learning rate scheduler configuration, as discussed in Sec. [ we train 15 models for 100 epochs, one from each distinct DE, All other training config- urations, such data splitting and augmentation, are kept consistent with those used for the other models. Additionally, temperature annealing, outlined in the original EDD paper (Malinin et al, 2019) is also applied during the EDD training process.
In terms of the evaluation using the large-scale datasets, we mainly use two Tesla P100-SXM2- 16GB GPUs devices to contract deep ensembles by independently training 15 standard neural
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networks (SNNs) under different random seeds. The input shape of the networks is (224, 224, 3). The Adam optimizer is employed, with a learning rate scheduler set at 0.001 and reduced to 0.0001 during the final 5 training epochs. Concerning the CIFARIO dataset, SNNs are trained using 15, 15, and 25 epochs for ResNet-50, EffB2, and ViT-B backbones, respectively. As for the CIFAR100 100 dataset, SNNs are trained using 20, 20, and 25 epochs for ResNet-50, EffB2, and ViT- B backbones, respectively. Note that models based on ViT-B backbones are trained on one single Nvidia A100-SXM4-00GG GPU. In the ImageNet experiments, we employ one single Nvidia A SXM4-40GB GPU to retrain SNNs based on pre-trained ResNet-50 model for 3 epochs, using the Adam optimizer with an initialized learning rate of 1e-6, under different random seeds. Standard data split is applied to all training processes. In ImageNet experiments, we apply three epochs as we observed that increasing the number of epochs further leads to lower validation loss and accuracy. The choice of not training them from scratch is mainly due to computational constraints. Although this setting may have implications for the performance of ensembles, regardless of the absolute baseline performance of ensembles under different training settings, our method exhibits consistent improvement.
Table 19: Poor ID prediction and OOD detection performance of EDD-Fair. CIFARIO as ID data.
OOD Detection Performance using EU
Model
ID Prediction Performance
SVII (OOD)
Tiny Imt (OOD)
ACC (9)
ECR
AUROC
AUPRO
AUROC
AUPRC
VGG16
EDD Jair
$6.57+12.56
0,249=00047
$6.60>12.35
71.10e7.51
55.7354.75
51 51.3223.19
ResNet-18
EDD-fair
85.68+3.48
0.102-0.056
85.30-785
90.16+5.55
76.84+4.59
69 69.84+4.29
OOD Detection Process In this paper, the OOD detection process is treated as binary classifica- tion. We label ID and OOD samples as O and 1, respectively. The model's uncertainty estimation (using the BU or TU) for each sample is the 'prediction' for the detection. In terms of performance indicators, the applied AUROC quantifies the rates of true and false positives. The AUPRC evalu- ates precision and recall trade-offs, providing valuable insights into the model's effectiveness across different confidence levels.
ECE Evaluation Process In the context of ECE, a 'well-calibrated" prediction is expected have a confidence value of 80% and be correct in approximately 80% of the test cases. To calculate ECE, predictions are split into a predetermined number Q of bins of equal confidence range. The ECE is then calculated by summing the absolute difference between the average accuracy and confidence within each bin (Mehrtens et al. 2023):
G
|Bal
ECE:
acc(B,)) conf
(16)
where [Byl is the number of samples in the g-th bin and n is the total number of samples.
C DISCUSSION ON GENERATING CREDAL SETS FROM INDIVIDUAL PROBABILITY DISTRIBUTIONS
Given a finite set of probability predictions. a possible alternative method for generating a credal set, termed convex hull approach, also exists. This method involves directly constructing the convex hull from the finite set of predictions. In our credal wrapper, the probability intervals define the credal set by injecting upper and lower bounds per class in eq. (5) and eq. whereas the convex hull method requires computing all vertices of the credal sets.
The theoretical underpinning for the convex hull method when reasoning with coherent lower prob- abilities (and, therefore, the corresponding credal sets) is that it allows us to comply with the co- herence principle (Walley 1991). In Bayesian context, individual predictions (such those of networks with specified weights) can be interpreted as subjective pieces of evidence about a fact (e.g., what is the true class of an input observation). Coherence ensures that one realizes the full implications of such partial assessments (Walley) [991] Cuzzolin 2008). Figure I7 conceptually shows the differences between two methods in 2D simplex.
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Figure 17: Different credal generation methods. Left: credal wrapper; right: convex hull.
Compared to the convex hull method, our probability interval systems exhibit a more conservative nature. Another practical difference is that the convex hull method is highly computationally com- plex, preventing it from being practically implemented multi-class classification tasks. In the following, aim to explain the associated complexity of the calculation process.
Given such a coherent lower probability, denoted as P. the associated credal set K(P) comprises all probability measures (P(W) in the target space (Y) that be defined as follows:
K(P)= {P P(Y) P(A) 2 P(A) S W}.
(17)
Further, P is considered coherent if it can be computed as:
P(A)
P(A) Y PEK(P)
min
(18)
where denotes any possible subset of the target space Y.
Mapping the finite set probabilistic predictions to a credal set generally requires the following steps:
i) Compute the lower probability for any subset of the target space P(A).
ii) Compute the masses over the subsets by using Möbius inverse CChateauneuf & Jaffray 1989
mp(A) = Σ (-1)AA33P(()) VAC
(19)
BCA
iii) Compute the vertices of the credal set K(P). Its vertices are all the distributions pa induced by a permutation π = {IagI)...... (x())) } of the elements of its sample space W, of the form
(Chateauneuf & Jaffray 1989 Cuzzolin 2008
p" P (Ix())
Σ
mp(A)
(20)
A3 A3 Visi
The vertex (the extreme probability) assigns to each class, put in position (i)) by the permutation π, the mass of all the focal elements (sets classes with non-zero mass) containing it, but not containing any elements preceding it in the permutation order (Wallner 2005).
The above procedure is quite computationally expensive, because i) When the number of classes |Y| = C' is large, it can be highly inefficient to compute probabilities for all 2c events in the power set of Y). ii) Compute the vertices in eq. (20 is extremely computationally difficult.
On the contrary, the probability interval systems applied do not suffer such computational burden, as shown eq. (5) and eq. (6). the resulting credal set directly defined by C' pairs probability bounds over classes. The extensive experimental validation also strongly demonstrates the consistently improved uncertainty performance of our proposed credal wrapper.
D DISCUSSION ON CREDAL WRAPPER IN CASES OF INFINITE SAMPLES
In this section, we aim to discuss the behaviour of the credal wrapper and the associated intersection probability in the case of infinitely many samples from individual probability distributions. In this
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context, assume that the support of the sampled distribution spans the entire simplex, without imposing any additional distributional constraints.
If assume the predictions sampled from 'sccond-order' distribution (as in the BNN case), when one keeps collecting samples the corresponding credal set in eq. (6) expands (as it relies on min/max probability calculations in eq. (5)) Therefore, as p is function of the probability interval bounds of the credal set (and not the actually collected samples), for that goes to infinity will tend the p" of the probability simplex A, i.e., the uniform distribution. Indeed, this does not happen to the BMA theoretically, which averages the samples. However:
(i) The argument for the rationality of the representative p" of an interval probability system is compelling (Cuzzolin) 2009 2022). The same asymptotic effect is shared by the center of mass of the credal set, indicating that this is feature inherent the credal representation itself.
(ii) In practice, N is small even in the general cases of BNNs (N = 5). In the ablation study in Sec. A.3 we examine the performance of our credal wrapper in cases of N 10, N 50, and even 1000, our credal wrapper consistently shows an improved uncertainty estimation and calibration performance, as shown Tables and Figures 3 14 p is better than BMA, it appears that BMA can be biased/less representative at N never going to infinite w.r.t. p. whereas it enjoys better convergence properties.
the practical performance of
(iii) This does not apply to ensembles, ensemble members are not 'sampled' from any distribution.
suggest that the consistent superior performance of p" of calibration performance corrupted data.
The ablation study Appendix $A.2 as well as Figures 3
5
15
29


Reviews:
[{"review_id": "PN0JwMA5oA", "reviewer": "Reviewer_TRkv", "summary": "This paper introduces the \"intersection probability\" to model averaging, to improve uncertainty estimation in OOD detection. This introduced method is an alternative for aggregating predictive distributions by different models/particles, and considers the lower/upper bounds of the probability interval.", "questions": "Please refer to weaknesses.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The idea of using probability intervals to reflect the epistemic uncertainty is appealing.\n\n2. The intersection probability transformation has a deterministic solution, and thus it is as efficient as other alternatives in model averaging.\n\n3. Experimental results are strong and comprehensive.", "weaknesses": "1. Except for table results, I expect a more direct demonstration of why intersection probability is better than naive averaging. For example, can you visualize the uncertainty on simple regression tasks?\n\n2. Computing epistemic uncertainty using probability intervals is costly. I expect discussions on the extra computational cost compared with entropy.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "90mrzdzE3b", "reviewer": "Reviewer_3c79", "summary": "This article proposes a computationally feasible method for improving the quality of Uncertainty Quantification (UQ) estimations from sample-based models (in a broader sense), such as BNNs and DEs. The proposed method utilizes credal sets to compute the entropy bounds within the probability interval. Intense experiments are conducted on a collection of datasets and models, and the proposed method shows a clear advantage over baseline methods.", "questions": "As I stated above, I'd like to know the following (even simple, intuitive answers will be helpful):\n- Why can the simple-looking method drastically improve the baseline results?\n- How well does the proposed method perform compared to methods listed in, e.g., lines 116-125?\n- Are there any visualizations/etc. of the credal sets in practice? (real-world datasets with well-trained classifier samples)", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 2, "contribution": 3, "strengths": "- The proposed method is simple yet effective, as being demonstrated in the widely-conducted experiments.\n- The proposed approximation algorithm, PIA, effectively approximates the original optimization problem and ultimately makes the idea practically tractable.\n- The proposed method is easy to \"wrap\" around various methods as long as they provide multiple samples of model predictions.\n- Experimental details are well documented.", "weaknesses": "Overall speaking, the paper is a bit tricky to follow as lots of details are densely presented. It will be clearer if some parts (e.g., detailed experiment setups, ablation studies on # of predictive samples, etc.) could be moved to the appendix. Some parts of the appendix, e.g., A.6 ablation studies on PIA parameter $J$, feel more important to me compared to some of the ablation studies listed in the main body.\n\nBesides, although I am unaware of existing similar works, it would be beneficial if the authors could provide clear intuitions/reasonings on why this simple method can drastically improve baseline results, and clarify their contributions (for example, the approximation algorithm). Visualizations of real experiments can be provided to let readers grasp a general idea of what the credal sets are like in practice, as well as the prediction produced by the intersection probability transform. The paper provides a tremendous number of experiment results, which is good and validates the proposed method, yet some intuitions will still be very helpful on top of those results.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "5qkXX4HZDH", "reviewer": "Reviewer_kVbE", "summary": "The paper proposes a simple method to estimate epistemic uncertainty and make predictions based on sampled predictions of Bayesian Neural Networks (BNNs). For that, it determines a credal set representation using an interval defined by lower and upper bounds. These bounds are derived from sampled predictions of a BNN or Deep Ensemble (DE). Using this credal set representation, the authors compute the epistemic uncertainty and a single prediction (termed intersection probability). Their experiments demonstrate that these epistemic uncertainty estimates are more effective for out-of-distribution (OOD) detection compared to baseline BNNs and DEs. Furthermore, they show improved calibration across different distributional shifts of the intersection probability.", "questions": "Can you provide feedback regarding the stated weaknesses?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 2, "strengths": "- The paper is really well-written and easy to follow.\n- The proposed idea is really simple and is highly applicable in many settings.\n- The experiments are nicely structure and follows the evaluation protocol of the literature.\n- The authors clearly highlight the approach's limitations stemming from its computational complexity.", "weaknesses": "- **Related Work:** The approach is presented as innovative but lacks clear placement within existing literature. While there is significant discussion on credal sets in ML, the authors only briefly touch on their application in deep learning, mentioning computational complexity as a key challenge. After a short research I noticed some papers focusing on credal sets in deep learning [1, 2] and was wondering why these are not discussed in the related work. More importantly, how is this paper improving the state-of-the-art, and what are the differences to other credal set papers that are used in deep learning? Given these facts, I can not accurately determine the novelty of the proposed method.\n- **Metric inconsistencies with OOD literature:** I noticed some inconsistencies in the evaluation metrics compared to related papers, particularly with AUROC values for baseline methods. For instance, Table 1 reports an AUROC of 88.63 for CIFAR-10 vs. SVHN using ResNet-18, which is significantly lower than the 93 reported in related work, including [3]. It’s unclear why this discrepancy exists—perhaps due to differences in experimental setup, data preprocessing, or other implementation details. Clarifying these points would help verify the reliability of comparisons and ensure the evaluation is aligned with the literature.\n- **Hyperparameter:** The choice of the hyperparameter $J$ remains unclear; the authors simply state $J=20$ for CIFAR-100 and $J=50$ for ImageNet, without explaining the basis for these values. It would be helpful if they clarified the selection process.  \nAdditionally, while the authors discuss the method's computational complexity and propose an approximation to mitigate it, they moved the PIA experiments to the appendix. Given the main paper includes the algorithm as a contribution, the experimental results on PIA should also appear in the main text to reflect the algorithm’s practical impact.\n\n\n[1] Michele Caprio, Souradeep Dutta, Kuk Jang, Vivian Lin, Radoslav Ivanov, Oleg Sokolsky, and Insup Lee. Credal Bayesian Deep Learning. arXiv preprint arXiv:2302.09656, 2023.  \n[2] Wang, Kaizheng, et al. \"CreINNs: Credal-Set Interval Neural Networks for Uncertainty Estimation in Classification Tasks.\" arXiv preprint arXiv:2401.05043, 2024.  \n[3] Van Amersfoort, Joost, et al. \"Uncertainty estimation using a single deep deterministic neural network.\" ICML, 2020.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JyHtKe6VaV", "reviewer": "Reviewer_Err5", "summary": "This paper proposes the Credal Wrapper, an ensemble aggregation method designed to improve uncertainty estimation. By applying the concept of credal set representation to neural network ensembles, the algorithm estimates probability intervals and intersection probabilities. Experimental results demonstrate that the proposed method enhances the performance of conventional Bayesian neural networks and deep ensemble methods, particularly in terms of uncertainty estimation and out-of-distribution (OOD) detection.", "questions": "Please see the weaknesses.", "limitations": "", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 1, "strengths": "* The experimental results demonstrate that the proposed method consistently improves uncertainty estimation and OOD detection.\n* The proposed method is straightforward and convincing. Additionally, Figure 1 effectively provides an intuitive overview of the method.", "weaknesses": "* I am not yet fully convinced that the problem addressed in this paper is of breakthrough significance. While it is beneficial to develop a better way to aggregate probabilities, I believe such a method may not have a substantial impact on the Bayesian neural network (BNN) community, as deep ensembles and BNN methods already provide sufficient performance. In my view, focusing on methods that reduce inference or training time (e.g., model parameter ensemble methods) or those that scale well with large datasets (e.g., last-layer Bayesian methods) would be more important topics, as the paper itself discusses. Moreover, modern neural networks no longer suffer significantly from overconfidence [1, 2].\n* Even if I acknowledge the importance of the stated problem, the performance gain is consistent but somewhat marginal. I suspect this performance gap could even decrease when larger training datasets or modern data augmentation methods are used (see Figure F1 in [3]).\n* The computational cost is significantly higher than simple probability averaging, which may be a barrier to using this method in real-world applications.\n\nIn summary, while I recognize that the proposed method consistently outperforms simple probability averaging with solid evidence, in my humble opinion, the contributions are somewhat marginal. Additionally, the increased computational cost may hinder its application to real-world problems.\n\n\n\n[1] Minderer, Matthias, et al. \"Revisiting the calibration of modern neural networks.\" Advances in Neural Information Processing Systems 34 (2021): 15682-15694.\n[2] Vishniakov, Kirill, Zhiqiang Shen, and Zhuang Liu. \"Convnet vs transformer, supervised vs clip: Beyond imagenet accuracy.\" arXiv preprint arXiv:2311.09215 (2023).\n[3] Park, Namuk, and Songkuk Kim. \"How do vision transformers work?.\" arXiv preprint arXiv:2202.06709 (2022).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Spotlight)