Credal Wrapper of Model Averaging for Uncertainty Estimation in Classification

## Abstract

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, capable of improving uncertainty estimation in classification tasks. Given a finite collection of single distributions derived from BNNs or deep ensembles, the proposed approach extracts an upper and a lower probability bound per class, acknowledging the epistemic uncertainty due to the availability of a limited amount of
sampled predictive 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 multiple 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, Res18/50, EfficientNet B2, and ViT Base). Compared to BNN and deep ensemble baselines, the proposed credal representation methodology exhibits superior performance in uncertainty estimation and achieves lower expected calibration error on OOD samples.

## 1 Introduction

Despite their success in various scientific and industrial areas, deep neural networks often generate inaccurate and overly confident predictions when faced with uncertainties induced by, e.g. out-of-distribution (OOD) samples, natural fluctuations or adversarial disruptions ; ; ; . Properly estimating the uncertainty associated with their predictions is key
to improve the reliability and robustness of neural networks
; ; .

Researchers in the field of uncertainty theory commonly distinguish two types of uncertainty in the context of neural networks (and machine learning models in general).
Aleatoric Uncertainty, also known as ‘data’ uncertainty, arises from inherent randomness such as data noise and is irreducible. Epistemic Uncertainty stems from the lack of knowledge of the data generation process and can be reduced with increased availability of training data . The distinction between AU and EU can be beneficial in applications such as active learning or OOD detection, particularly in safety-critical and practical fields such as autonomous driving and medical diagnosis . For instance, in active learning, the objective is to avoid inputs that exhibit high aleatoric uncertainty but low epistemic uncertainty. Similarly, an effective EU estimation can prevent the misclassification of ambiguous in-distribution (ID) examples as OOD instances .

In this paper, we propose a new method for accurate uncertainty estimation and compare this with widely used methods, such as Bayesian Neural Networks and Deep ensembles.

The Bayesian neural network (BNN) framework is a prominent method for uncertainty estimation ; ; ; . BNNs learn the posterior distributions of a network’s weights, and thus predict a ‘second-order’ distribution (a distribution of distributions ) over the target space of the network.
While BNNs are straightforward to construct, their practical training presents a significant challenge due to inherent computational complexity. Consequently, several approximation algorithms, such as sampling methods ; ; ; ; ; and variational inference approaches ; , have been developed . As computing 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 .
The latter entails sampling a finite set of deterministic weight values from the posterior (obtained after training) to generate a range of single (softmax) distributions via the network.
BMA averages such distributions taken as the final prediction. Still, in practice, BNNs are challenging to scale to large datasets and architectures due to high computational complexity , so a limited number of samples is often employed prior to BMA to reduce complexity at inference time.

Deep ensembles, an established alternative approach for quantifying uncertainty, have been recently serving as the state-of-the-art uncertainty estimation baseline in deep learning ; ; ; .
Deep ensembles marginalize multiple deterministic models to obtain a predictive distribution instead of explicitly inferring a distribution over the parameters like BNNs ; . At prediction time, deep ensembles average a finite set of single probability distributions to derive a predictive distribution for classification .
Despite their successes in uncertainty quantification, a recent study has shown that the EU estimates of deep ensembles may demonstrate relatively low quality . They have also been criticized due to their substantial demand for memory resources and computational power ; ; .

Two issues are commonly associated with model averaging for BNNs and deep ensembles at prediction time. Firstly, the number of single distributions generated for approximating the exact predictive distribution is quite limited (e.g., most papers use up to five), due to computational complexity and memory constraints ; . Secondly, model averaging can result in an overly confident and erroneous prediction when the sampled distributions exhibit a conflict opinion regarding the class and a minority of the individual distributions ‘lead’ to the correct decision.

Numerous mathematical theories, including subjective probability , possibility theory ; , credal sets ; , probability intervals ; , random sets , and imprecise probability theory , have been devised in response to the challenge
posed by the limited availability of in-domain training data (in our case, a limited number of sample precise 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.

Probability intervals represent one of the simplest approaches and have garnered significant interest among researchers ; ; ; . Instead of providing point-valued probabilities, probability intervals assume that the probability values $p(y)$ of the elements (e.g., the classes) of the target space
(e.g., the set of classes)
belong to an interval $p_{L}(y)\!\leq\!p(y)\!\leq\!p_{U}(y)$, delimited by a lower probability bound $p_{L}(y)$ and an upper probability bound $p_{U}(y)$. Researchers have claimed that probability intervals
may express uncertainty more appropriately than single probabilities ; ; , particularly in situations in which: (i) limited information is available for estimating the 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 employing BNN model averaging or deep ensembles using a limited number of predictive samples, as we discuss here.

Novelty and Contribution  This paper presents an innovative methodology for formulating a credal set representation for both Bayesian Model Averaging in BNNs and deep ensembles,
capable of improving these models’ 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 deep ensembles, the proposed approach computes a lower and an upper probability bound per class, to acknowledge the ‘epistemic’ uncertainty about the sought exact predictive distribution due to limited information availability.
Such
lower and upper bounds on each class’ probability induce, in turn,
a convex set of probabilities, known as a *credal set* ; ; . From such a credal prediction (the output of our credal wrapper), a precise predictive distribution can be derived using a transform called
*intersection probability* ,
in whose favor theoretical arguments exist as the natural way of mapping a credal set to a single probability .

Extensive experiments are performed on several OOD detection benchmarks including different dataset pairs (CIFAR10/100 vs SVHN/Tiny-ImageNet, CIFAR10 vs CIFAR10-C, CIFAR100 vs CIFAR100-C, and ImageNet vs ImageNet-O) and utilizing various network architectures (VGG16, Res18/50, EfficientNet B2 and ViT Base). Compared to BNN and deep ensemble baselines, the proposed credal representation approach demonstrates improved uncertainty estimation and lower expected calibration error on the corrupted OOD samples when
using the intersection probability.

Other Related Work  Credal sets have
attracted
increasing attention of late, as shown by research efforts within the broader field of machine learning for uncertainty quantification ; ; ; ; ; ; . The advantage of adopting credal sets is the integration of notions of set and probability distribution in a single, unified framework. Using credal sets, rather than individual distributions, arguably allows models to more naturally express epistemic uncertainty ; . Concerning deep neural networks, ‘imprecise’ BNNs modeling 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, a recent work has proposed a method for predicting credal sets from training data labeled by probability distributions in the conformal learning framework.

Paper Outline  The paper is structured as follows. Section 2 presents how uncertainty estimation works in different model classes. Section 3 introduces our credal wrapper in full detail. Section 4 describes validation and results. Section 5 summarizes our conclusions and future work. Appendices report additional experiments in §A and implementation details in §B, respectively.

## 2 Uncertainty Estimation in Different Model Classes

Bayesian Neural Networks  BNNs model network parameters, i.e., weights and biases, as probability distributions.
The resulting predictive distributions can thus be seen as
‘second-order’ distributions, i.e., probability distributions of distributions . As mentioned, for computational reasons,
BMA is often applied for BNN inference . Namely, in a classification context, we obtain:

$$ $\tilde{{\bm{p}}}\!=\!\frac{1}{N_{p}}\sum\nolimits_{i=1}^{N_{p}}{\bm{p}}_{i}\!= \!\frac{1}{N_{p}}\sum\nolimits_{i=1}^{N_{p}}h_{\text{bnn}}^{\bm{\omega}_{i}}( \bm{x}),$ (1) $$

where ${\bm{p}}_{i}$ is the sampled prediction from the deterministic model ($h_{\text{bnn}}^{\bm{\omega}_{i}}$) parametrized by $\bm{\omega}_{i}$,
${N_{p}}$ is the number of sample predictions, and
$\tilde{{\bm{p}}}$ is the averaged probability vector. The vector of parameters $\bm{\omega}_{i}$ is obtained as the $i$-th sampling instance of the parameter posterior distribution of the BNN .

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 , respectively, as follows:

$$ $\text{TU}\!:=\!H(\tilde{{\bm{p}}})\!=\!-\!\!\sum\nolimits_{k}^{C}\!\!\tilde{q} _{k}\!\log_{2}\!{\tilde{q}_{k}},\ \text{AU}\!:=\!\tilde{H}({\bm{p}})\!=\!\frac {1}{N_{p}}\!\!\sum\nolimits_{i=1}^{N_{p}}\!\!H({\bm{p}}_{i})\!=\!-\frac{1}{N_{ p}}\!\!\sum\nolimits_{i=1}^{N_{p}}\!\!\sum\nolimits_{k}^{C}\!\!p_{i_{k}}\!\! \log_{2}\!{p_{i_{k}}},$ (2) $$

where $\tilde{q}_{k}$ and $p_{i_{k}}$ are the $k$-th elements of the probability vectors $\tilde{{\bm{p}}}$ and ${\bm{p}}_{i}$ across the $C$ classes, respectively. An estimate of epistemic uncertainty can then be derived as
$\text{EU}\!:=\!H(\tilde{{\bm{p}}})\!-\!\tilde{H}({\bm{p}})$ , which can be interpreted as an approximation of ‘mutual information’ ; .

Deep Ensembles Deep ensembles generate a prediction from a set of $M$ individually-trained standard neural networks, as follows:

$$ $\tilde{{\bm{p}}}\!=\!\frac{1}{M}\!\sum\nolimits_{m=1}^{M}\!h_{m}(\bm{x})\!=\! \!\frac{1}{M}\!\sum\nolimits_{m}^{M}\!{\bm{p}}_{m},$ (3) $$

where ${\bm{p}}_{m}$ denotes the single probability vector provided by the $m^{\text{th}}$ member $h_{m}$ of the ensemble. Considered by some an approximation of BMA ; ,
DEs also
quantify TU, AU, and EU as in Eq. (2), where the sampled single probability vector ${\bm{p}}_{i}$ is replaced by ${\bm{p}}_{m}$.

Credal Sets  Uncertainty quantification for credal sets is an active research subject ; . To that extent, researchers have developed the concepts of generalized entropy and generalized Hartley (GH) measure ; . Applying the GH measure in practice, however, is challenging due to the high computational cost of solving constrained optimization problems involving $2^{C}$ subsets ; , particularly
when the number of classes
$C$ is large (e.g., $C=100$).
Probability interval width has also been proposed as a measure of EU ,
but is only applicable to binary classification.
As a result, in this paper we opt for generalized entropy. Given a credal set prediction, denoted by ${\mathbb{P}}$, its upper and lower entropy $\overline{H}({\mathbb{P}})$ and $\underline{H}({\mathbb{P}})$ can be calculated as a generalization of the classical Shannon entropy, allowing us to estimate the TU and AU for the credal prediction , as follows:

$$ $\color[rgb]{0,0,0}\text{TU}\!:=\!\overline{H}({\mathbb{P}})\!=\!\max_{{\bm{p}} \in{\mathbb{P}}}\!H({\bm{p}}),\quad\text{AU}\!:=\!\underline{H}({\mathbb{P}}) \!=\!\min_{{\bm{p}}\in{\mathbb{P}}}\!H({\bm{p}}).$ (4) $$

Such measures capture the maximal and the minimal Shannon entropy within the credal set, respectively. The level of EU can then be measured by their difference, namely $\text{EU}\!:=\!\overline{H}({\mathbb{P}})\!-\!\underline{H}({\mathbb{P}})$.

## 3 Methodology

As previously stated, providing a probability interval rather than a single probability value is more informative when the probabilities are uncertain, due to limited or conflicting evidence ; ; . Studies on the combination, marginalization, conditioning or decision-making with interval probabilities ; ; ;
often assume
that such intervals are known, or provided by experts.
Inspired by the use of probability intervals for decision-making ; , 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 (Section 4).
E.g., consider again the task of predicting weather conditions (rainy, sunny, or 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 $[0.1,0.7]$.

Each probability interval determines
a convex set of probabilities over the set 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 to
constraints on the unknown exact distribution ; ; .
Nevertheless, a precise predictive distribution, termed *intersection probability*, can still be derived from a credal set to generate a unique class prediction
for classification purposes.
Our credal wrapper framework is depicted in Figure 1.

The remainder of the section discusses the credal representation approach, a method for computational complexity reduction of uncertainty quantification, and the intersection probability,
in this order.

Figure: Figure 1: Credal wrapper framework for a three-class (A, B, D) classification task. Given a 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. (5). Such lower and upper probability intervals induce a convex set of probabilities on {A, B, D}, known as a credal set (${\mathbb{P}}$, light blue convex hull in the triangle). A precise single intersection probability is computed from the credal set using the transform in Eq. (5). Uncertainty is estimated in the mathematical framework of credal sets.
Refer to caption: x1.png

Credal Representation Approach  Given a set of $N$ individual distributions from BNNs or deep ensembles, an upper and a lower probability bound for $k$-th class element, denoted as $p_{U_{k}}$ and $p_{L_{k}}$, respectively, can be obtained from

$$ $p_{U_{k}}\!=\!\operatorname*{max}_{n=1,..,N}{p_{n,k}},\quad p_{L_{k}}\!=\! \operatorname*{min}_{n=1,..,N}{p_{n,k}},$ (5) $$

where $p_{n,k}$ denotes the $k$-th element of the $n$-th single probability vector ${\bm{p}}_{n}$. Such probability intervals over $C$ classes determine a non-empty credal set ${\mathbb{P}}$,
as follows ; :

$$ ${\mathbb{P}}\!=\!\{{\bm{p}}\!\mid p_{k}\!\in\![p_{L_{k}},p_{U_{k}}],\ \forall k \!=\!1,2,...,C\}\ \ \text{s.t.}\ \sum\nolimits_{k=1}^{C}p_{L_{k}}\leq 1\!\leq \!\sum\nolimits_{k=1}^{C}p_{U_{k}}.$ (6) $$

Individual probabilities in the credal set satisfy the normalization condition, and their probability value per class is constrained to belong to the given probability interval.

It can be readily proven that the probability intervals in Eq. (5) meet the above condition, as follows:

$$ $\sum\nolimits_{k}^{C}\!p_{L_{k}}\!=\!\sum\nolimits_{k}^{C}\!\operatorname*{min }_{n=1,..,N}{p_{n,k}}\!\leq\!\sum\nolimits_{k}^{C}\!{{p_{n^{*},k}}}\!=\!1\! \leq\!\sum\nolimits_{k}^{C}\operatorname*{max}_{n=1,..,N}{p_{n,k}}\!=\!\sum \nolimits_{k}^{C}p_{U_{k}},$ (7) $$

where $n^{*}$ is any index in $1,\ldots,N$.
Computing $\overline{H}({\mathbb{P}})$ and $\underline{H}({\mathbb{P}})$ in Eq. (4) for uncertainty estimation, on the other hand, requires solving the following optimization problems:

$$ $\begin{aligned} \overline{H}({\mathbb{P}})\!&=\!\operatorname*{maximize}\!\sum \nolimits_{k}^{C}\!\!\!-p_{k}\!\cdot\!\log_{2}p_{k}\\ \underline{H}({\mathbb{P}})\!&=\!\operatorname*{minimize}\!\sum\nolimits_{k}^{ C}\!\!\!-p_{k}\!\cdot\!\log_{2}p_{k}\\ \end{aligned}\text{s.t.}\sum\nolimits_{k}^{C}\!p_{k}\!=\!1\ \text{and}\ p_{k} \!\in\![p_{L_{k}},p_{U_{k}}],\ \forall k,$ (8) $$

which can be addressed by using a standard solver, such as the SciPy optimization package .

Computational Complexity Reduction Method  The convex optimization problem in Eq. (8) may present a computational challenge for a large value of $C$ (e.g., $C\!=\!1000$). To
mitigate this issue
we propose an approach, termed *Probability Interval Approximation* (PIA), in Algorithm 1. The PIA method initially identifies the top $J-1$ relevant classes by sorting the probability values of the intersection probability ${\bm{p}}^{*}$ (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 a result, the dimension of the approximate probability interval is reduced from $C$ to $J$.

Figure: Algorithm 1 Probability Interval Approximation

Intersection Probability  In classification tasks, it is desirable to eventually map a credal set to a single probability prediction, in order to make a unique class prediction. A rational criterion for doing so is provided by the *intersection probability* .
Given the probability interval system defined in Eq. (5), the criterion is that, when computing the single predictive probability, the probability interval for each class
should be treated in the same way,
as we have no grounds for distinguishing
one class from another
. Mathematically,
this translates into seeking the
single probability vector ${\bm{p}}^{*}$ such that

$$ $p^{*}_{k}\!=\!p_{L_{k}}+\alpha\cdot(p_{U_{k}}\!-\!p_{L_{k}})\quad\forall k\!= \!1,...,C$ (9) $$

where $p^{*}_{k}$ is the $k$-th element of ${\bm{p}}^{*}$,
under the constant value $\alpha\!\in\![0,1]$.

The concept of intersection probability is pictorially
illustrated in Figure 2.

Figure: Figure 2: An illustration of the intersection probability for a probability interval system of three classes {A, B, D} ; .
Refer to caption: x2.png

By definition, Eq. (9) belongs to the probability interval for any $\alpha\in[0,1]$.
The unique $\alpha$ that satisfies the normalization constraint (making ${\bm{p}}^{*}$ a proper probability distribution)
can be computed as :

$$ $\alpha\!=\!\bigg{(}1-\sum\nolimits_{k}^{C}p_{L_{k}}\bigg{)}/\bigg{(}{\sum \nolimits_{k}^{C}(p_{U_{k}}\!-\!p_{L_{k}})}\bigg{)}.$ (10) $$

The intersection probability transform follows clear rationale principles and does not suffer from the drawbacks of other potential transforms. Let us see this in detail.
Taking (i) the average probability value of the original sampled distributions produces a valid probability vector that falls within the
probability interval system.
However,
such a transform does not leverage the
probability bounds, thus violating the semantics of probability intervals
; (while also exhibiting inferior performance, see Section 4.2).
Selecting (ii) the midpoint of each probability interval $[p_{L},p_{U}]$, instead, does not generally result in a valid normalized probability vector. Finally, (iii) normalizing the lower or the upper probability vectors, i.e., taking
$\hat{p}_{U_{k}}\!=\!p_{U_{k}}/\sum\nolimits_{k}^{C}p_{U_{k}}$ or $\hat{p}_{L_{k}}\!=\!p_{L_{k}}/\sum\nolimits_{k}^{C}p_{L_{k}}$, yields a probability that is
not guaranteed to be consistent with the interval system .

## 4 Experimental Validation

### 4.1 Uncertainty Estimation Evaluation via OOD Detection

The OOD detection task is a widely applied benchmark for EU quantification assessment . In OOD detection, a model is expected to exhibit high EU estimates on OOD samples in comparison to in-domain (ID) instances.
Consequently, superior OOD detection performance provides evidence of enhanced uncertainty estimation quality.
To assess OOD detection performance, we use the AUROC (Area Under the Receiver Operating Characteristic curve) and AUPRC (Area Under the Precision-Recall curve) scores. Greater scores indicate a higher quality of uncertainty estimation. In addition, we also evaluate TU estimation in this setting, as TU estimates are widely employed for OOD detection within BNNs and deep ensembles ; .

#### 4.1.1 Evaluation using Small-Scale Datasets

Setup  For this first experiment, we use dataset pairings of the type (ID samples vs OOD data), including CIFAR10 vs SVHN /Tiny-ImageNet and CIFAR10 vs CIFAR10-C .
As BNN baselines, we choose two standardized variational BNNs: BNNR (Auto-Encoding variational Bayes with the local reparameterization trick ) and BNNF (Flipout gradient estimator with the negative evidence lower bound loss ).
We do not consider
BNNs using sampling approaches because of
their generally high computational resource requirements ; . As for deep ensembles,
our baselines, denoted as DEs-5,
aggregates five SNNs trained using distinct random seeds.
All models are implemented on the established VGG16 and Res18 architectures using the CIFAR10 dataset.
The number of sample predictions for BNNs is set to
${N_{p}\!=\!5}$.
More implementation details are given in Appendix §B.

Results Table 1 reports the OOD detection performance of BNNR, BNNF, and DEs-5, as measured in the CIFAR10 (ID) vs SVHN and Tiny-ImageNet (OOD) settings, based on VGG16 and Res18 backbones. Regarding the CIFAR10-C dataset, which applies 15 corruption modes to the CIFAR10 instances, with 5 intensities per corruption type, Figure 3 shows the OOD detection results of CIFAR10 vs CIFAR10-C against increased corruption intensity. The results over the 15 corruption types are averaged.
The consistently enhanced OOD detection performance on diverse data pairs and distinct baselines across different backbones provides strong evidence that our proposed credal wrapper method does indeed improve EU and TU estimation in a model averaging framework.

**Table 1: OOD detection AUROC and AUPRC performance (%) comparison between classical and credal set representation of BNNs and deep ensembles-5 using EU (left) and TU (right) as uncertainty metrics. All models are implemented on VGG16/Res18 backbones and tested on CIFAR10 (ID) vs SVHN (OOD) and Tiny-ImageNet (OOD). The results are from 15 runs. The best scores per uncertainty metric are in bold.**
|  |  |  | SVHN | Tiny-ImageNet |  | SVHN | Tiny-ImageNet |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  | Model | EU | AUROC | AUPRC | AUROC | AUPRC | TU | AUROC | AUPRC | AUROC | AUPRC |  |
|  |  | Baseline | $H(\tilde{{\bm{p}}})\!-\!\tilde{H}({\bm{p}})$ | 86.65±1.26 | 90.61±0.88 | 84.62±0.28 | 80.06±0.40 | $H(\tilde{{\bm{p}}})$ | 88.57±1.47 | 93.26±1.08 | 85.50±0.31 | 82.60±0.41 |
|  | BNNR | Ours | $\overline{H}({\mathbb{P}})\!-\!\underline{H}({\mathbb{P}})$ | 87.85±1.45 | 92.32±1.05 | 85.55±0.30 | 82.76±0.46 | $\overline{H}({\mathbb{P}})$ | 88.79±1.57 | 93.44±1.14 | 85.89±0.31 | 83.53±0.43 |
|  |  | Baseline | $H(\tilde{{\bm{p}}})\!-\!\tilde{H}({\bm{p}})$ | 86.79±0.47 | 90.76±0.55 | 84.54±0.20 | 79.91±0.35 | $H(\tilde{{\bm{p}}})$ | 88.32±0.50 | 93.06±0.51 | 85.30±0.20 | 82.32±0.30 |
|  | BNNF | Ours | $\overline{H}({\mathbb{P}})\!-\!\underline{H}({\mathbb{P}})$ | 87.66±0.50 | 92.26±0.51 | 85.26±0.20 | 82.33±0.30 | $\overline{H}({\mathbb{P}})$ | 88.47±0.52 | 93.27±0.52 | 85.62±0.20 | 83.23±0.28 |
|  |  | Baseline | $H(\tilde{{\bm{p}}})\!-\!\tilde{H}({\bm{p}})$ | 89.74±1.31 | 93.58±0.97 | 88.49±0.17 | 85.79±0.35 | $H(\tilde{{\bm{p}}})$ | 91.20±1.29 | 94.87±0.78 | 89.55±0.08 | 87.89±0.14 |
| VGG16 | DEs-5 | Ours | $\overline{H}({\mathbb{P}})\!-\!\underline{H}({\mathbb{P}})$ | 91.96±1.33 | 95.24±0.79 | 89.80±0.08 | 87.99±0.16 | $\overline{H}({\mathbb{P}})$ | 92.12±1.32 | 95.41±0.75 | 90.00±0.07 | 88.49±0.17 |
|  |  | Baseline | $H(\tilde{{\bm{p}}})\!-\!\tilde{H}({\bm{p}})$ | 88.32±1.22 | 93.03±0.74 | 85.83±0.25 | 81.43±0.51 | $H(\tilde{{\bm{p}}})$ | 88.18±1.30 | 92.68±0.86 | 86.40±0.24 | 83.16±0.40 |
|  | BNNR | Ours | $\overline{H}({\mathbb{P}})\!-\!\underline{H}({\mathbb{P}})$ | 88.40±1.25 | 93.16±0.79 | 85.99±0.20 | 82.51±0.39 | $\overline{H}({\mathbb{P}})$ | 88.36±1.30 | 92.95±0.87 | 86.52±0.22 | 83.66±0.38 |
|  |  | Baseline | $H(\tilde{{\bm{p}}})\!-\!\tilde{H}({\bm{p}})$ | 88.62±0.80 | 93.26±0.52 | 85.94±0.30 | 81.64±0.34 | $H(\tilde{{\bm{p}}})$ | 88.43±0.83 | 92.90±0.51 | 86.45±0.33 | 83.22±0.33 |
|  | BNNF | Ours | $\overline{H}({\mathbb{P}})\!-\!\underline{H}({\mathbb{P}})$ | 88.71±0.81 | 93.41±0.51 | 86.06±0.31 | 82.69±0.27 | $\overline{H}({\mathbb{P}})$ | 88.65±0.81 | 93.19±0.51 | 86.58±0.32 | 83.75±0.31 |
|  |  | Baseline | $H(\tilde{{\bm{p}}})\!-\!\tilde{H}({\bm{p}})$ | 87.40±0.68 | 91.82±0.51 | 87.94±0.15 | 84.86±0.24 | $H(\tilde{{\bm{p}}})$ | 88.47±0.80 | 92.82±0.64 | 88.96±0.14 | 87.22±0.18 |
| Res18 | DEs-5 | Ours | $\overline{H}({\mathbb{P}})\!-\!\underline{H}({\mathbb{P}})$ | 88.63±0.80 | 92.76±0.64 | 89.16±0.15 | 87.41±0.19 | $\overline{H}({\mathbb{P}})$ | 88.98±0.86 | 93.13±0.70 | 89.53±0.14 | 88.23±0.17 |

Figure: Figure 3: OOD detection on CIFAR10 vs CIFAR10-C of BNNs and deep ensembles against increased corruption intensity, using VGG16 and Res18 as backbones. The metrics include EU and TU.
Refer to caption: x3.png

#### 4.1.2 Evaluation using Large-Scale Datasets

Setup  In this second experiment, we exclusively employ deep ensembles (DEs-5) to assess the quality of uncertainty estimation of our proposed method on large-scale datasets and architectures.
This is because, in practice, BNNs are typically unable to scale to large datasets and model architectures due to the high computational complexity .
For instance, training a Res50-based BNNs on CIFAR-10 (resized to (224, 224, 3)) failed in our experiment due to exceeding the memory capacity of a single A100 GPU.
The dataset pairs (ID vs OOD) considered include CIFAR10/CIFAR100 vs SVHN/Tiny-ImageNet, ImageNet vs ImageNet-O , CIFAR10 vs CIFAR10-C, and CIFAR100 vs CIFAR100-C . DEs-5 are implemented on the well-established Res50 . All data samples have a shape of (224, 224, 3).
More training details are given in Appendix §B.
The PIA algorithm (Algorithm 1) is applied using the settings $J\!=\!20$ and $J\!=\!50$ to calculate the generalized entropy ($\overline{H}({\mathbb{P}})$ and $\underline{H}({\mathbb{P}})$) on dataset pairs involving CIFAR100 and ImageNet, respectively.

Results  Compared to vanilla model averaging applied to deep ensembles, our wrapper’s enhanced OOD detection performance on diverse data pairs shown in Table 2 provides compelling evidence that our proposed method can indeed improve EU and TU estimation.

**Table 2: OOD detection AUROC and AUPRC performance (%) of the classical and credal set representation of DEs-5 using EU (left) and TU (right) as uncertainty metrics. The results are from 15 runs, based on the Res50 backbone. Best scores per uncertainty metric in bold.**
| Uncertain Metrics | CIFAR10 (ID) | CIFAR100 (ID) | ImageNet (ID) |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  |  |  | SVHN (OOD) | Tiny-ImageNet (OOD) | SVHN (OOD) | Tiny-ImageNet (OOD) | ImageNet-O (OOD) |  |  |  |  |  |
|  |  |  | AUROC | AUPRC | AUROC | AUPRC | AUROC | AUPRC | AUROC | AUPRC | AUROC | AUPRC |
|  | Baseline | $H(\tilde{{\bm{p}}})$ | 94.80±0.43 | 97.26±0.29 | 88.80±0.19 | 87.21±0.29 | 78.53±1.94 | 88.83±1.01 | 80.75±0.15 | 77.65±0.19 | 50.20±0.07 | 50.44±0.06 |
| TU | Ours | $\overline{H}({\mathbb{P}})$ | 95.44±0.37 | 97.57±0.23 | 89.30±0.17 | 87.97±0.25 | 80.71±1.96 | 89.97±0.99 | 81.46±0.14 | 78.29±0.17 | 54.87±0.08 | 52.27±0.05 |
|  | Baseline | $H(\tilde{{\bm{p}}})\!-\!\tilde{H}({\bm{p}})$ | 89.58±0.93 | 92.29±1.00 | 86.87±0.20 | 83.02±0.16 | 73.83±1.97 | 84.96±1.25 | 78.80±0.20 | 74.68±0.27 | 65.70±0.41 | 63.20±0.35 |
| EU | Ours | $\overline{H}({\mathbb{P}})\!-\!\underline{H}({\mathbb{P}})$ | 93.77±0.60 | 96.06±0.46 | 88.78±0.15 | 86.83±0.23 | 80.22±1.96 | 89.40±1.03 | 81.00±0.16 | 77.16±0.23 | 66.20±0.38 | 63.23±0.34 |

#### 4.1.3 Ablation Study on Network Architectures

Setup  We also perform an ablation study using EfficientNetB2 (EffB2) and Virtual Transformer Base (ViT-B) as architecture backbones, involving CIFAR10/100 vs SVHN and Tiny-ImageNet, CIFAR10 vs CIFAR10-C, and CIFA100 vs CIFAR100-C.

Results  The key findings, presented in Table 3 and Figure 4, demonstrate that the proposed credal representation approach significantly enhances the EU and TU estimation quality of deep ensembles, and is robust against both the dataset pairs and the architecture backbones. In Table 3, a 0.1% drop in AUPRC using the TU metric (EffB2 backbone, CIFAR100 vs Tiny-ImageNet) is observed. TU ($\overline{{\mathbb{P}}}$) is computed by solving a constrained optimization problem (Eq. 8) using a numerical solver from SciPy. The slight performance decrease is likely due to numerical errors during the optimization process.

**Table 3: OOD detection AUROC and AUPRC performance (%) of the classical and credal set representation of DEs-5 using EU (left) and TU (right) as metrics. Results are from 15 runs, based on EffB2 and ViT-B backbones. Best scores per uncertainty metric in bold.**
|  |  |  | SVHN (OOD) | Tiny-ImageNet (OOD) |  | SVHN (ID) | Tiny-ImageNet (OOD) |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| ID | Backbone | EU | AUROC | AUPRC | AUROC | AUPRC | TU | AUROC | AUPRC | AUROC | AUPRC |  |
|  |  | Baseline | $H(\tilde{{\bm{p}}})\!-\!\tilde{H}({\bm{p}})$ | 95.76±0.59 | 97.43±0.47 | 92.32±0.14 | 90.72±0.22 | $H(\tilde{{\bm{p}}})$ | 97.55±0.27 | 98.78±0.16 | 93.41±0.15 | 93.01±0.17 |
|  | EffB2 | Ours | $\overline{H}({\mathbb{P}})\!-\!\underline{H}({\mathbb{P}})$ | 97.60±0.25 | 98.74±0.14 | 93.55±0.15 | 93.17±0.16 | $\overline{H}({\mathbb{P}})$ | 97.91±0.21 | 98.98±0.10 | 93.74±0.15 | 93.55±0.16 |
|  |  | Baseline | $H(\tilde{{\bm{p}}})\!-\!\tilde{H}({\bm{p}})$ | 77.71±1.67 | 85.82±1.14 | 82.27±0.79 | 78.85±0.81 | $H(\tilde{{\bm{p}}})$ | 79.80±1.75 | 87.97±1.17 | 83.81±0.81 | 81.67±0.89 |
| CIFAR10 | ViT-B | Ours | $\overline{H}({\mathbb{P}})\!-\!\underline{H}({\mathbb{P}})$ | 80.71±1.83 | 88.43±1.26 | 84.28±0.76 | 82.82±0.85 | $\overline{H}({\mathbb{P}})$ | 81.08±1.82 | 88.48±1.48 | 84.62±0.75 | 82.82±0.85 |
|  |  | Baseline | $H(\tilde{{\bm{p}}})\!-\!\tilde{H}({\bm{p}})$ | 87.52±1.52 | 93.81±0.80 | 85.29±0.15 | 82.98±0.24 | $H(\tilde{{\bm{p}}})$ | 88.46±1.33 | 94.40±0.59 | 86.45±0.10 | 84.88±0.14 |
|  | EffB2 | Ours | $\overline{H}({\mathbb{P}})\!-\!\underline{H}({\mathbb{P}})$ | 89.20±1.19 | 94.58±0.56 | 86.23±0.10 | 83.69±0.21 | $\overline{H}({\mathbb{P}})$ | 89.35±1.20 | 94.83±0.52 | 86.61±0.09 | 84.87±0.12 |
|  |  | Baseline | $H(\tilde{{\bm{p}}})\!-\!\tilde{H}({\bm{p}})$ | 81.41±1.33 | 89.00±0.96 | 81.18±0.31 | 77.46±0.49 | $H(\tilde{{\bm{p}}})$ | 84.10±1.12 | 91.41±0.72 | 82.64±0.28 | 79.94±0.43 |
| CIFAR100 | ViT-B | Ours | $\overline{H}({\mathbb{P}})\!-\!\underline{H}({\mathbb{P}})$ | 84.87±1.06 | 91.40±0.67 | 82.53±0.32 | 79.04±0.39 | $\overline{H}({\mathbb{P}})$ | 85.40±1.05 | 92.02±0.70 | 83.06±0.29 | 80.34±0.41 |

Figure: Figure 4: OOD detection on CIFAR10/100 vs CIFAR10-C/100-C of deep ensembles against increased corruption intensity, using Res50, EffB2, and ViT-B as backbones. The metrics include EU and TU.
Refer to caption: x4.png

### 4.2 ECE Performance Evaluation

Setup  *Expected calibration error* (ECE) ; is an indicator for measuring the model calibration performance. A lower ECE value signifies a closer alignment between the model’s confidence scores and the true probabilities of the events. The OOD samples CIFAR10-C and CIFAR100-C have the same target space (set of class labels) as in the CIFAR10 and CIFAR100 datasets, respectively. This allows us to evaluate ECE performance in the case of facing corrupted data, using both the proposed intersection probability and the average consensus probability as the final model prediction.

Results  Figure 6 reports the ECE values of BNNR, BNNF, and DEs-5 on the CIFAR10-C dataset against the increased corruption intensity, using VGG16 and Res18 backbones. Furthermore, Figure 6 shows the ECE values of deep ensembles on both CIFAR10-C and CIFAR100-C datasets, using Res50, EffB2, and Vit-B backbones.
The findings 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.

Figure: Figure 5: ECE values of BNNR, BNNF, and DEs-5 on CIFAR10-C against increased corruption intensity, using the averaged probability (Prob.) and our proposed intersection probability (Prob.). VGG16 and Res18 are backbones. Results are from 15 runs.
Refer to caption: x5.png

### 4.3 Additional Results

Appendix §A.1 performs an ablation study on the number $N$ of probability distributions sampled at prediction time, using Res50-based deep ensembles on the OOD benchmark (CIFAR10 vs CIFAR10-C/SVHN/TinyImageNet).
It demonstrates that our credal wrapper approach consistently achieves the higher quality of EU and TU estimation and lower ECE (using the intersection probability).
A larger $N$ results in better uncertainty estimation and lower ECE values.
Appendix §A.2 analyzes the effect of the hyperparameter $J$ of the PIA algorithm 1. The findings suggest that increasing the value of $J$ improves the OOD detection performance, but may lead to an increase in execution time.

## 5 Conclusion and Future Work

Conclusion  This paper presents an innovative approach, called a *credal wrapper*, to formulating a credal set representation for model averaging in BNNs and deep ensembles, capable of improving their uncertainty estimation in classification tasks. Given a limited collection of single distributions derived at prediction time from either BNNs or deep ensembles, the proposed approach computes a lower and an upper probability bound per class, to acknowledge the epistemic uncertainty about the sought exact predictive distribution due to limited information availability.
Such lower and upper probability bounds on each class induce a credal set. From such a credal prediction (the output of our wrapper), an intersection probability distribution can be derived, for which there are theoretical arguments that it is the natural way to map a credal set to a single probability.

Extensive experiments are performed on several OOD detection benchmarks including CIFAR10/100 vs SVHN/Tiny-ImageNet, CIFAR10/100 vs CIFAR10/100-C, and ImageNet vs ImageNet-O and utilizing various network architectures (VGG16, Res18/50, EfficientNet B2 and ViT Base). Compared to BNN and deep ensemble baselines, the proposed credal wrapper approach demonstrates improved uncertainty estimation and lower expected calibration error on corrupted OOD samples.

Limitation and Future work  Despite the evident efficacy of our credal wrapper approach, uncertainty quantification in the credal set framework requires a greater computational investment, as shown in Appendix §A.2.
As a result, if there are limitations on memory usage and computational resources, the proposed approach might not be the optimal solution. A primary objective of our forthcoming research is to apply and benchmark our approach on real-world applications, such as medical image analysis.

## Acknowledgments

This work was supported by granted H2020 FETOPEN-2018-2019-2020-01 European project, *Epistemic AI* under grant agreement No. 964505 (E-pi).

## Appendix A Additional Experiments

### A.1 Ablation Study on Numbers of Predictive Samples

In this experiment, we train 30 Res50-based SNNs individually using the CIFAR10 dataset and under different random seeds. Then, we construct deep ensembles using the different numbers of ensemble members $N\!=\!3,5,10,15,20,25$. Each type of deep ensemble includes 15 instances using distinct seed combinations.

Tables 4 and 5 report the OOD detection performance comparison under different numbers of samples and using the EU and TU estimates as the metrics, respectively. The findings suggest that

- •
Increasing the number of samples can improve the EU and TU estimation of our proposed credal wrapper approach.
- •
Compared to classical MBA of deep ensembles, our method consistently shows superior performance on EU and TU quantification, robust against the number of samples.

Figure 7 shows the ECE performance evaluation using the different number of samples on the CIFAR10-C samples. We can observe that

- •
Increasing the number of samples overall can lead to a lower ECE value.
- •
Compared to the classical averaged probability of deep ensembles, our intersection probability consistently achieves the lower ECE value on the corrupted instances.

**Table 4: OOD detection AUROC and AUPRC performance (%) comparison between classical and credal set representation of DEs-5 using EU as the uncertainty metrics, 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.**
|  | CIFAR10 vs SVHN | CIFAR10 vs Tiny-ImageNet |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  | AUROC | AUPRC | AUROC | AUPRC |  |  |  |  |
| $N$ | Baseline | Ours | Baseline | Ours | Baseline | Ours | Baseline | Ours |
| 3 | 89.64±1.16 | 91.97±1.38 | 92.13±1.12 | 94.91±1.08 | 86.31±0.24 | 87.57±0.25 | 81.76±0.40 | 85.02±0.32 |
| 5 | 89.26±1.02 | 93.50±0.79 | 91.96±0.97 | 95.79±0.70 | 86.83±0.25 | 88.65±0.25 | 83.12±0.38 | 86.77±0.37 |
| 10 | 90.54±0.72 | 95.03±0.40 | 93.22±0.70 | 96.80±0.35 | 87.75±0.13 | 89.62±0.14 | 84.99±0.25 | 88.04±0.20 |
| 15 | 90.78±0.68 | 95.31±0.47 | 93.23±0.62 | 96.91±0.43 | 88.10±0.10 | 89.94±0.10 | 85.59±0.16 | 88.44±0.13 |
| 20 | 91.09±0.50 | 95.69±0.34 | 93.49±0.48 | 97.15±0.30 | 88.33±0.08 | 90.11±0.08 | 85.96±0.11 | 88.59±0.10 |
| 25 | 91.31±0.27 | 95.84±0.15 | 93.62±0.24 | 97.24±0.12 | 88.46±0.04 | 90.22±0.05 | 86.17±0.08 | 88.72±0.07 |

**Table 5: OOD detection AUROC and AUPRC performance (%) comparison between classical and credal set representation of DEs-5 using TU as uncertainty metrics, involving CIFAR10 (ID) vs SVHN (OOD) and Tiny-ImageNet (OOD). The results are from 15 runs. The best scores per uncertainty metric are in bold.**
|  | CIFAR10 vs SVHN | CIFAR10 vs Tiny-ImageNet |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  | AUROC | AUPRC | AUROC | AUPRC |  |  |  |  |
| $N$ | Baseline | Ours | Baseline | Ours | Baseline | Ours | Baseline | Ours |
| 3 | 94.30±0.99 | 94.68±1.01 | 97.03±0.62 | 97.21±0.65 | 88.24±0.23 | 88.54±0.23 | 86.46±0.33 | 86.98±0.32 |
| 5 | 94.78±0.54 | 95.40±0.55 | 97.19±0.38 | 97.49±0.40 | 88.71±0.25 | 89.17±0.26 | 87.10±0.37 | 87.86±0.37 |
| 10 | 95.47±0.34 | 96.18±0.31 | 97.61±0.23 | 97.91±0.23 | 89.27±0.13 | 89.86±0.14 | 87.90±0.19 | 88.63±0.19 |
| 15 | 95.55±0.32 | 96.25±0.37 | 97.63±0.23 | 97.88±0.29 | 89.48±0.08 | 90.11±0.10 | 88.19±0.12 | 88.87±0.12 |
| 20 | 95.72±0.23 | 96.46±0.26 | 97.74±0.15 | 97.98±0.23 | 89.60±0.07 | 90.24±0.08 | 88.34±0.09 | 88.95±0.11 |
| 25 | 95.80±0.12 | 96.53±0.12 | 97.78±0.08 | 98.01±0.10 | 89.67±0.04 | 90.32±0.05 | 88.42±0.05 | 89.01±0.07 |

Figure: Figure 7: ECE values of deep ensembles with various numbers of samples ($N$) on CIFAR10-C against increased corruption intensity, using the averaged probability and our proposed intersection probability.
Refer to caption: x7.png

### A.2 Ablation Study on Hyperparameter of PIA Algorithm

In this experiment, we investigate the effect of the hyperparameter $J$ of the PIA algorithm 1. The OOD detection tasks include CIFAR100 (ID) vs SVHN (OOD) and Tiny-ImageNet (OOD) and deep ensembles are implemented on the Res50 backbone. Table 6 shows OOD detection performance and the time cost for uncertainty quantification per sample, using different settings of $J$ for the PIA algorithm. The findings suggest that increasing the value of $J$ improves the OOD detection performance, but can lead to an increase in execution time. This is because solving the constrained optimization problem in Eq. (8) involves more variables and constraints.

Computational Cost  The reported time in Table 6 cost is measured on a single Intel Xeon Gold 8358 CPU@2.6 GHz, without optimization in the calculation process. As a reference, the time cost of the classical method for uncertainty quantification per sample is 0.007 ± 0.001s. We believe a more efficient code implementation of our approach could significantly mitigate the computational cost.

**Table 6: OOD detection AUROC and AUPRC performance (%) of credal set representation of DEs-5 using EU (left) and TU (right) as uncertainty metrics, and the time cost, using different setting of $J$ of PIA algorithm. The OOD detection involves CIFAR100 (ID) vs SVHN (OOD) and Tiny-ImageNet (OOD). The results are from 15 runs, based on the Res50 backbone.**
| $J$ | EU as Metrics | TU as Metrics | Time Cost<br>per Sample<br>(s) |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| SVHN | Tiny-ImageNet | SVHN | Tiny-ImageNet |  |  |  |  |  |  |
| AUROC | AUPRC | AUROC | AUPRC | AUROC | AUPRC | AUROC | AUPRC |  |  |
| 10 | 78.81±1.87 | 87.96±1.09 | 80.17±0.19 | 75.25±0.26 | 80.05±1.92 | 89.30±0.97 | 81.24±0.14 | 77.60±0.18 | 7.182±0.087 |
| 20 | 80.22±1.96 | 89.40±1.03 | 81.00±0.16 | 77.16±0.23 | 80.71±1.96 | 89.97±0.99 | 81.46±0.14 | 78.29±0.17 | 11.346±0.038 |
| 100 | 80.55±1.99 | 89.68±1.04 | 81.16±0.16 | 77.76±0.21 | 80.90±1.98 | 90.07±1.00 | 81.52±0.14 | 78.50±0.15 | 109.469±1.330 |

## Appendix B Experimental Implementation Details

### B.1 Training Details

In terms of the evaluation using the small-scale datasets in Section 4.1.1, all models are implemented on the established VGG16 and Res18 architectures using the CIFAR10 dataset. The Adam optimizer is applied with a learning rate scheduler, initialized at 0.001, and subjected to a 0.1 reduction at epochs 80 and 120. Standard neural networks and BNNs are trained for 100 and 150 epochs, respectively. Standard data augmentation is uniformly implemented across all methodologies to enhance the training performance quality of training data and training performance. The training batch size is set as 128. The device is a single Tesla P100-SXM2-16GB GPU. The standard data split is applied.

In terms of the evaluation using the large-scale datasets and the ablation study in Sections 4.1.2 and 4.1.3,
We mainly use two Tesla P100-SXM2-16GB GPUs as devices to contract deep ensembles by independently training 15 standard neural 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 CIFAR10 dataset, SNNs are trained using 15, 15, and 25 epochs for Res50, EffB2, and ViT-B backbones, respectively. As for the CIFAR100 dataset, SNNs are trained using 20, 20, and 25 epochs for Res50, EffB2, and ViT-B backbones, respectively. Note that models based on ViT-B backbones are trained on one single NVIDIA A100-SXM4-40GB GPU. In the ImageNet experiments, we employ one single NVIDIA A100-SXM4-40GB GPU to retrain SNNs based on a pre-trained Res50 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.

The experiment codes are provided in the supplementary files.

### B.2 OOD Detection Process

In this paper, the OOD detection process is treated as a binary classification. We label ID and OOD samples as 0
and 1, respectively. The model’s uncertainty estimation (using the EU 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 evaluates precision and recall trade-offs, providing valuable insights into the model’s effectiveness across different confidence levels.

The detailed implementation for OOD detection is shown in Algorithm 2.

Figure: Algorithm 2 AUROC and AUPRC Scores for OOD Detection

### B.3 ECE Evaluation Process

In the context of ECE, a ‘well-calibrated’ prediction is expected to have a confidence value of 80% and be correct in approximately 80% of the test cases. In ECE calculation, predictions are split into a predetermined number $Q$ of bins $B$ of equal confidence range. The ECE is then calculated by summing the absolute difference between the average accuracy and confidence within each bin []:

$$ $\text{ECE}\!:=\!\sum\nolimits_{g=1}^{G}\frac{|B_{g}|}{n}\bigg{|}\text{acc}(B_{ g})-\text{conf}(B_{g})\bigg{|},$ (11) $$

where $|B_{g}|$ is the number of samples in the $g$-th bin and $n$ is the total number of samples.

## Peer Reviews

confidence: 3
contribution: 3
initial_rating: 6
presentation: 2
questions: Please refer to weaknesses.
soundness: 3
strengths: 1. The idea of using probability intervals to reflect the epistemic uncertainty is appealing.

2. The intersection probability transformation has a deterministic solution, and thus it is as efficient as other alternatives in model averaging.

3. Experimental results are strong and comprehensive.
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.
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?

2. Computing epistemic uncertainty using probability intervals is costly. I expect discussions on the extra computational cost compared with entropy.

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 2
questions: As I stated above, I'd like to know the following (even simple, intuitive answers will be helpful):
- Why can the simple-looking method drastically improve the baseline results?
- How well does the proposed method perform compared to methods listed in, e.g., lines 116-125?
- Are there any visualizations/etc. of the credal sets in practice? (real-world datasets with well-trained classifier samples)
soundness: 4
strengths: - The proposed method is simple yet effective, as being demonstrated in the widely-conducted experiments.
- The proposed approximation algorithm, PIA, effectively approximates the original optimization problem and ultimately makes the idea practically tractable.
- The proposed method is easy to "wrap" around various methods as long as they provide multiple samples of model predictions.
- Experimental details are well documented.
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.
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.

Besides, 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.

---

confidence: 4
contribution: 2
initial_rating: 6
presentation: 4
questions: Can you provide feedback regarding the stated weaknesses?
soundness: 4
strengths: - The paper is really well-written and easy to follow.
- The proposed idea is really simple and is highly applicable in many settings.
- The experiments are nicely structure and follows the evaluation protocol of the literature.
- The authors clearly highlight the approach's limitations stemming from its computational complexity.
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.
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.
- **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.
- **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.  
Additionally, 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.


[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.  
[2] Wang, Kaizheng, et al. "CreINNs: Credal-Set Interval Neural Networks for Uncertainty Estimation in Classification Tasks." arXiv preprint arXiv:2401.05043, 2024.  
[3] Van Amersfoort, Joost, et al. "Uncertainty estimation using a single deep deterministic neural network." ICML, 2020.

---

confidence: 3
contribution: 1
initial_rating: 3
presentation: 3
questions: Please see the weaknesses.
soundness: 2
strengths: * The experimental results demonstrate that the proposed method consistently improves uncertainty estimation and OOD detection.
* The proposed method is straightforward and convincing. Additionally, Figure 1 effectively provides an intuitive overview of the method.
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.
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].
* 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]).
* The computational cost is significantly higher than simple probability averaging, which may be a barrier to using this method in real-world applications.

In 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.



[1] Minderer, Matthias, et al. "Revisiting the calibration of modern neural networks." Advances in Neural Information Processing Systems 34 (2021): 15682-15694.
[2] Vishniakov, Kirill, Zhiqiang Shen, and Zhuang Liu. "Convnet vs transformer, supervised vs clip: Beyond imagenet accuracy." arXiv preprint arXiv:2311.09215 (2023).
[3] Park, Namuk, and Songkuk Kim. "How do vision transformers work?." arXiv preprint arXiv:2202.06709 (2022).

Meta-review: The paper introduces a new approach called the "credal wrapper" to improve uncertainty estimation in classification tasks using Bayesian neural networks and deep ensembles. The method constructs a probability interval per class and maps that interval to a credal set.  The authors show in experiments that their method maintains high accuracy while being better calibrated than the baselines, and that the method is effective for performing out-of-distribution detection.  

The reviewers mostly agreed to accept the paper although one reviewer thought it was borderline reject (8, 8, 8, 5).  The dissenting reviewer seemed to find the approach sensible, but mostly voiced concerns over the impact of the paper / method.  They seemed to suggest that existing methods were sufficient to improve the calibration of deep neural networks, or that modern networks are well-calibrated already, and that the current goal of the literature should be to improve efficiency.  The reviewers agreed the paper was well-written, the approach novel, simple and sensible and the experiments convincing.  As weaknesses, the reviewers noted lack of comparison to existing literature, limited theory, and high computational cost.

Overall, the reviewers seem to find this a strong paper.  There doesn't seem to be a reason to overrule this assessment.  While the criticism of the dissenting reviewer is noted (there is an extensive literature on improving calibration already), I would disagree that this problem seems at all solved by existing tools such as ensembling (or that existing models are well-calibrated).  Therefore the recommendation is to accept.

Decision: Accept (Spotlight)