Credal Deep Ensembles for Uncertainty Quantification

Dear Reviewer,

We thank you for your valuable feedback.

Response to W1: We believe that lines 172-175 in the original paper are also an important result, which shows that cross-entropy loss (CE) can be applied to upper probability vectors as representatives of part of the boundary of the predicted credal set.

However, in the original paper, indeed, the only validation of our hypothesis that credal prediction leads to better accuracy, calibration, and OOD detection is empirical. Below we outline a more rigorous argument.

Epistemic uncertainty in predictions is induced by the probability distribution(s) that generated the training data may be very different from the distribution that generates a test point. It is well known that using CE as a loss leads to the maximum likelihood estimator of the model parameters, under the assumption that all training points are drawn i.i.d. from a single, unknown distribution $P$:

$$\hat{\theta} = \arg \min_\theta\text{CE} = \theta_{MLE}.$$

What does Distributionally Robust Optimization (DRO) do? It takes a cautious stance, and assumes training points are generated by one of many potential distributions $\mathcal{U}$, and minimizes the upper bound of the expected risk (Eq. (6) of the paper). In practice, this is approximated using Adversarially Reweighted Learning as in Eq. (9). Therefore, it learns $\hat{\theta} = \arg \min_{P \in \mathcal{U}} E_P (Loss)$. However, if we apply DRO to a traditional network, the network will learn a single parameter value (Eq. (9) of the paper); for a new test data point $x_t$ it will then produce a single predicted probability vector, the one produced by the model that, in the long run, minimizes the worst-case risk.

This, though, does not express at all the (epistemic) uncertainty about which of the distributions in $\mathcal{U}$ has generated the particular test datapoint $x_t$; as a result, it does not express how uncertain the prediction may be for that particular data point.

Allowing a network to output lower and upper probability vectors for each input, instead, allows this epistemic uncertainty to be explicitly modeled.

What is the best-case scenario? That in which all data (including the test point $x_t$) are indeed generated by the same distribution – in this case, $\theta_{MLE}$ is the best choice, in a maximum likelihood sense.

What is the worst case? That in which $x_t$ is generated by a distribution that is far away from the groups of distributions $P_g$ (see lines 140-141) that generated the training data. In this case, the DRO solution will still be far from perfect, but still the best one can do, given the evidence in the training data.

Therefore, if I minimize the CE versus $\boldsymbol{q}_U$ (first component of Eq. (10)), I get a credal set Cre of predictions whose upper probability is such that, for each class $i = 1,…,C$, there exists a probability vector $\boldsymbol{q}\in \mathbb{Q}$ such that $q(i)$ is the MLE prediction for that class (please remember the discussion of lines 172-175).

Similarly, if I minimize the DRO loss versus $\boldsymbol{q}_L$ (the second component of Eq. (10)), for each class $i = 1,…,C$, there exists a probability vector $\boldsymbol{q} \in \mathbb{Q}$ such that $q(i)$ is the DRO prediction for that class.

Minimizing the two components together, as in Eq. (10), thus encourages the predicted credal set to extend from the best-case to the worst-case prediction.

Response to W2, W4, and W6: We kindly invite you to follow the detailed response in our global rebuttal. Based on your suggestions, we further reported additional inference complexity comparison in Table 1 of the rebuttal file and included MC Dropout and two TensorFlow-standardized BNNs.

Response to W3: Yes, our method is for classification and we provide a roadmap for extending it to regression tasks in Appendix E.3. Our training includes a hyperparameter $\delta$. We performed the ablation study (lines 287-312), showing that our CreDEs are not particularly sensitive to $\delta$. $K$ is a design choice by the end user. The ablation study in Appendix B.2 shows that increasing the value of $K$ improves performance and increases cost. In addition, the reduction algorithm has a strong mathematical background in probability intervals [17], which can rigorously guarantee a valid credal set.

Response to W5, Q1, Q2: Based on your advice, we reported the entropy plots and reliability diagrams in the rebuttal file. Regarding Table 5, the EU estimates of DEs-5 are 0.0996 (CIFAR10), 0.4758 (SVHN), and 0.4753 (Tiny-ImageNet). Although the EU estimates of DEs-5 and CreDEs-5 are not comparable in values due to different representations, CreDEs-5 exhibits much higher EU estimates of OOD samples than those of ID data and more reliable predictions on ID samples, qualitatively. We could not report $Var_i[\mu_i(x)]$ in our cases because the calculation is for models with a custom softmax layer, according to Eq. (6-8) in [10].

Response to Q3: In our settings, the corrupted CIFAR10 samples will be detected as OOD data and be to abstain. Therefore, we did not report the accuracy and ECE values on CIFAR10-C. For completeness, we reported the numbers in Table 3 of the rebuttal file, showing the visibly improved performance of CreDEs compared to DEs.

Response to Q4: $K$ is used to reduce the complexity of uncertainty estimation. We do not employ Alg. 2 for accuracy and ECE calculation. The difference is most likely because the training settings are significantly different. As the ImageNet experiments are conducted fairly for DEs and CreDEs (shown in Appendix C), we believe the results can properly reflect the performance improvements.

Response to Q5: We randomly selected five members from the 15 models to construct the ensemble. The member list of each ensemble was strictly different. Could you please specify the settings that you are curious about? We are eager to continue the discussion.

Dear Reviewer,

We express our sincere appreciation for your recognition of our additional results and your effort in reviewing our paper. Please allow us to address your remaining concerns.

Lacking of a generalization bound or robustness certificate from a theoretical perspective

Thank you for clarifying your question. This is true, CreDEs currently do not provide coverage guarantees/robustness certificates. However, the extensive experiments across different datasets, architectures, and tasks do empirically show that our CreDEs generally achieve consistent improvements over DEs. We have acknowledged lacking theoretical guarantees of our CreDEs in our conclusion (lines 349-350) and detailed our future research plan in Appendix E.2. Incorporating your comments, we will expand our emphasis on this point in the main body of the revised paper.

Computationally inefficient than other baselines

Regarding more hardware settings to evaluate the inference cost

Following your comments, we report the inference cost measured on a single AMD EPYC 7643 48-core CPU in Table R1. From the result in Table R1, we did not observe any significant overhead. In addition, we can also observe that using VGG16 (a lighter model compared to ResNet50) can visibly reduce the inference cost of DEs and CreDEs.

Table R1: Inference cost comparison between SNN and CreNet per single CIFAR10 input of different architectures.

Since our main evaluation scope focuses on image classification of different deep neural network architectures across various tasks, good-performance GPUs are a favorable choice for implementation.

We appreciate your consideration of the complexity of applying our models on smaller hardware such as mobile and IoT devices. To our knowledge, in the context of smaller hardware, lightweight neural networks are generally used, and there are also techniques such as quantization to further improve computational efficiency. We believe that these factors, as well as optimized code implementation of our CreDEs, could help mitigate the added complexity compared to DEs. If practitioners are already using or planning to use DEs in their applications, we believe it is worth considering our CreDEs to improve performance with an acceptable increase in complexity.

Regarding further comparisons.

We fully agree that there are uncertainty-aware models that are more computationally efficient than DEs and CreDEs. As you rightly pointed out, achieving comparable or slightly lower performance of DEs while being computationally cheaper is the main objective of this line of research. However, since DEs are active strong baselines [1, 2] and are actively applied in practice, such as [3] [4] [5], we aim to improve the performance of DEs (e.g., the quality of uncertainty estimation) to benefit safety-critical applications in particular.

Following your comments, we tried to implement the Rank-1 BNNs in our environment to fairly compare the performance between Rank-1 BNNs and CreDEs. Implementing Rank-1 BNNs requires installing a package of Edward2's Bayesian Layers from their official GitHub repository. Unfortunately, although we followed the instructions, we could not import the edward2 package correctly due to dependency conflicts, software version incompatibility, and limited time. There are the same unresolved issues posted by others on the official GitHub repository.

Nevertheless, from the main results of Tables 1-4 in the original paper on Rank-1 BNNs, we could observe that Rank-1 BNNs achieve comparable or slightly lower performance compared to DE baselines while being computationally efficient. Combined with our observations in our paper that CreDEs generally outperform DEs, we believe, there is a high probability that our CreDEs could achieve better performance than Rank-1 BNNs.

The practical message we could deliver to the community is that our CreDEs have a high potential to improve the performance of DEs in real-world applications with an acceptable increase in complexity. If DEs are already out of consideration in practice due to computational constraints, our CreDEs would indeed not be an ideal option. As promised, we will extend our original discussion on the computational limitations of CreDEs (lines 347-348) in the revised paper.

[1] Benchmarking uncertainty disentanglement: Specialized uncertainties for specialized tasks. 2024.

[2] Deep deterministic uncertainty: A new simple baseline. CVPR 2023.

[3] De-tgn: Uncertainty-aware human motion forecasting using deep ensembles. IEEE Robotics and Automation Letters 2024

[4] Flood uncertainty estimation using deep ensembles. Water, 2022.

[5] Benchmarking common uncertainty estimation methods with histopathological images under domain shift and label noise. Medical image analysis, 2023.

hyperparameter $\delta$

Throughout the main experimental evaluations (except the ablation study on $\delta$), we set the default $\delta=5$, to reflect a balanced assessment of the train-test divergence and show how such a value allows our model to outperform the baselines. Upon this setting of $\delta$, the extensive comparisons consistently show our CreDEs outperform DEs.

The ablation study on Hyperparameter $\delta$ (lines 287-312) verifies the robustness of CreDEs across hyperparameter setups in terms of test accuracy, ECE, and OOD detection performance, consistently better than the performance of DEs. For a visible sense, we copied the results of Table 4 of the paper as follows in Table R2.

Table R2: Performance comparison of CreDEs trained on the CIFAR10 dataset under different settings of $\delta$.

Table 5 of the paper reports the averaged EU estimate under different settings of $\delta$. You are correct, we could see significant differences between $\delta=0.5$ and $\delta=96875$ (The EU estimates using $\delta=96875$ are much smaller than those using $\delta=0.5$).

From Table 5, we could also observe that increasing the value of $\delta$ (i.e., giving less importance to the divergence between test and training distributions) leads to a decreasing trend in the average EU estimates per dataset (particularly for ID CIFAR10 samples). This aligns with the intuition that, if the model is more uncertain about the divergence of the distributions (smaller $\delta$), it should express a larger EU. Despite smaller uncertainty values at high $\delta$’s, the difference between In-domain (ID) and OOD samples remains noticeable. This explains why a $\delta$ closer to 1 does not always lead to low-performance OOD detection and why our model’s OOD detection performance is robust against the choice of $\delta$.

The ablation study also shows that we further improve the performance of CreDEs by finding the 'best' $\delta$. One possible way is to conduct standard cross-validation on specific test scenarios. Considering the performance of CreDEs robust against $\delta$, an alternative interesting option, in the presence of multiple datasets (e.g., acquired over time in a continual learning setting), could be applying the DRO loss component to different components of the training set, and assessing the results to robustly select $\delta$. (lines 294-300).

Regarding the trade-off parameter $K$

The parameter $K$ represents a trade-off between uncertainty estimation and computational efficiency. As demonstrated in the ablation study in Appendix B.2, $K\leq10$ is a practical range for measuring epistemic uncertainty using the Generalized Hartley (GH) measure. As shown in Tables 7 and 8 and Figure 7 of Appendix B.2, the application of $K=4$ for OOD detection involving the CIFAR10 dataset (reducing the dimension from 10 to 4 classes and costing 0.02 ms per input sample) and the application of $K=10$ for the OOD detection involving the CIFAR100 datasets (reducing the dimension from 100 to 10 classes and costing 17 ms per input sample) can guarantee the outperformance of CreDEs over the DEs.

Choosing $K$ for computing the upper and lower entropy measures is more flexible. For instance, in Tables 2 and 7 and Figure 8 of the paper, $K=20$ for the ImageNet experiment dataset (reducing the original 1k to 20 classes and costing 6.1ms per sample) shows an improved OOD detection performance compared to DEs.

We appreciate your efforts to review our work and your active engagement in this discussion. We benefit from the process. We hope that the increased clarity of our responses has addressed your concerns.

Dear Reviewer,

We sincerely appreciate your high recognition of our work and valuable feedback and suggestions. Below we address your concerns.

Regarding W1

Response: We fully agree that moving the additional results of DEs$^*$-5 (standard ensemble with DRO loss) from the appendix to the main body is a better presentation. Following your suggestion, we include the MC Dropout model in Table 2 of the rebuttal file, which is claimed to be a good choice for uncertainty quantification across the board in the evaluation of [A]. The additional comparison further demonstrates the outperformance of our CreDEs. We did not report the result in the rebuttal because the Mahalanobis method is a distance-based method for uncertainty quantification and was outside the original scope, which was to address uncertainty quantification using ''second-order'' representations. However, we will properly extend the above discussion in the limitations section of the revised paper. For a more detailed response to the baseline choice, we invite you to read our global rebuttal (part 2).

Regarding W2

Response: In our experiments, we randomly selected five members from the 15 trained models to construct the ensemble. The ensemble member lists of each ensemble were strictly guaranteed to be different. Therefore, we believe that this setting could reflect the error bars of the results. Based on your comments, we will highlight the setting to improve clarity.

Regarding Q1

Response: Yes, we reported the accuracy of the models using $i_{max}$ and $i_{min}$ in Table 1 and Table 3 of the paper. They both show improvement in test accuracy.

Regarding Q2

Response: Due to the high dimensionality, it is hard to visualize and compute the size of the credal set when there are $C>3$ elements. However, we can indirectly judge whether the generated credal set resembles a Dirac credal set by calculating the maximum reachable upper bound probability of each sample, namely $\max{(q^*_{U_1}, ..., q^*_{U_C})}$. The closer the probability value is to 1, the closer it is to an almost Dirac credal set. We show the results of ResNet50-based CreDEs-5 on the CIFAR10, SVHN, and Tiny-ImageNet datasets in Figure 1 of the rebuttal file. We can conclude that our method does not always generate almost Dirac credal sets, especially on OOD samples. For CIFAR10 samples, there are a large number of quasi-Dirac credal sets (but not all), which is reasonable considering the high test accuracy of CreDEs and the low ECE (shown in Table 1 of the paper).

Regarding Q3

Response: If we first calculate the probability vector based on the midpoint $\boldsymbol{m}$ and then calculate the cross-entropy loss, we observed that we could not make the model reasonably learn to generate half-length $\boldsymbol{h}$ to generate the probability interval from the training process, as the $\boldsymbol{h}$ is not incorporated in the training. In another case, we could not calculate the midpoint of the generated probability intervals as $\boldsymbol{q}_U-\boldsymbol{q}_L$ for cross-entropy calculation, it is not normalized. In Appendix E.1, we discuss another sensible way of generalizing cross-entropy for lower/upper probability as our future exploration.

Regarding Q4

Response: No, the inference cost in Table 6 does not include the uncertainty calculation cost. We reported the cost of calculating the generalized Hartley (GH) measure and the upper entropy in Figures 7 and 8 of the paper. For instance, the time cost for GH calculation for CIFAR10 without approximation is 17 ms (0.02 ms in the reduced case considering 4 out of 10 classes) while calculating EU in deep ensembles for CIFAR10 takes $4.1\times 10^{-4}$ ms, measured on the same single CPU. Though higher CreDEs remain practical without actual computational constraints. Besides, the numbers reported are without code efficiency optimization: a more efficient code implementation could significantly reduce the cost.

Regarding the training complexity, the CreNet uses a custom training loop, unlike the TensorFlow-standardized training standard neural networks, precluding a fair comparison. Given the evidence that CreNets only marginally increases the inference time (single forward pass) in Table 6, we are optimistic that by standardizing and optimizing the custom training loop and adopting a more efficient code implementation of Algorithm 1, the training effort could be significantly reduced.

Regarding Q5

Response: Thank you for pointing the typo out. We will revise it accordingly.

[A] Benchmarking uncertainty disentanglement: Specialized uncertainties for specialized tasks. 2024

Dear Reviewer,

We appreciate your efforts in reviewing our work and your recognition of our novelty. Below we address your concerns.

Response to W1: We believe that we have provided a comprehensive evaluation of our approach, which consistently shows significantly improved performance, uncertainty quality, OOD detection, and model calibration in several popular and important setups. For instance, in Table 3 of our paper, we can observe the OOD detection improvements of about 5% and 11% for CIFAR 10 vs SVHN (AUROC) on VGG16 and ViT base architectures, respectively. Such visible improvements can be consistently observed in, for example, Table 2, Figures 5, 6, and Tables 8-12 of the paper.

Response to W2 and Q2: We value the opportunity to provide a more rigorous argument of the theoretic analysis and loss design.

Epistemic uncertainty in predictions is induced by the probability distribution(s) that generated the training data may be very different from the distribution that generates a test point. It is well known that using CE as a loss leads to the maximum likelihood estimator of the model parameters, under the assumption that all training points are drawn i.i.d. from a single, unknown distribution $P$:

$$\hat{\theta} = \arg \min_\theta\text{CE} = \theta_{MLE}.$$

What does Distributionally Robust Optimization (DRO) do? It takes a cautious stance, and assumes training points are generated by one of many potential distributions $\mathcal{U}$, and minimizes the upper bound of the expected risk (Eq. (6) of the paper). In practice, this is approximated using Adversarially Reweighted Learning as in Eq. (9). Therefore, it learns $\hat{\theta} = \arg \min_{P \in \mathcal{U}} E_P (Loss)$. However, if we apply DRO to a traditional network, the network will learn a single parameter value (Eq. (9) of the paper); for a new test data point $x_t$ it will then produce a single predicted probability vector, the one produced by the model that, in the long run, minimizes the worst-case risk.

This, though, does not express at all the (epistemic) uncertainty about which of the distributions in $\mathcal{U}$ has generated the particular test datapoint $x_t$; as a result, it does not express how uncertain the prediction may be for that particular data point.

Allowing a network to output lower and upper probability vectors for each input, instead, allows this epistemic uncertainty to be explicitly modeled.

What is the best-case scenario? That in which all data (including the test point $x_t$) are indeed generated by the same distribution – in this case, $\theta_{MLE}$ is the best choice, in a maximum likelihood sense.

What is the worst case? That in which $x_t$ is generated by a distribution that is far away from the groups of distributions $P_g$ (see lines 140-141) that generated the training data. In this case, the DRO solution will still be far from perfect, but still the best one can do, given the evidence in the training data.

Therefore, if I minimize the CE versus $\boldsymbol{q}_U$ (first component of Eq. (10)), I get a credal set Cre of predictions whose upper probability is such that, for each class $i = 1,…,C$, there exists a probability vector $\boldsymbol{q}\in \mathbb{Q}$ such that $q(i)$ is the MLE prediction for that class (please remember the discussion of lines 172-175).

Similarly, if I minimize the DRO loss versus $\boldsymbol{q}_L$ (the second component of Eq. (10)), for each class $i = 1,…,C$, there exists a probability vector $\boldsymbol{q} \in \mathbb{Q}$ such that $q(i)$ is the DRO prediction for that class.

Minimizing the two components together, as in Eq. (10), thus encourages the predicted credal set to extend from the best-case to the worst-case prediction.

Response to W3 and Q1: We kindly invite you to follow the detailed response in our global rebuttal (Part 2) for the comparison baselines.

Response to Q3: Let us consider a numerical example in a three-element case. If we have three intervals such as $[1.1, 3.1]$, $[1, 3]$, and $[1, 2]$ as the inputs of the Interval SoftMax, the Interval SoftMax returns the following probability intervals as the outcomes: $[0.15612615, 0.71893494]$, $[0.12732467, 0.70053512]$, $[0.09810377, 0.43282862]$. We can observe that the Interval SoftMax also generates overlapped probability intervals to correctly reflect the overlap in the input. In extreme cases, If all the probability intervals over all classes are significantly overleaped, the resulting credal set could include the uniform probability. However, this does not prevent us from using the maximin and maximax criteria in Eq. (11) to derive a class prediction according to the conservativeness of the decision maker. Uncertainty estimation methods for the credal sets are also valid.

Response to Q4: The main difference is that getting upper/lower probabilities e.g. from an ensemble of NNs would be a post-processing procedure only in the prediction space, no training process is involved. In our CreDEs, the model learns the probability interval from data and could encourage the predicted credal set to extend from the best-case to the worst-case prediction.

Response to Q5: The Interval SoftMax Eq.(4) can ensure a valid credal set regardless of the dimension of the tasks. To reduce the computational complexity for high-dimensional classes, we performed Alg. 2 to reduce the dimension from high $C$ to $K$. Alg. 2 has a strong mathematical background in probability intervals [17] (Eq. (12) of the paper) and can rigorously guarantee a valid credal set. In addition, the ablation study in Appendix B.2 and the OOD detection cases for ImageNet in Table 1 show the effectiveness of the reduction in uncertainty quantification of credal sets.

Response to L1: Following your comments, we will duly extend the discussion on the computational complexity and comparison baselines in the revised version, as we discussed in the global rebuttal.

Dear reviewer,

We deeply appreciate your recognition of our response and your engagement in the discussion with us. In the following, we address your remaining questions.

Interpretation of the epistemic uncertainty

In supervised learning, we assume an underlying data-generating process in the form of an unknown probability distribution $P$ on the input space $\mathbb{X}$ and the target space $\mathbb{Y}$. The lack of knowledge regarding the underlying process is the source of the epistemic uncertainty [A, B, C]. In an ideal scenario, if the model could learn the exact distribution $P$ from the training data, there would be no epistemic uncertainty about a prediction given a new input. Nevertheless, in practice, given that the training dataset represents only a subset of the full space and that the test data may differ from the training data, it is not possible to guarantee that the distribution $\hat{P}$ learned from the training data will be identical to the true distribution $P$. Therefore, the epistemic uncertainty is induced. In our paper, we proposed CreDEs that output lower and upper probability vectors for each input to express this epistemic uncertainty in predictions.

OOD data are significantly different from the training data, but not limited to the OOD data. For example, active learning aims to efficiently train models using minimal data by selectively acquiring additional samples, which are then labeled by experts. In this context, a larger EU indicates that such samples are significantly different from those used in the previous active learning iteration. Using the effective epistemic uncertainty (EU) measure as the acquisition function to obtain samples with a higher EU allows for an efficient active learning procedure [D].

The motivation of our paper is to improve the uncertainty estimation using Credal Deep Ensembles, not limited to the OOD case. We chose OOD detection benchmarks as the main evaluation because they are widely used to assess the quality of epistemic uncertainty estimation. We also performed an active learning case study in Appendix B.5 of the paper, which also shows that our CreDEs provide improved uncertainty quantification.

If we consider only the in-domain samples, effective uncertainty estimation can also improve classification performance through a rejection operation [C, E]. For example, when processing a batch of input data, those with higher epistemic uncertainty are first rejected, and then the accuracy of the remaining test samples is calculated. If the estimation of the prediction uncertainty is valid, the accuracy shows a monotonic increase. To illustrate, we show the results of the CreDEs and DEs on the CIFAR100 dataset in Table 1.

The result shows that our CreDEs provide valid EU estimates and better accuracy than DEs.

Table 1. Test accuracy on the CIFAR100 dataset using EU estimates as rejection metrics under different rejection rates.

[A] A review of uncertainty quantification in deep learning: Techniques, applications and challenges. Information Fusion, 2021

[B] Aleatoric and epistemic uncertainty in machine learning: An introduction to concepts and methods. Machine Learning, 2021

[C] Quantification of credal uncertainty in machine learning: A critical analysis and empirical comparison, UAI, 2022.

[D] Deep deterministic uncertainty: A new simple baseline. CVPR 2023.

[E] Benchmarking common uncertainty estimation methods with histopathological images under domain shift and label noise." Medical image analysis, 2023.

Ablation study on comparing our CreDEs and the post-processing approach

Thank you for your enlightenment. Inspired by your comments, we did an ablation study to compare the methods. Regarding the post-processing approach, we compute the upper and the lower bound per class $k$ from the five single predictions from DEs-5, as follows:

$$q_{U_k} = \max_{n=1,..,5}{q_{n,k}}; p_{L_k} = \min_{n=1,..,5}{q_{n,k}}.$$

Then, we formulated a credal set by post-processing for uncertainty representation. Table 2 reports the performance comparison on CIFAR10 vs SVHN and Tiny-ImageNet benchmarks.

Table 2: OOD performance comparison on CIFAR10 vs SVHN and Tiny-ImageNet benchmarks.

From Table 2, we can observe that the post-processing approach can improve the performance of classical DEs. Our method still performs better than others. In addition, due to the post-processing nature (representing the DEs predictions in credal sets), our CreDEs also achieve better performance in test accuracy and ECE values. For instance, test accuracy: 93.75% ($\delta=0.5$), 93.87% ($\delta=0.875$) vs 93.32% (DEs-5) and ECE: 0.0092 ($\delta=0.5$), 0.0093 ($\delta=0.875$) vs 0.0131.

We deeply thank your comments and inspiration, we believe incorporating more extensive comparisons, especially in real-world applications such as medical image analysis could strengthen our future work.

Following questions about the loss Eq. (10)

For a training batch, our CreNet first forward propagates via the Interval SoftMax activation (Eq. (4) of the paper) to generate meaningful upper and lower probability vectors to define the credal sets (lines 105-166). Because of the functional structure of Interval SoftMax activation, the upper and lower probability vectors are not computed independently but are correlated.

In backward propagation, we apply the upper and lower probability vectors to the vanilla and DRO components, respectively. Following the rationale of the design choices (as we explained in lines 157-166 of the paper and the previous responses), the learned prediction interval can represent the boundary cases of the credal set by considering the optimistic scenario (Vanilla component) and the pessimistic case (DRO component), respectively. Namely, assuming that the true class of the label is index $j^*$, the Vanilla and DRO components in Eq. (10) minimize the Cross-Entropy loss $-log_2q_U(j*)$ and $-log_2q_L(j^*)$, respectively, in a mini-batch optimization process (lines 167-171). Thanks to Interval SoftMax, the trained probability intervals are valid for generating credal sets.

In addition, because of the correlated computation between the upper and lower probability vectors in Interval SoftMax, minimizing the DRO component also affects the upper probability, driving the solution away from the all-one upper probability vectors.

Once again, Thank you very much for your effort and your engaging in this discussion! We hope the increased clarity in our responses has addressed your concerns.

Dear Reviewer,

We express our sincere gratitude for acknowledging our work and your valuable feedback. In the following, we address your concerns.

Regarding W4 and Q3: A more detailed explanation of the source of the improvements

Response: We would like to make a clearer explanation below.

Epistemic uncertainty in predictions is induced by the probability distribution(s) that generated the training data may be very different from the distribution that generates a test point. It is well known that using CE as a loss leads to the maximum likelihood estimator of the model parameters, under the assumption that all training points are drawn i.i.d. from a single, unknown distribution $P$:

$$\hat{\theta} = \arg \min_\theta\text{CE} = \theta_{MLE},$$

where $\hat{\theta}$ denotes the learned parameters.

What does Distributionally Robust Optimization (DRO) do? It takes a cautious stance, and assumes training points are generated by one of many potential distributions $\mathcal{U}$, and minimizes the upper bound of the expected risk (Eq. (6) of the paper). In practice, this is approximated using Adversarially Reweighted Learning as in Eq. (9). Therefore, it learns using the following equation: $\hat{\theta} = \arg \min_{P \in \mathcal{U}} E_P (Loss)$

However, if we apply DRO to a traditional network, the network will learn a single parameter value (Eq. (9) of the paper); for a new test data point $x_t$ it will then produce a single predicted probability vector, the one produced by the model that, in the long run, minimizes the worst-case risk.

This, though, does not express at all the (epistemic) uncertainty about which of the distributions in $\mathcal{U}$ has generated the particular test datapoint $x_t$; as a result, it does not express how uncertain the prediction may be for that particular data point.

Allowing a network to output lower and upper probability vectors for each input, instead, allows this epistemic uncertainty to be explicitly modelled.

What is the best-case scenario? That in which all data (including the test point $x_t$) are indeed generated by the same distribution – in this case, $\theta_{MLE}$ is the best choice, in a maximum likelihood sense.

What is the worst case? That in which $x_t$ is generated by a distribution that is far away from the groups of distributions $P_g$ (see lines 140-141) that generated the training data. In this case, the DRO solution will still be far from perfect, but still the best one can do, given the evidence in the training data.

Therefore, if I minimize the CE versus $\boldsymbol{q}_U$ (first component of Eq. (10)), I get a credal set Cre of predictions whose upper probability is such that, for each class $i = 1,…,C$, there exists a probability vector $\boldsymbol{q}\in \mathbb{Q}$ such that $q(i)$ is the MLE prediction for that class (please remember the discussion of lines 172-175).

Similarly, if I minimize the DRO loss versus $\boldsymbol{q}_L$ (the second component of Eq. (10)), for each class $i = 1,…,C$, there exists a probability vector $\boldsymbol{q} \in \mathbb{Q}$ such that $q(i)$ is the DRO prediction for that class.

Minimizing the two components together, as in Eq. (10), thus encourages the predicted credal set to extend from the best-case to the worst-case prediction.

Regarding Q1: any strategies to mitigate these complexities ($\delta$ parameters for training or $K$ parameter for the optimized inference)

Response: In our main evaluation, we set by default $\delta=0.5$ to reflect a balanced assessment of the train-test divergence and show how such a value allows our model to outperform the baselines. Most importantly, the ablation study on $\delta$ (Lines 287-312) demonstrates that our CreDEs are not particularly sensitive to $\delta$.

One possible way to find the best $\delta$ in practice is to conduct standard cross-validation on specific test scenarios. Perspectively, an interesting option, in the presence of multiple datasets (e.g.acquired over time in a continual learning setting), could be applying the DRO loss component to different components of the training set, and assessing the results to robustly select $\delta$.

$K$ is a trade-off parameter between uncertainty estimation and computational efficiency. The ablation study in Appendix B.2 shows that increasing the value of $K$ improves the OOD detection performance, but leads to an increase in execution time. Applying $K<=10$ to the Generalized Hartly (GH) measure for EU estimation is more practical. Choosing $K$ for upper and lower entropy measures is more flexible. For instance, in Table 2 of the paper, $K=20$ for the ImageNet dataset (original 1k classes) shows an improved OOD detection performance compared to DEs.

Regarding concerns about the complexity (W1 and W3) and the comparison baselines (W2 and Q2)

Response: We kindly invite you to follow the detailed response in our global rebuttal part 1 and part 2, respectively.

Regarding W5: Expecting evidence in other domains

Response: We fully agree with your comments that showing more evidence of performance improvement of our CreDEs in other domains can further strengthen our work. Given the consistent performance improvements of CreDEs on a wide range of image classification tasks (which you acknowledged as our strength), we are optimistic about successfully extending our work to other domains. We will include your suggestion in the discussion of our future work and find its proper place within the expanded scope of that work.