Estimating Epistemic and Aleatoric Uncertainty with a Single Model

Thank you for your thoughtful feedback. We have summarized and responded to your questions and concerns below. Please let us know if you have any additional questions or comments and we will do our best to respond promptly.

1. Missing citation where diffusion models are used to estimate epistemic uncertainty.

Thank you for bringing this to our attention. We weren’t able to include a citation to DECU [C] in our original submission since it was published after the NeurIPS submission deadline. However, we will certainly update our paper and include a reference and discussion in Section 2.3.

2. Modern methods ensemble only certain parts of the network, so training ensembles is not necessarily as expensive as stated.

We agree with the reviewer; this statement should be further clarified. We have attempted to make this point more apparent by adding references to relevant ensembling papers [A, B] in Section 2 and updating our statements in Section 5 to state that our method achieves a reduction in training cost up to $M\times$ that of deep ensembling methods, depending on the number of ensembled parameters.

3. How does scaling of the BHN compare to MC Dropout? Both have issues with scalability.

In our experiments, we chose to set the BHN just large enough that it produces reasonable weights for the diffusion model. In doing so, we avoid adding a significant increase in the number of trainable parameters. Our BHN incurred only a moderate 10% overhead compared to a baseline diffusion model; from 44.1 min to 48.5 min. In comparison, we observed a slightly smaller training overhead of 6.64% with MC-Dropout; from 44.1 min to 47.0 min.

4. The subscript on $x$ is overloaded. It refers to both the number of denoising steps and the number of samples in the training set.

Thank you for pointing this out. We have clarified this notation in the main text by denoting the denoising step $t$ with a superscript and the dataset index $i$ with a subscript. Specifically, a sample $i$ from the dataset at denoising step $t$ is now expressed as $x_i^{(t)}$.

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References

[A] Osband, Ian, et al. "Deep exploration via bootstrapped DQN." Advances in neural information processing systems 29 (2016).

[B] Berry, Lucas, and David Meger. "Normalizing flow ensembles for rich aleatoric and epistemic uncertainty modeling." Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 37. No. 6. 2023.

[C] Berry, Lucas, Axel Brando, and David Meger. "Shedding Light on Large Generative Networks: Estimating Epistemic Uncertainty in Diffusion Models." The 40th Conference on Uncertainty in Artificial Intelligence.

Thank you for your thoughtful feedback. We have summarized and responded to your questions and concerns below. Please let us know if you have any additional questions or comments and we will do our best to respond promptly.

1. The paper doesn’t have a dedicated contributions section.

Thank you for the suggestion. We have described the technical novelty and specific contributions of our paper in bullet #2 of our global response.

2. The paper lacks a straightforward comparison of computational overhead of baselines and the proposed method during training and evaluation.

We have added a table in our global response that shows the computational runtime required by all methods for training and evaluation. This result will be added to the main paper.

3. In Equation 1, the marginal distribution on the left side of the equation should provide a better estimate of the aleatoric component, given that it's marginalized over model weights distribution. This can be further confirmed by the way the authors approach estimating aleatoric uncertainty in practice, described in Equation 10.

We acknowledge that future work should explore a better estimate of aleatoric uncertainty in Equation 1 and will mention this in the conclusion. However, in the main paper, we derive Equation 1 from Ekmekci et al. [9], which is supported by numerous works [24, A, B], in which the predictive distribution contains both an aleatoric and epistemic component. Likewise, our derivation for Equation 10 stems from prior works [49, C, D] which apply the law of total variance to disentangle epistemic and aleatoric uncertainties.

4. In Appendix A, authors describe training of BHN as: compute an L2 loss, and back-propagate the gradients to update the hyper-network weights. But in the next paragraph the following is said: "The BHN is trained with the same L2 loss and an additional Kullback-Leibler (KL) divergence loss."

We apologize for the typo in Appendix A, which has been corrected in the text: The Bayesian neural network (BNN), not Bayesian hyper-network (BHN), is trained with an additional KL divergence loss.

5. How is BHN training performed? How often is the noise vector changed during training (e.g., per-input, per-batch, etc.)? Why did the BHN not collapse to a small optimal mode?

The BHN is trained only with an L2 loss. During training, the noise vector is changed per-batch, and for the majority of our experiments we set the batch size to 32. The BHN doesn’t collapse to a mode because there are a variety of network weights that yield plausible predictions with respect to L2 distance.

6. The toy experiment should be done by independently varying noise level and training dataset size and plotting results on AU and EU axes.

This is a great suggestion. We have provided these figures in the general response PDF, which show that our method performs similarly to deep ensembles. We note, however, both methods struggle to fully disentangle EU and AU when one type of uncertainty is very high. In Figure R1, both DPS-UQ and HyperDDPM estimate a similar AU across different dataset sizes, indicating that the AU estimate is relatively unaffected by the increasing EU of the problem. Similarly, in Figure R2, both methods estimate a similar EU across different noise levels, except in the extreme noise scenario. Prior work [E] suggests that one should disregard the AU estimate when EU is high (i.e., the measurement is out-of-distribution). We will incorporate these figures and corresponding descriptions into the main text.

7. The paper would benefit from a more in-depth analysis on real world problems. For example, how can EU help improve reconstruction quality (e.g., by filtering out low-confidence outputs / rejection verification)?

We will update the main text to provide more concrete examples of use cases for EU and AU estimation (e.g., active learning for weather station placement, predicting long-tailed events with high EU, and cost mitigation strategies). We note that our current experimental results demonstrate that our approach can be used for rejection verification. For instance, in both real-world experiments, we provide in-distribution and out-of-distribution measurements to the model, and we are able to identify which parts of the prediction are unreliable and should be investigated more thoroughly by a trained expert.

8. Were the drop-out masks kept consistent during training and evaluation of the MC Dropout baseline? Any differences could explain the performance drop.

We generate random activation masks with the same drop-out probability $p=0.3$ for both training and evaluation. During training, the activation masks are randomly generated per-batch. At evaluation time, we use the same procedure to randomly generate $M=10$ masks and compute AU and EU according to Equations 10 and 11. We will clarify this procedure in the revised paper.

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References

[A] Fellaji, Mohammed, and Frédéric Pennerath. "The Epistemic Uncertainty Hole: an issue of Bayesian Neural Networks." arXiv preprint arXiv:2407.01985 (2024).

[B] Kwon, Yongchan, et al. "Uncertainty quantification using Bayesian neural networks in classification: Application to biomedical image segmentation." Computational Statistics & Data Analysis 142 (2020): 106816.

[C] Schreck, John S., et al. "Evidential deep learning: Enhancing predictive uncertainty estimation for earth system science applications." arXiv preprint arXiv:2309.13207 (2023).

[D] Joshi, Shalmali, Sonali Parbhoo, and Finale Doshi-Velez. "Pre-emptive learning-to-defer for sequential medical decision-making under uncertainty." arXiv preprint arXiv:2109.06312 (2021).

[E] Mukhoti, Jishnu, et al. "Deep deterministic uncertainty: A new simple baseline." Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2023.

Thank you for your thoughtful feedback. We have summarized and responded to your questions and concerns below. Please let us know if you have any additional questions or comments and we will do our best to respond promptly.

1. The paragraph starting at L29 is misleading and vague. A useful uncertainty estimate does not necessarily have to differentiate between aleatoric and epistemic uncertainty. Also, consider providing concrete use cases for the separate aleatoric and epistemic estimates and how they can improve the model's performance.

This statement was indeed too strong. We have changed “To be useful…” to “To be most useful…” and added the following motivating text to the paper: “In applications such as weather forecasting, epistemic uncertainty can be used to inform the optimal placement of new weather stations. Additionally, in medical imaging, decomposition of uncertainty into its aleatoric and epistemic components is important for identifying out-of-distribution measurements where model predictions should be verified by trained experts.”

2. More complex models introduce more degrees of freedom into the learning problem, leading to a larger set of models consistent with the training data and an increase, not decrease, of epistemic uncertainty.

We agree with the reviewer’s statement and will remove the referenced statement from the main text.

3. The claim that the “estimate converges to the true aleatoric AU of the inverse problem” is imprecise. This is only true under the assumption that (i) the score estimator is expressive enough to model the true score of the generative process perfectly, (ii) is trained on the entire data distribution, and (iii) the global minimum is reached during optimization.

Thank you for pointing this out. We will specify the exact assumptions under which our statement is true, as listed by the reviewer.

4. Why should we trust AU estimates when EU is high? Prior work suggests that we should only consider AU when EU is below a threshold.

Our results illustrate that our single-model ensembling approach produces similar estimates of AU and EU as multi-model deep ensembling approaches. It's true$\textemdash$and important to acknowledge$\textemdash$that AU estimates are only meaningful when there's sufficient data that EU estimates are low. We will highlight this point in the revision.

5. The precise formulation of Eq. (7) would be $\mathbb{E}\_{(x, y) \sim \mathcal{D}}\left[\mathbb{E}\_{z \sim \mathcal{N}(0, \sigma^2)}\left[\mathcal{L}(f(y \mid \phi(z)), x)\right]\right]$. The sampling is not captured by the original formula.

Thank you for the correction. As suggested, we have updated the formula in Equation 7 to more accurately denote the sampling of noise vectors $z\sim\mathcal{N}(0,\sigma^2)$.

6. Move Appendix C into the main paper for the final version.

Thanks for the suggestion. We have integrated Appendix C into the revised paper.

7. Make figures vector-graphical.

Thank you for the suggestion. We have updated all figures in the revision to use vector graphics.

Thank you for your thoughtful feedback. We have summarized and responded to your questions and concerns below. Please let us know if you have any additional questions or comments and we will do our best to respond promptly.

1. Provide error bars / standard deviations.

We have trained five sets of ensembles and computed the variance across the EU and AU estimates. The uncertainty estimates are quite consistent, but due to space constraints we are not able to fit these figures in our response. We will include them in the revised text. We have also provided RMSE errors and variances (i.e., total uncertainty) for our experimental results in Figures 8 and 9 of the supplement.

2. Provide a summary figure comparing the computational overhead of baselines and the proposed method during training and evaluation.

We have added a figure in our global response that shows the computational runtime required by all methods for training and evaluation.

3. The technical novelty of the paper is not described precisely. The abstract should highlight the application of existing methods in a new setting.

Thank you for the suggestion. We have described the technical novelty and specific contributions of our paper in Question 2 of the global response.

4. What's the training overhead for the hyper-network? How should one set the complexity of the hyper network? If the network is too large / small, would the uncertainty vanish?

We chose to set the hyper-network just large enough that it produces reasonable weights for the diffusion model. In doing so, we avoid adding a significant increase in the number of trainable parameters. Specifically, this amounted to around a 10.05% increase in training time (i.e., from 44.1 min to 48.5 min) for our weather forecasting and CT reconstruction experiments. If the hyper-network architecture is too small, we found that it outputs poor quality weights for the diffusion model. This text will be incorporated into the revised paper.

5. How does the number of samples, i.e. M and N, affect performance?

In general, we found that under-sampling M (i.e., the number of samples from the implicit weight distribution) leads to uncertainty maps which underestimate uncertainty around out-of-distribution features and overestimate uncertainty around in-distribution features. As we continually sample network weights, we observe increased uncertainty in areas around the abnormal feature and suppressed uncertainty around in-distribution features. Similarly, under-sampling N (i.e., the number of samples from the likelihood) leads to irregular peaks in the predicted aleatoric uncertainty maps. As we sample more from the diffusion model and the sample mean converges, the aleatoric uncertainty becomes more uniform. Please see our ablation study in Section B.1 of the supplement for additional figures and details.

6. What does BNN mean in the main text? Does it refer to MC-dropout or DPS-UQ or both?

BNN stands for Bayesian neural network. This acronym is specified in Section 2.1, and the network model for the BNN used in our experiment is described in Section A of the supplement. We have attempted to clarify this by placing a reference in the toy experiment which points to the definition of BNNs in Section 2.1.

7. Provide stronger BNN baselines (e.g., cyclical SGLD, linearized Laplace, SWAG).

In our non-toy experiments, we did not include a BNN baseline because we wanted to focus on evaluating our technique against methods which are able to predict both AU and EU for a more direct comparison. We will include SQR [B] and Kendall et al. [A] as additional baselines in our benchmark.

8. In Figure 1, hyper-network is written as “hypernetwork” in the figure body. The term should be spelled in a consistent way.

Thank you for bringing this to our attention. We have corrected the text in Figure 1 to more consistently reflect the spelling of the term “hyper-network” throughout the paper.

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References

[A] Kendall, Alex, and Yarin Gal. "What uncertainties do we need in bayesian deep learning for computer vision?." Advances in neural information processing systems 30 (2017).

[B] Tagasovska, Natasa, and David Lopez-Paz. "Single-model uncertainties for deep learning." Advances in neural information processing systems 32 (2019).