REVIEW 3 major objections 5 minor 29 references
DeepUQ: Assessing the Aleatoric Uncertainties from two Deep Learning Methods
T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read Deep learning uncertainty estimates—Deep Ensembles and Deep Evidential Regression—scale with injected noise but are miscalibrated on toy regression tasks, with only 7 of 12 DE and 2 of 12 DER experiments meeting the paper's calibration…
desk verdict A transparent toy-data comparison of DE and DER aleatoric uncertainties, whose headline miscalibration counts rest on a non-standard tolerance criterion and no seed variability. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing mechanism is the benchmark's ability to know the true aleatoric uncertainty exactly. For output-uncertainty experiments, noise is injected directly on $y$ with known $\sigma_y$; for input-uncertainty experiments, noise is injected on $x$ and propagated through the known map $y = mx$ (0D) or $y = \sum_{i=1}^{32\times32} x_i$ (2D) via standard error propagation, giving $\sigma_y = |m|\sigma_x$ and $\sigma_y = 32\sigma_x$. The predicted aleatoric uncertainty is then compared with that known value through the distribution of $\sigma_{\rm al}$ over the test set, using the paper's criterion that the true $\sigma_y$ must fall inside the mean $\pm$ one standard deviation of that distribution. The two methods define $\sigma_{\rm al}$ differently: Deep Ensembles average the predicted variances of $K=10$ mean-variance networks, while Deep Evidential Regression uses the width of the Student-$t$ distribution implied by its normal-inverse-gamma evidential priors. Training losses are modified—$\beta$-NLL for the ensembles, and NIG with a $t$-width regularization term for evidential regression—to keep variance estimates from collapsing.
What would settle it
Recompute the paper's 24 experiments but score calibration by the empirical coverage of the predicted $\pm 1\sigma$ (or $\pm 2\sigma$) intervals on held-out data, or by the probability integral transform of the true values under each model's predictive distribution; if most experiments then look calibrated, the paper's miscalibration conclusion is an artifact of the $\mathrm{std}(\sigma_{\rm al})$ criterion rather than a property of DE and DER.
Extended reading notes
Core claim
On its own terms, the paper establishes that two popular uncertainty quantification methods do not return quantitatively trustworthy aleatoric uncertainty estimates, even on simple toy data. The central result is a calibration count: requiring the true output uncertainty $\sigma_y$ to lie within one standard deviation of the mean of the distribution of predicted $\sigma_{\rm al}$ values on the test set, only seven of twelve Deep Ensemble experiments and two of twelve Deep Evidential Regression experiments are well calibrated. Deep Ensembles pass the criterion for the 0D medium- and high-noise experiments but deviate most on the 2D input-uncertainty setup; Deep Evidential Regression overestimates uncertainty at low noise and underestimates it at medium and high noise, with only the 0D output low-noise and 2D output medium-noise runs calibrated. Both methods satisfy the weaker requirement that predicted uncertainty scales with injected noise. The authors explicitly caution that these conclusions apply only to their toy, homoskedastic Gaussian datasets, not to real-world data.
Load-bearing premise
The headline calibration counts depend on the study's definition of 'well-calibrated'—the true uncertainty lying within one standard deviation of the mean of the predicted-uncertainty distribution—rather than on a standard statistical calibration measure such as prediction-interval coverage or the probability integral transform; under a stricter criterion the 7/12 and 2/12 proportions would change.
Editorial extensions
If this is right
- If the central claim is right, DE and DER uncertainty values should not be read as physical noise levels on these tasks without a calibration check, even though the underlying regressions fit well.
- Because the predicted uncertainties scale with injected noise, the methods can still be used to rank relative noise levels across datasets.
- Deep Evidential Regression would need calibration before use in downstream physics analyses, since it fails the paper's criterion in nearly all tested configurations.
- The 2D input-uncertainty, high-noise regime is where both methods are least reliable, so applications with noisy image inputs and large noise should expect the largest errors.
- Post-facto calibration (for example, training an auxiliary isotonic model on uncertainty values) is the natural next step, as the paper itself suggests.
Reading between the lines
- Beyond the paper: because the calibration criterion uses the spread of the predicted-uncertainty distribution as its tolerance, a method that produces highly variable $\sigma_{\rm al}$ estimates can pass even when its typical estimate is far from the truth; the 7/12 and 2/12 counts are therefore not a general statement about which method is better calibrated in the standard probabilistic sense.
- Beyond the paper: the same exact-truth benchmark could be applied to heteroskedastic and non-Gaussian noise, where the Normal-likelihood assumptions inside both methods are violated and calibration failures would likely be larger.
- Beyond the paper: a direct testable extension is to score the same 24 configurations with a proper coverage-based metric—for instance, the fraction of test points inside the predicted $\pm 1\sigma$ band—to see whether the qualitative ordering (2D input high-noise worst, DER worse than DE) survives.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper compares aleatoric uncertainty estimates from Deep Ensembles (DE) and Deep Evidential Regression (DER) on two toy datasets—0D linear regression and 2D galaxy images—with noise injected on either the output or input variables and three noise levels. The authors propose three desiderata: predicted uncertainty should scale with injected noise, should be calibrated to the true uncertainty within one standard deviation of the predicted σal distribution, and should satisfy both properties across data dimensionality and injection type. They report that scaling holds across all models, but only 7 of 12 DE experiments and 2 of 12 DER experiments pass the calibration criterion, with the worst performance in the 2D input-uncertainty high-noise setting. They conclude that post-hoc calibration deserves further study and explicitly limit their claims to their toy, homoskedastic Gaussian setting.
Significance. The paper's main strength is its construction of benchmarks with known ground-truth aleatoric uncertainty, including a clear propagation of input noise to output uncertainty, and its explicit statement of limitations. The scaling result (desideratum i) is plausible and visually supported by Figure 2. The release of the DeepUQ code and notebooks is a useful community resource. However, the central calibration claim, expressed as the 7/12 and 2/12 counts, rests on a non-standard metric whose properties are problematic for homoskedastic data, and the quantitative values behind those counts are not reported. If the requested reanalysis with standard calibration diagnostics confirms the qualitative trend, the paper would provide a useful cautionary benchmarking result; without it, the headline conclusion is not yet established.
major comments (3)
- [Section 3, desideratum (ii)] The calibration criterion used to produce the headline counts—that the true uncertainty σy lies within one standard deviation of the distribution of predicted σal values over the test set—is not a standard calibration measure and is pathological for the homoskedastic data studied here. A perfectly calibrated model would predict a constant σal = σy, giving std(σal) = 0 and requiring exact equality; a model that outputs overdispersed σal values gets a larger tolerance and can pass despite substantial bias. Because the 7/12 and 2/12 counts in Section 4 are computed from this criterion, they may reflect the width of the predicted σal distribution rather than actual calibration. Please redo the analysis with a standard diagnostic such as empirical coverage of the central 68% predictive interval or a PIT-based uniformity test.
- [Section 4, Figure 2 and Tables 1–2] The quantitative basis for the 7/12 and 2/12 counts is not reported. Figure 2 shows distributions of predicted σal values, but the text does not give the per-condition values of mean σal, std(σal), or the pass/fail determination, so the counts cannot be independently checked or reproduced from the paper. A table listing these quantities for each of the 12 conditions per method would make the central claim verifiable.
- [Section 3, experimental design] Each of the 12 experimental conditions per method appears to be run with a single training initialization. Since the comparison between DE and DER rests on the discrete counts 7/12 versus 2/12, the absence of repeated seeds means the headline difference has no statistical uncertainty and could be within run-to-run variability. Please report results across multiple seeds (e.g., mean and standard deviation of the calibration metrics) or at least confirm that the qualitative miscalibration pattern is stable across seeds.
minor comments (5)
- [References] References [22] and [23] are the same paper (Seitzer et al., arXiv:2203.09168); they should be consolidated into a single citation.
- [Appendix D] Appendix D mentions several β schedules that were experimented with, but only the final choice β = 0.5 is presented in the main text; please state whether the other schedules are omitted for space or documented in the accompanying repository.
- [Abstract and Section 4] The phrase "miscalibrated to std(σal)" is awkward and unclear; suggest rewording to something like "miscalibrated relative to the true uncertainty when judged by the std(σal) criterion."
- [Section 4] The sentence "the 0D experiments are calibrated for the medium- and high-noise models" is ambiguous because it does not specify whether both output- and input-injection 0D experiments are included; please clarify which of the four 0D conditions pass.
- [Appendix F] The DeepUQ-neurIPS-WS-2024 repository is mentioned, but no URL or DOI is provided; please include an availability statement so the code can be located.
Circularity Check
No significant circularity: the ground-truth uncertainties are set externally by noise injection and analytic error propagation, and the calibration comparison is against that independent benchmark.
full rationale
The paper's load-bearing empirical claim is that predicted aleatoric uncertainties from Deep Ensembles (DE) and Deep Evidential Regression (DER) can be checked against known injected noise levels. This comparison is self-contained: the true output uncertainty sigma_y is set externally by injecting noise of known amplitude and, for the input-uncertainty experiments, by propagating that noise analytically (Appendix C gives sigma_y = |m| sigma_x for 0D and sigma_y = 32 sigma_x for 2D). These reference values do not depend on the trained models or on any fitted parameter, so the predicted sigma_al values are being compared with an independent benchmark. The hyperparameters beta = 0.5 and lambda = 0.01 are taken from prior literature ([22] and [16]) and are not fitted to the test data, so no fitted input is later renamed as a prediction. The headline counts (7/12 DE and 2/12 DER) are computed using the paper's own stated calibration criterion, namely that the true sigma_y lies within one standard deviation of the distribution of predicted sigma_al values over the test set. That criterion is non-standard and model-dependent, and it could be criticized on statistical grounds, but it is an explicitly adopted evaluation rule rather than a result smuggled in by construction. The paper does not claim to derive the criterion from first principles; it defines it in Section 1 desideratum (ii) and applies it in Section 3. The only self-citations are to related work by Nord ([6]) and to the DeepBench simulation package ([28], authored by Nevin and Nord), which is used only as a tool to generate 2D toy images; neither citation supplies a load-bearing mathematical result or forbids alternative methods. No equation in the paper reduces to itself, and no fitted parameter is presented as an independent prediction. Therefore there is no significant circularity; any concerns about the calibration metric or the lack of repeated seeds are correctness and robustness issues, not circularity.
Assumptions & free parameters
free parameters (3)
- β (DE loss exponent) =
0.5
- λ (DER regularization weight) =
0.01
- Network architecture (two hidden layers of 64 neurons, five convolutional layers)
assumptions (4)
- standard math First-order error propagation with neglected covariance terms yields the true output uncertainty for input-injected noise (Eq. C.1).
- domain assumption The aleatoric uncertainty definitions for DE (mean of ensemble variances) and DER (width of the t-distribution) are the correct operationalizations of aleatoric uncertainty.
- domain assumption Data are generated so that y is uniform in [0,2] for both 0D and 2D cases, with the stated ranges for Sérsic parameters.
- domain assumption The injected noise is exactly Gaussian and homoskedastic, so the true σy is exactly known.
Cite this review
Pith. "Pith review of DeepUQ: Assessing the Aleatoric Uncertainties from two Deep Learning Methods." pith.science (2026). https://pith.science/paper/6XMCPBZ2
@misc{pith2026241108587,
author = {Pith},
title = {Pith review of: DeepUQ: Assessing the Aleatoric Uncertainties from two Deep Learning Methods},
year = {2026},
howpublished = {\url{https://pith.science/paper/6XMCPBZ2}},
note = {Machine review of arXiv:2411.08587}
}
abstract
Assessing the quality of aleatoric uncertainty estimates from uncertainty quantification (UQ) deep learning methods is important in scientific contexts, where uncertainty is physically meaningful and important to characterize and interpret exactly. We systematically compare aleatoric uncertainty measured by two UQ techniques, Deep Ensembles (DE) and Deep Evidential Regression (DER). Our method focuses on both zero-dimensional (0D) and two-dimensional (2D) data, to explore how the UQ methods function for different data dimensionalities. We investigate uncertainty injected on the input and output variables and include a method to propagate uncertainty in the case of input uncertainty so that we can compare the predicted aleatoric uncertainty to the known values. We experiment with three levels of noise. The aleatoric uncertainty predicted across all models and experiments scales with the injected noise level. However, the predicted uncertainty is miscalibrated to $\rm{std}(\sigma_{\rm al})$ with the true uncertainty for half of the DE experiments and almost all of the DER experiments. The predicted uncertainty is the least accurate for both UQ methods for the 2D input uncertainty experiment and the high-noise level. While these results do not apply to more complex data, they highlight that further research on post-facto calibration for these methods would be beneficial, particularly for high-noise and high-dimensional settings.
Figures
Reference graph
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as a means for avoiding a commonly observed problem in MVEs, where the variance artificially enlarges resulting in a poor estimate of the mean. The β parameter helps ensure convergence of the network predictions for µ(xi) and σ2(xi): Lβ−NLL = 1 N NX i=0 σ2β(xi) 1 2 log σ2(xi) ...
2024
Reviewed August 12, 2026 · model on record in the stance chip above.
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