PaperlayerView exact revision on arXiv

Source-linked AI summary

Structural Limits of the Information-Theoretic Uncertainty Decomposition

Jakob Lønborg Christensen, Christian F. Baumgartner, Morten Rieger Hannemose, Anders Bjorholm Dahl, Vedrana Andersen Dahl

arXiv:2609.39591v2cs.CV

TL;DR

The standard information-theoretic decomposition faces reported AU–EU entanglement and epistemic collapse. This paper analyzes attainable uncertainty pairs for finite posterior samples and finds structural boundaries that depend on sample and class counts. Those boundaries help explain uncertainty patterns and motivate caution when interpreting AU and EU in low-AU regimes.

  • Problem

    Reported AU–EU entanglement and decreases in EU as model capacity grows raise questions about whether the standard decomposition isolates its intended uncertainty sources.

    Source: §1 · p. 2

  • Method

    The paper analyzes the functional dependence and attainable AU–EU region for finite posterior samples, using theory, simulations, and experiments.

    Source: §1 · p. 2, §6 · p. 10

  • Results

    The infeasible region depends on posterior sample count N and class count C, with an AU ≤ log(2)/N bound; its boundary helps explain epistemic collapse.

    Source: §1 · p. 2, §5 · p. 9

  • Takeaways & Limitations

    In low-AU regimes, finite-sample constraints mean AU and EU cannot be interpreted as independently varying quantities.

    Source: §5 · p. 8

    The analysis is limited to the information-theoretic decomposition and does not provide a way to eliminate the boundary effect at fixed posterior sample count.

    Source: §5 · p. 9

Abstract

from arXiv · show

Uncertainty estimation in machine learning typically decomposes uncertainty into aleatoric uncertainty (AU) and epistemic uncertainty (EU) using the standard information-theoretic framework. However, in practice, two critical issues arise: entanglement (AU and EU are highly correlated) and epistemic collapse (EU magnitude shrinks with increasing model capacity). We analyze this framework on a functional level and discover that significant portions of the assumed AU, EU range are infeasible in finite settings, and cannot be attained with any class probabilities. We characterize how this infeasible region scales with the number of classes and Monte Carlo samples $N$ (e.g., from ensembles with $N$ members), revealing it is bounded by $\text{AU} \leq \log(2)/N$. Crucially, the infeasible region's boundary helps explain epistemic collapse: when model confidence is high, $\text{AU} > \text{EU}$ is guaranteed by this fundamental structural limitation. Our findings show that increasing ensemble size mitigates epistemic collapse by reducing the infeasible area. Lastly, we caution against interpreting AU and EU as independent quantities in low AU regimes, since we show they are coupled when $\text{AU} \leq \log(2)/N$.

1 INTRODUCTION

The introduction frames entanglement and epistemic collapse as concerns for the standard AU–EU decomposition. It presents finite-sample structural limits as a way to analyze these issues.

1 INTRODUCTION

  • Prior work reports strong AU–EU correlation and questions whether the measures consistently isolate their intended uncertainty sources.The introduction also describes epistemic uncertainty collapse: EU can decrease as model capacity grows despite persistent training-data limitations or distribution shift.Source: §1 · pp. 1–2
  • The paper motivates its analysis by showing that finite-sample AU–EU pairs follow a parametric form and that some regions are infeasible.The plotted examples include simulated probabilities and predictions from a trained ensemble.Source: §1 · p. 2
  • The attainable (AU, TU) region has sample-count- and class-count-dependent infeasible areas, including a bound of AU ≤ log(2)/N.The paper supports its theoretical analysis with simulations and trained ensembles and Monte Carlo dropout models on CIFAR-10 and MNIST.Source: §1 · p. 2
  • The infeasible boundary helps explain why the epistemic-uncertainty fraction shrinks for larger, more confident models.The paper relates both the growth of the epistemic fraction with N and its shrinkage in more confident models to the first boundary curve.Source: §1 · p. 2

2 BACKGROUND

The background defines the finite-sample information-theoretic decomposition and reviews concerns about how AU and EU behave in practice. It also positions the paper alongside work on entropy-bounded information maximization.

2.1 INFORMATION-THEORETIC DECOMPOSITION

  • For N equally weighted posterior samples, AU is mean prediction entropy, TU is entropy of the mean prediction, and EU is TU minus AU.EU is also the mutual information between the uniformly sampled posterior-sample index and predicted class; these quantities depend on the predictions, not their generation method.Source: §2.1 · pp. 2–3

2.2 RELATED WORK

  • Prior studies report AU–EU correlations, limits to disentanglement across settings, and epistemic collapse as model size increases.The cited work also discusses task-specific evaluation, annotator disagreement, and implicit ensembling as a possible mechanism for reduced disagreement.Source: §2.2 · p. 3
  • Ebrahimi et al. (2024) study mutual-information maximization under an output-entropy constraint, while this paper analyzes the attainable region for equally weighted finite posterior samples.Their global optimality conjecture remains conjectural; this paper examines dependence on sample count and class count and links the geometry to epistemic collapse.Source: §2.2 · p. 3

3 THEORY

The theory characterizes which finite-sample (AU, TU) pairs are attainable and describes the infeasible region through a conjectured boundary construction. It also establishes that boundary’s AU extent is at most log(2)/N and proves the boundary conjecture for two classes, while leaving the general case open.

3 THEORY

  • When AU = 0, attainable TU values come from integer partitions of N into at most C parts; other zero-AU pairs are infeasible.These partitions correspond to deterministic posterior-sample assignments and their mean predictions.Source: §3 · p. 4, Figure 3 · p. 4
  • The set D contains configurations with N − 1 deterministic posterior samples and at most one sample on an edge between simplex vertices.Under the conjecture, its image supplies the lower envelope defining the infeasible boundary.Source: §3 · p. 4
  • For C = 2, numerical and empirical evidence supports the boundary conjecture, and the authors prove it; the general case remains open.The proof shows that, at fixed TU, configurations with at least two non-deterministic predictions are suboptimal for minimizing AU.Source: §3 · p. 5
  • For fixed TU, transferring probability mass between two non-deterministic predictions preserves the mean prediction and TU while decreasing their summed entropy.This perturbation supports the C = 2 proof that an AU-minimizing configuration lies in D.Source: §3 · p. 5
  • The infeasible region lies below a boundary formed, under the conjecture, by the lower envelope of curves interpolating between zero-AU points.The region contains pairs unattainable with finitely many posterior samples but attainable in the infinite-sample limit.Source: §3 · pp. 4–5
  • AU along simplex edges is bounded by log(2)/N, so the boundary’s extent in the AU direction decreases as N increases.The AU maximum occurs at s = 1/2.Source: §3 · pp. 5–6
  • The first bubble is the boundary curve connecting [N, 0, 0, . . .] and [N − 1, 1, 0, . . .], with the lowest TU values and unique interior TU values among those curves.It is the smallest-TU transition from a fully deterministic configuration to another deterministic configuration.Source: §3 · p. 6

4 EXPERIMENTS

Experiments vary posterior-sample count and network capacity to test theoretical constraints and their relation to epistemic collapse. Increasing sample count improves OOD detection for both AU and EU, while higher-capacity models show lower AU and EU alongside improved accuracy and OOD detection.

4.1 EXPERIMENTAL SETUP

  • The experiments systematically vary posterior-sample count and model capacity, evaluating OOD detection with the final three classes of each dataset as OOD.Evaluation uses CIFAR-10 and MNIST with EfficientNet models, deep ensembles or MC dropout, and validation data.Source: §4.1 · pp. 6–7

4.2 ADHERENCE TO THE THEORETICAL PREDICTIONS

  • Predictions from trained models respect the predicted infeasible boundary in the (AU, EU) plane, with many examples near the origin.The adherence test is illustrated with an ensemble of 10 EfficientNet-B0 networks trained on CIFAR-10.Source: §4.2 · p. 7, Figure 7 · p. 7

4.3 VARYING THE POSTERIOR SAMPLE COUNT

  • Increasing N improves OOD detection performance for both AU and EU, and the relative EU fraction increases, though entanglement shows no clear trend.The reported Figure 8 estimates are jackknife-corrected because the plug-in EU estimate is downward biased for independent predictions from a fixed network distribution.Source: §4.3 · p. 7, Figure 8 · p. 8
  • As N increases, fewer examples concentrate near the origin, and the infeasible boundary’s extent in the AU direction decreases.Figure 9 shows the boundary extent is constrained by AU ≤ log(2)/N.Source: §4.3 · p. 7, Figure 9 · p. 8

4.4 VARYING THE NETWORK CAPACITY

  • As capacity increases, AU and EU decrease while accuracy and OOD detection improve; highly confident predictions approach the origin, where the boundary ensures AU > EU.The relative EU fraction reaches approximately 20% for the largest models.Source: §4.4 · p. 8

5 DISCUSSION

Finite-sample predictions cannot realize every AU–EU combination: in low-AU regimes, the boundary restricts attainable pairs to AU > EU. This geometry contributes to epistemic collapse as model capacity grows, while larger posterior sample counts can lessen that limitation at additional computational cost.

5 DISCUSSION

  • In low-AU regimes, the infeasible boundary restricts attainable uncertainty pairs to AU > EU, so AU and EU cannot vary independently.This is a structural constraint of the information-theoretic decomposition for finite posterior samples.Source: §5 · p. 8
  • As model capacity grows, confident predictions accumulate near the first infeasible boundary, which contributes to—but is not the only cause of—the decreasing EU fraction.The boundary geometry restricts the relative sizes of AU and EU near that region.Source: §5 · pp. 8–9
  • Increasing posterior sample count can lessen this geometric limitation and reduce epistemic collapse, but incurs additional computational cost.The derived bound is AU ≤ log(2)/N.Source: §5 · p. 9
  • The analysis covers the information-theoretic decomposition, does not generalize to frameworks such as EPKL, and offers no way to eliminate the boundary effect at fixed sample count.Alternative frameworks and practical mitigation strategies are left for future work.Source: §5 · p. 9

6 CONCLUSION

The paper characterizes finite-sample structural limits on the information-theoretic decomposition and examines how they relate to observed uncertainty patterns. Its findings motivate interpreting estimates in light of these constraints, while extensions beyond the binary-class proof and practical mitigation remain open.

6 CONCLUSION

  • The attainable AU–EU region for finite posterior samples has an infeasible boundary that depends on posterior sample count and class count.The paper characterizes these structural limits on the information-theoretic decomposition.Source: §6 · p. 10
  • Simulations and experiments on CIFAR-10 and MNIST show how this geometry relates to uncertainty patterns, including changes in epistemic uncertainty with model capacity.Source: §6 · p. 10
  • The findings motivate interpreting information-theoretic estimates in light of finite-sample constraints; extending the analysis beyond the binary-class proof and developing practical mitigation remain open directions.Source: §6 · p. 10

A LOGIT SIMULATION

The logit simulation introduces dependence across posterior samples and classes through correlation parameters. Its resulting probabilities illustrate how uncertainty regions change across settings, including a visualization with five classes and five samples.

A LOGIT SIMULATION

  • Gaussian noise is added to logits, with dependence across posterior samples and classes controlled by correlation parameters ρN and ρC.Source: A LOGIT SIMULATION · p. 13
  • The simulated probabilities illustrate how the attainable uncertainty region changes with the simulation settings.Source: A LOGIT SIMULATION · p. 13
  • Algorithm 1 converts logits to probabilities with a softmax over axis 2.Source: A LOGIT SIMULATION · p. 13
  • Figure 12 visualizes simulated AU and TU values for C = N = 5.Source: Figure 12 · p. 13

B FINITE-SAMPLE BIAS AND JACKKNIFE CORRECTION

The finite-sample EU estimator is downward biased because entropy is concave, and its expected value increases with sample count toward population EU. The appendix uses a jackknife correction, which reduces bias in experiments but remains approximate and has limitations.

B FINITE-SAMPLE BIAS AND JACKKNIFE CORRECTION

  • The appendix considers independent posterior samples from the same predictive distribution for a fixed input.Source: B FINITE-SAMPLE BIAS AND JACKKNIFE… · p. 13
  • Entropy concavity makes the entropy of the sample-mean prediction, and consequently the usual EU estimate, downward biased.The expected EU estimate is nondecreasing with N and converges to population EU, so it can increase with N even when the underlying predictive distribution is fixed.Source: B FINITE-SAMPLE BIAS AND JACKKNIFE… · p. 13
  • The jackknife correction is used because KL divergence lacks a universal multiplicative correction of the form N/(N − 1).Source: B FINITE-SAMPLE BIAS AND JACKKNIFE… · p. 13
  • The jackknife correction cancels the leading 1/N bias term when the estimator has a smooth expansion in powers of 1/N.For EU, the leave-one-out AU terms equal the full-sample AU estimate, so the correction acts only on TU.Source: B FINITE-SAMPLE BIAS AND JACKKNIFE… · p. 14
  • The correction is approximate, may retain bias at small sample counts near the simplex boundary, and does not guarantee an unbiased EU/TU ratio.Correcting TU and EU separately does not ensure that their ratio is unbiased; corrected estimates target population rather than finite-sample uncertainty, so the main-text bounds do not directly apply.Source: B FINITE-SAMPLE BIAS AND JACKKNIFE… · p. 14
  • After jackknife correction, mean EU curves are mostly flat across sample counts, but EU still increases with N when N < C.The maximum achievable EU is limited to log(min(C, N)); Figure 13 compares corrected estimates across N on CIFAR.Source: B FINITE-SAMPLE BIAS AND JACKKNIFE… · p. 14, Figure 13 · p. 14

C BOUNDARY CURVE MINIMUM LEMMA

For any boundary curve (AU(s), TU(s)), TU reaches its minimum at an endpoint. This follows because TU is concave in s.

C BOUNDARY CURVE MINIMUM LEMMA

Related papers

Loading 2609.39591v2…