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Quantifying Aleatoric and Epistemic Uncertainty in Machine Learning: Are Conditional Entropy and Mutual Information Appropriate Measures?
Lisa Wimmer, Yusuf Sale, Paul Hofman, Bern Bischl, Eyke Hüllermeier
TL;DR
The paper asks whether conditional entropy and mutual information appropriately quantify aleatoric and epistemic uncertainty in machine learning. It analyzes their behavior and additive decomposition theoretically and empirically, finding incoherencies across learning settings and concerns about current uncertainty-quantification practice.
Problem
Conditional entropy and mutual information are widely used for aleatoric and epistemic uncertainty, but their suitability and additive decomposition remain questionable for statistical learning.
Method
The paper examines entropy-based uncertainty measures, their additive decomposition, and empirical behavior across computer-vision learning situations.
Results
Experiments find that total uncertainty often behaves sensibly, whereas its aleatoric and epistemic attribution can be implausible, with very low epistemic uncertainty emerging globally.
Takeaways & Limitations
The findings support critically reassessing entropy-based uncertainty measures and additive decompositions in machine-learning applications.
Takeaways & Limitations
In finite-sample settings, aleatoric estimates remain entangled with epistemic uncertainty, and the absence of ground truth prevents determining the exact effect.
Abstract
from arXiv · showhide
The quantification of aleatoric and epistemic uncertainty in terms of conditional entropy and mutual information, respectively, has recently become quite common in machine learning. While the properties of these measures, which are rooted in information theory, seem appealing at first glance, we identify various incoherencies that call their appropriateness into question. In addition to the measures themselves, we critically discuss the idea of an additive decomposition of total uncertainty into its aleatoric and epistemic constituents. Experiments across different computer vision tasks support our theoretical findings and raise concerns about current practice in uncertainty quantification.
1 INTRODUCTION
The paper distinguishes aleatoric uncertainty, arising from stochastic data-generation, from epistemic uncertainty about the gap between the true and estimated conditional distributions. It questions whether widely used entropy-based measures and additive decompositions appropriately quantify these components.
- Aleatoric uncertainty reflects irreducible randomness in the mapping from inputs to outcomes, including measurement errors and unexplained variation.
- Epistemic uncertainty concerns limited knowledge about the discrepancy between the true conditional distribution and its empirical estimate.
- Epistemic uncertainty can be reduced by collecting more training samples and can inform decisions to invoke human intervention.
- A common framework uses Shannon entropy for total uncertainty, conditional entropy for aleatoric uncertainty, and mutual information for epistemic uncertainty.
- The paper identifies incoherencies in these measures, questions additive disaggregation of total uncertainty, and provides evidence from practical applications.
- Model misspecification is recognized as an additional source of epistemic uncertainty but is often omitted because it is difficult to analyze formally.
2 BACKGROUND
The paper represents predictions with a second-order distribution over first-order class-probability distributions, then quantifies uncertainty using entropy-based quantities and finite ensemble approximations. Conditional entropy is treated as aleatoric uncertainty, while mutual information is treated as epistemic uncertainty.
- 2.1 UNCERTAINTY REPRESENTATIONS: Bayesian inference models Q as a posterior distribution, while alternative approaches can estimate epistemic uncertainty directly through a Dirichlet distribution’s concentration parameter.
- 2.1 UNCERTAINTY REPRESENTATIONS: A second-order distribution Q over first-order probability distributions represents the learner’s epistemic state, with greater concentration indicating more certainty about the true distribution.
- 2.1 UNCERTAINTY REPRESENTATIONS: For binary classification, the first-order distribution is determined by the probability vector θ over the two classes, while Q expresses uncertainty about that distribution.
- 2.2 ENTROPY-BASED INFORMATION MEASURES: The standard information-theoretic approach uses Shannon entropy as total uncertainty and decomposes it additively into conditional entropy and mutual information.
- 2.2 ENTROPY-BASED INFORMATION MEASURES: Conditional entropy measures uncertainty remaining in Y when θ is known and is therefore interpreted as aleatoric uncertainty.
- 2.2 ENTROPY-BASED INFORMATION MEASURES: Mutual information is interpreted as epistemic uncertainty because it measures the potential reduction in uncertainty about Y from observing Θ.
- 2.3 FINITE-ENSEMBLE APPROXIMATION: Because expectations over Q are generally intractable, Monte Carlo integration approximates them using M sampled predictions, corresponding in practice to ensemble learning.
- 2.3 FINITE-ENSEMBLE APPROXIMATION: Finite ensembles provide coarse approximations, and small deep ensembles can systematically undersample low-density regions, particularly affecting epistemic-uncertainty estimates.
3 RELATED WORK
Prior work widely applies entropy decomposition in Bayesian learning and related machine-learning applications. The paper argues that mathematical correctness does not guarantee that the individual quantities or their additive aggregation are suitable measures of predictive uncertainty.
- Entropy decomposition has been used for robust prediction, optimization procedures, neural-network analysis, and filtering unreliable predictions.
- The paper distinguishes the mathematical correctness of the entropy identities from the suitability of their individual quantities for evaluating predictive uncertainty.
- It argues that additive aggregation of the entropy-based components may produce unintended results and calls for scrutiny of both the measures and the decomposition.
4 CRITICAL ASSESSMENT
The paper identifies incoherencies in entropy-based total, aleatoric, and epistemic uncertainty measures and questions whether total uncertainty can be additively decomposed into reliable aleatoric and epistemic parts.
- Axiomatic assessment: The proposed axioms require uncertainty measures to be non-negative and impose behavior under concentration, spreading, and location shifts of second-order distributions.These properties include vanishing epistemic uncertainty for Dirac measures and monotonicity under mean-preserving spreads.
- Total uncertainty: Shannon entropy violates the strict spread-monotonicity property because total uncertainty depends on Q only through its expectation.Consequently, distinct second-order distributions can induce the same total uncertainty despite differing concentrations.
- Total uncertainty: Uniform U[0, 1] and a 50–50 mixture of Dirac measures at θ = 0 and θ = 1 receive the same total uncertainty, despite representing different learner knowledge states.The entropy calculation marginalizes Q, so its symmetry around θ = 1/2 drives the result while Q’s concentration is largely ignored.
- Aleatoric uncertainty: Conditional entropy is not generally a true expectation or bound for aleatoric uncertainty, because its estimate depends on the learner’s epistemic state.With epistemic uncertainty present, the estimate can be compromised, and the unknown ground truth prevents determining the direction of the error.
- Epistemic uncertainty: Mutual information violates the proposed epistemic properties: a uniform prior yields AU = 0.72 and EU = 0.28, while a deterministic Dirac mixture reaches EU = 1.00.It is also location-sensitive: U[0.3, 0.7] and U[0.45, 0.85] have equal spread but EU values of 0.04 and 0.08, respectively.
- Additive decomposition: Additive decomposition is problematic because the estimated aleatoric component depends on epistemic uncertainty and need not equal ground-truth aleatoric uncertainty.At the beginning of learning, TU − EU can at best be interpreted as a lower bound to true AU; additivity may need replacement by sub-additivity.
5 EXPERIMENTS
Experiments across image, synthetic, and tabular classification tasks show that total entropy often decreases sensibly, while its aleatoric and epistemic allocations can behave inconsistently and depend strongly on the probabilistic learner.
- Experimental setup: Experiments varied training sample size, image resolution, class overlap, label noise, class interpolation, and out-of-distribution inputs across image, synthetic, and tabular tasks.The study used CIFAR10, MNIST, synthetic rectangles and polygons, and a four-class tabular problem.
- Increasing sample size: Total uncertainty and epistemic uncertainty generally decreased as sample size increased, but aleatoric estimates also decreased in three of four cases instead of approaching a constant.The CIFAR10 results made this behavior especially visible; MNIST deep-ensemble uncertainty was initially ultra-low but showed the same trend on close inspection.
- Increasing data noise: Increasing noise raised aleatoric uncertainty in all cases, whereas epistemic uncertainty remained basically constant and very low even when predictive performance plummeted.The exception was the MNIST Laplace approximation, whose epistemic uncertainty increased at the lowest resolutions while aleatoric uncertainty remained small and fell from 10% to 5%.
- Test distribution shift: Under class interpolation, the deep ensemble behaved roughly as expected, while the Laplace learner assigned most uncertainty to the epistemic component and left aleatoric uncertainty almost unchanged.For OOD polygons, both classifiers reported higher uncertainty, but Laplace raised epistemic uncertainty almost exclusively whereas the deep ensemble raised both components.
- Overall findings: The experiments found that entropy-based component allocations often appear implausible, that epistemic uncertainty is globally very low, and that learner design can substantially affect results despite similar predictive performance.The overall uncertainty behaved fairly sensibly, but the attribution to aleatoric and epistemic components did not consistently do so.
6 CONCLUDING REMARKS
The paper concludes that Shannon entropy, conditional entropy, and mutual information can behave incoherently as uncertainty measures and that their additive decomposition warrants reassessment.
- Conclusion: The authors report that Shannon entropy, conditional entropy, and mutual information behave neither coherently nor always as expected in uncertainty quantification.They identify concerns with both the individual measures and their additive aggregation.
- Conclusion: In finite-sample settings, aleatoric uncertainty is derived from and depends on subjective epistemic uncertainty, contrasting with the ground-truth scenario of objective aleatoric uncertainty and zero epistemic uncertainty.This discrepancy leads to incoherencies and calls the additivity of the decomposition into question.
- Conclusion: The paper recommends critical assessment of the existing framework and cautious use of these measures.It also points to alternative measures and more expressive uncertainty representations as possible directions.
A EXPERIMENTAL DETAILS
The experiments used standard software and training configurations across image, synthetic, and tabular classification tasks, with ensemble-based probabilistic classifiers and fixed optimization settings.
- Software: The codebase relied chiefly on PyTorch, PyTorch Lightning, Laplace Redux, and scikit-learn.The full experimental code was hosted in a public repository.
- Base learners: Probabilistic classifiers combined base learners through explicit deep or random-forest ensembles or an implicit Laplace ensemble.EfficientNet-B7 was used for CIFAR10, a small convolutional network for MNIST and rectangle/polygon images, and a depth-ten random forest for tabular data.
- Training configurations: Training used SGD with momentum 0.9, an initial learning rate of 10^-2, cosine annealing, weight decay of 5×10^-4, and up to 200 epochs.Batch size was 256, with early stopping after five validation-loss evaluations without improvement.
B.1 INCREASING DATA NOISE
In tabular experiments with increasing class overlap and label noise, random forests responded in both uncertainty components, while MLP epistemic uncertainty remained extremely low.
- Increasing class overlap: When class overlap increased, the random forest reacted in both aleatoric and epistemic uncertainty components compared with the MLP ensemble.The comparison used the tabular classification task shown in the corresponding overlap figures.
- Increasing label noise: Label noise was simulated by randomly changing classes for 1%–75% of tabular observations.The resulting datasets varied the share of observations assigned incorrect labels.
- Increasing label noise: As label noise increased, aleatoric uncertainty rose, but remained moderate for the random forest even when three out of four labels were assigned randomly.Epistemic uncertainty rose slightly for the random forest but remained ultra-low across every ablation for the MLP ensemble.
B.2 NUMBER OF ENSEMBLE MEMBERS
Across tabular and computer-vision experiments, uncertainty estimates are generally stable as ensemble size changes, although particularly noisy settings can be exceptions. The overall uncertainty levels still differ considerably between probabilistic learners.
- Ensemble uncertainty estimates remain fairly stable as the tabular ensemble size varies from 2 to 50.The study expected possible volatility mainly for very small ensembles because finite-ensemble estimators may have larger bias.
- Overall uncertainty levels differ considerably between probabilistic learners despite stability across ensemble sizes.
- Computer-vision uncertainty results are generally robust when using ensembles of M = 5.The values average over all possible five-member ensembles formed from ten original predictions.
- Particularly noisy settings, including 1% sample size, produce some exceptions to this robustness.
B.3 BASE LEARNER COMPLEXITY
Increasing base-learner complexity substantially reduces aleatoric uncertainty, especially for random forests, while epistemic uncertainty changes little. These effects can be missed when performance remains relatively stable, so accuracy, calibration, and uncertainty should be considered jointly.
- Aleatoric uncertainty decreases considerably as random forests and MLP ensembles use more complex base learners.Capacity is varied through maximum tree depth for random forests and hidden-layer size for MLPs.
- The reduction in aleatoric uncertainty is especially strong for random forests, whose simple base learners overstate it and have high calibration error.
- Epistemic uncertainty changes little when base-learner capacity increases.The expected increase from greater freedom for model disagreement is not observed to a substantial degree.
- Relatively stable performance can overlook strong uncertainty effects, underscoring the need to assess accuracy, calibration, and uncertainty jointly.