Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods

Eyke Hüllermeier; Willem Waegeman

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Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods

Eyke Hüllermeier, Willem Waegeman

arXiv:1910.09457v3Oct 21, 2019cs.LGstat.ML

TL;DR

Machine learning involves uncertainty from inductive inference, model assumptions, and noisy or imprecise data, yet standard probabilistic approaches do not explicitly distinguish its types. This paper surveys approaches to quantifying aleatoric and epistemic uncertainty, emphasizing that set-based representations may better capture epistemic lack of knowledge. However, quantifying model uncertainty remains difficult and often requires assuming a correctly specified hypothesis space.

Problem
Machine learning uncertainty arises from induction, model assumptions, and noisy or imprecise data, but its aleatoric and epistemic sources are often not distinguished.
Method
The paper introduces uncertainty concepts and surveys approaches for representing and quantifying aleatoric and epistemic uncertainty in supervised-learning predictions.
Results
The survey argues that graded set-based representations are a viable alternative for epistemic uncertainty because probability distributions can obscure ignorance.
Takeaways & Limitations
Aleatoric uncertainty concerns outcome predictability, whereas epistemic uncertainty reflects lack of knowledge and can decrease with increasing sample size.
Takeaways & Limitations
Quantifying model uncertainty is difficult because it requires considering uncertainty over which hypothesis space is correctly specified; practical analyses of this kind may be infeasible.

Abstract

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The notion of uncertainty is of major importance in machine learning and constitutes a key element of machine learning methodology. In line with the statistical tradition, uncertainty has long been perceived as almost synonymous with standard probability and probabilistic predictions. Yet, due to the steadily increasing relevance of machine learning for practical applications and related issues such as safety requirements, new problems and challenges have recently been identified by machine learning scholars, and these problems may call for new methodological developments. In particular, this includes the importance of distinguishing between (at least) two different types of uncertainty, often referred to as aleatoric and epistemic. In this paper, we provide an introduction to the topic of uncertainty in machine learning as well as an overview of attempts so far at handling uncertainty in general and formalizing this distinction in particular.

1 Introduction

Machine learning predictions are inherently uncertain because induction produces hypothetical models, while incorrect assumptions and noisy data add further uncertainty. The paper motivates distinguishing aleatoric randomness from epistemic uncertainty, especially when decisions can be deferred or information can be acquired.

Motivation: Inductive learning replaces observations with hypothetical models, so both the models and their predictions are necessarily uncertain.Incorrect model assumptions and noisy or imprecise data introduce additional uncertainty.
Motivation: Trustworthy uncertainty representations are particularly important in safety-critical applications such as medicine and sociotechnical systems.Uncertainty is also central to methods such as active learning, where uncertainty reduction guides information acquisition.
Aleatoric and epistemic uncertainty: Aleatoric uncertainty is irreducible randomness in outcomes, whereas epistemic uncertainty is reducible uncertainty arising from limited knowledge.For coin flipping, additional information cannot remove the stochastic component underlying the outcome.
Why the distinction matters: Distinguishing uncertainty sources matters when predictions can be rejected or delayed, or when actions can specifically reduce uncertainty.A source distinction may be irrelevant when an agent is forced to make a decision, but that condition does not always apply.
Prior work: Prior work quantifies aleatoric and epistemic uncertainty for applications including medical decision making and reliability maps.Reliability maps distinguish a predicted probability score from uncertainty in that prediction.
Paper scope: The paper surveys uncertainty-handling methods and focuses on formalizing aleatoric and epistemic uncertainty in supervised learning.It introduces the setting, examines set-based and distributional models of epistemic uncertainty, discusses concrete methods, and concludes with general background in Appendix A.

2 Sources of uncertainty in supervised learning

In supervised learning, uncertainty can arise from the data-generating relationship, the model class, and the learner’s estimate from finite data. These sources contribute to uncertainty about predictions for individual query instances and require context-sensitive interpretation.

Supervised learning and predictive uncertainty: The learner uses training data to induce a hypothesis that predicts outcomes for new query instances.The task is commonly formalized with i.i.d. training data, a hypothesis space, and a loss function, aiming for low expected loss.
Supervised learning and predictive uncertainty: Predictive uncertainty concerns the unknown outcome associated with a concrete query instance and can be represented by a predictive set rather than only a point prediction.Predictive sets may provide statistical guarantees such as marginal coverage or conditional coverage.
Sources of predictive uncertainty: Aleatoric uncertainty is irreducible uncertainty from the nondeterministic relationship between instances and outcomes, even when the underlying probability measure is known.It reflects uncertainty about the actual outcome rather than uncertainty about which hypothesis to use.
Sources of predictive uncertainty: Model uncertainty concerns whether the hypothesis space contains the appropriate predictor, whereas approximation uncertainty concerns how well the learned hypothesis estimates the best hypothesis in that space.The learned hypothesis’s approximation quality depends strongly on the amount and quality of training data.
Reducible versus irreducible uncertainty: The aleatoric–epistemic distinction is context-dependent because changing the instance, output, hypothesis, or probability setting can change which uncertainty source applies.Assuming a correctly specified hypothesis space also ignores model misspecification risk, which is difficult to capture and quantify.
Approaches to epistemic uncertainty: Version space learning represents uncertainty about the optimal hypothesis with a candidate set, while Bayesian learning represents it with a probability distribution over the hypothesis space.Both approaches distinguish uncertainty about the optimal hypothesis from its consequences for a query outcome.

3 Modeling approximation uncertainty: Set-based versus distributional representations

The paper contrasts set-based and distributional representations of approximation uncertainty. Version spaces encode possible hypotheses explicitly, whereas Bayesian model averaging aggregates hypotheses probabilistically and can obscure epistemic uncertainty.

3.1 Version space learning: Version space learning maintains the set V of hypotheses consistent with observed data, eliminating any hypothesis that makes a training mistake.Under the stated noise-free assumptions, the target function belongs to H and the optimal risk is 0.
3.1 Version space learning: For a query xq, the possible prediction set Y = {h(xq) | h ∈V} contains exactly the outcomes produced by at least one consistent hypothesis.The characteristic function π assigns 1 to possible outcomes and 0 otherwise.
3.1 Version space learning: Restrictive hypothesis spaces can yield more certain predictions than flexible spaces because the latter permit more hypotheses and divergent outcomes.The paper illustrates this dependence of predictive uncertainty on prior model assumptions with binary-classification examples.
3.1 Version space learning: The version space shrinks as training data increase, so model-level epistemic uncertainty decreases with sample size, while predictive uncertainty can vary across instances.The size of the candidate-outcome set measures how approximation uncertainty translates into prediction uncertainty for a particular query.
3.2 Bayesian inference: Bayesian inference represents knowledge with a posterior over hypotheses and produces predictions by averaging outcome probabilities across hypotheses.This probabilistic approach differs from version space learning, which qualifies hypotheses and predictions as possible or impossible.
3.2 Bayesian inference: Bayesian model averaging can erase epistemic distinctions: a fair predictive probability may arise from complete ignorance or certainty that the coin is fair.The paper also notes that MAP inference does not capture epistemic uncertainty.
3.3 Representing a lack of knowledge: Distributional uncertainty measures primarily describe distributional shape and outcome predictability, whereas candidate sets are better suited to representing epistemic uncertainty.The paper frames entropy-like measures as more akin to aleatoric uncertainty and set-based representations as suited to lack of knowledge.

4 Machine learning methods for representing uncertainty

The section surveys machine-learning methods for representing predictive uncertainty, contrasting frequentist and Bayesian approaches and asking whether they distinguish aleatoric from epistemic uncertainty.

Methods differ in the type of prediction produced, how uncertainty is represented, and whether aleatoric and epistemic uncertainty can be distinguished.
Frequentist methods: Frequentist methods include probability estimation, likelihood-based inference, density estimation, hypothesis testing, and set-valued prediction.
Probability estimation: Probability estimation typically learns a single hypothesis whose predictions primarily represent aleatoric uncertainty, without explicitly distinguishing uncertainty types.
Bayesian methods: Bayesian methods maintain uncertainty about the sought hypothesis as a distribution, set of distributions, or graded set of distributions updated with observations.
Taxonomy: The categorization is not unique because credal approaches connect Bayesian inference with set-valued prediction and other methods share conceptual links.
Likelihood-based inference: Likelihood-based inference uses Fisher information to characterize parameter uncertainty, with larger confidence regions indicating greater epistemic uncertainty.

4.4 Gaussian processes

Gaussian processes extend Bayesian inference to distributions over functions and provide posterior predictive uncertainty for new queries. Their uncertainty can reflect both observation noise and uncertainty about the latent function.

Gaussian processes generalize Bayesian inference from finite-dimensional random vectors to distributions over infinite-dimensional functions.
A Gaussian-process prior over functions is specified by a mean function and covariance kernel, then updated to a posterior using observed data.
For regression with Gaussian noise, the posterior predictive distribution for a new query is Gaussian with a mean and variance.
For discrete outcomes, link functions such as the logistic function connect class labels to latent real-valued functions, but inference requires approximation because the likelihood is non-Gaussian.
Uncertainty representation: The confidence band around the mean function narrows as the number of observations increases.
Uncertainty representation: The predictive variance represents total uncertainty, while observation-error variance corresponds to aleatoric uncertainty and the remaining component can represent epistemic uncertainty.

4.5 Deep neural networks

The section presents Bayesian uncertainty methods for neural networks and related ensemble approaches, separating predictive uncertainty into aleatoric and epistemic components. It also introduces credal sets as a distributional generalization for representing imprecise knowledge.

Neural networks: Neural networks output class probability distributions, while Bayesian extensions represent uncertainty about network weights.
Neural networks: Heteroscedastic aleatoric uncertainty can be learned by predicting residual error and attenuating the contribution of high-variance observations to the loss.
Neural networks: Fixing the weights removes epistemic uncertainty, so the expected entropy of conditional output distributions measures aleatoric uncertainty.
Neural networks: Epistemic uncertainty is obtained as the difference between total predictive uncertainty and aleatoric uncertainty, equaling mutual information between outcomes and weights.
Ensembles: Bayesian model averaging links Bayesian inference with ensembles, whose prediction variance indicates epistemic uncertainty.
Ensembles: Finite ensembles can approximate predictive uncertainty measures, including aleatoric and epistemic quantities, using the component hypotheses’ predictive distributions.
Credal sets: Credal sets replace a single probability distribution with a set of distributions, supporting generalized Bayesian inference and interval-valued posterior probabilities.
Credal sets: Credal representations distinguish conflict or randomness from non-specificity as two types of uncertainty contained in a credal set.

4.7 Reliable classification

Reliable classification derives graded plausibilities for candidate outcomes and uses them to quantify aleatoric and epistemic uncertainty. The resulting uncertainty degrees reflect ambiguity between outcomes and lack of support for either outcome.

Method: The method first assigns each candidate outcome a plausibility degree and then derives aleatoric and epistemic uncertainty from those degrees.
Modeling plausibility: Hypothesis plausibility is proportional to likelihood, with the maximum-likelihood hypothesis assigned plausibility 1.
Modeling plausibility: Support for the positive class is zero when the predicted probability is ≤1/2, increases linearly afterward, and reaches 1 at probability 1.
Uncertainty measures: Epistemic uncertainty measures the degree to which both outcomes are plausible, whereas aleatoric uncertainty measures the degree to which neither outcome is plausible.
Uncertainty measures: The total uncertainty is bounded by 1 because aleatoric and epistemic uncertainty satisfy ua + ue ≤ 1.
Special cases: Full epistemic uncertainty can occur when multiple equally plausible hypotheses support opposite classes, especially with small samples and a flat likelihood.
Special cases: Full aleatoric uncertainty occurs when all plausible hypotheses assign probability 1/2 to both classes, indicating agreement that the query is a boundary case.
Limitation: Computing the supremum used by the method may become complex because its optimization cost depends strongly on the hypothesis space.

4.8 Conformal prediction

Conformal prediction constructs prediction regions by testing candidate outcomes and retaining those not rejected at a chosen significance level. Its validity is guaranteed under exchangeability, while the nonconformity function controls prediction-region efficiency.

Conformal prediction tests each candidate outcome and collects the non-rejected candidates into a prediction region.The resulting set covers the true outcome with a prescribed probability such as 95%.
The method uses a nonparametric nonconformity function to score how strange each input/output tuple is.Higher scores indicate less conformity with expected observations.
Exchangeability makes all score-order permutations equally probable when the hypothetical candidate agrees with the data-generating process.This assumption is weaker than independence and treats observation order as irrelevant.
Candidates with p(y) < ϵ are rejected because the probability of such a p-value under the hypothesis is bounded by ϵ.The significance level ϵ is prespecified.
The error bounds are valid and calibrated by construction, but the nonconformity function determines prediction-region size.A more suitable function produces smaller regions while preserving validity.
Conformal scores also support uncertainty quantification through the notions of confidence and credibility.These scores are produced during prediction-region construction.

4.9 Set-valued prediction based on utility maximization

Utility-based set-valued prediction chooses prediction sets by balancing correctness against the cost of returning larger sets. Its risk-averse formulations support abstention when uncertainty makes a precise class decision inappropriate.

Utility maximization selects a predicted set using the ground-truth outcome, the candidate set, and estimated conditional class probabilities.The Bayes-optimal set maximizes expected set-based utility.
Brute-force optimization checks all subsets of the class set and therefore has exponential time complexity.More efficient solutions exist for many utility scores.
The common utility family rewards singleton correct predictions most, prefers smaller sets, and requires risk-averse behavior.Its defining sequence g(1), ..., g(K) governs utility by predicted-set size.
Precision, recall, and F1 are special cases of the set-based utility family with different choices of g(s).Precision uses gP(s) = 1/s, recall uses gR(s) = 1, and F1 uses gF1(s) = 2/(1+s).
Reject-option classifiers restrict predictions to either a singleton or the full class set, with abstention receiving a lower reward.The abstention penalty α satisfies 0 < α < 1 −1/K for risk-averse behavior.
Hierarchical classifiers can represent set-valued predictions as internal hierarchy nodes rather than arbitrary flat class subsets.Tree-distance loss may favor nodes near the hierarchy root and is not part of the general utility family.
Most utility-based methods derive decisions from frequentist conditional class probabilities and therefore represent primarily aleatoric uncertainty.The cited exceptions incorporate other uncertainty sources.

5 Discussion and conclusion

The paper frames aleatoric uncertainty as irreducible randomness and epistemic uncertainty as reducible lack of knowledge, while surveying representations and unresolved challenges. It argues that probabilistic averaging may obscure epistemic ignorance and that rigorous decomposition and evaluation remain open.

The paper surveys approaches for quantifying aleatoric and epistemic uncertainty about predictions in supervised learning.It associates epistemic uncertainty with lack of knowledge about the true or Bayes-optimal hypothesis.
Epistemic uncertainty is reducible through additional data, whereas aleatoric uncertainty arises from irreducible stochastic dependence between inputs and outcomes.
In Bayesian models, a less peaked posterior p(h | D) over hypotheses indicates greater epistemic uncertainty.
Bayesian model averaging can lose information about epistemic uncertainty, making graded set-based representations a possible alternative for expressing ignorance.Version-space size directly corresponds to epistemic uncertainty in version-space learning.
The field lacks a settled role for aleatoric and epistemic uncertainty in methods such as conformal prediction and continues to develop dynamically.The paper notes that proposals vary in how ad hoc or justified they are.
Empirical evaluation is difficult because data normally provide no ground-truth uncertainty, so methods are assessed indirectly through prediction and decision usefulness.Examples include utility maximization and accuracy-rejection curves.
Many approaches assume a correctly specified hypothesis space and therefore neglect model uncertainty.Non-normalized plausibility functions are proposed as one way to reflect inconsistency between the hypothesis space and data.

A Background on uncertainty modeling

Uncertainty has long been studied across scientific disciplines and became central to artificial intelligence through problems involving incomplete, inconsistent, imprecise, and vague information. Machine-learning uncertainty research includes both probabilistic and set-based representations.

Uncertainty has played a major role in economics, psychology, and the social sciences through applied statistics.
Artificial intelligence recognized uncertainty early, especially with expert systems that had to handle inconsistency, incompleteness, imprecision, and vagueness.
The contemporary uncertainty literature is broad and includes set-based and distributional probabilistic representations.Figure 14 presents various uncertainty calculi and frameworks, many of which generalize standard probability theory.
The paper treats the distinction between set-based and probabilistic representations as relevant to understanding aleatoric and epistemic uncertainty.

A.1 Sets versus distributions

Uncertain information can be represented either by subsets that constrain the possible truth or by distributions that assign weights to individual hypotheses. These approaches differ both in representation and processing, especially because probability requires normalization and additivity.

The reference set Ω contains exhaustive, mutually exclusive hypotheses, although open-world classification can allow the truth to lie outside Ω.
A subset C ⊆ Ω records which hypotheses remain possible, excluding those considered impossible.
A distribution assigns each hypothesis a weight interpretable as a degree of belief, generalizing subset indicators before probability constraints are imposed.
Probability distributions require nonnegative weights that sum or integrate to 1, so arbitrary set-based information is not generally a probability special case.
Probability processes disjoint events additively, whereas set-based processing treats information through constraints rather than competing probability masses.

A.2 Representation of ignorance

Probability distributions primarily model chance and outcome predictability, making them potentially unsuitable for representing epistemic uncertainty as lack of knowledge. Uniform probability can conflate genuine symmetry with complete ignorance and can depend on parameterization.

Probability distributions model chance more directly than imprecision, raising questions about their suitability for epistemic uncertainty.
Set-based uncertainty grows with |C|, whereas redistributing fixed probability mass changes relative beliefs rather than simply weakening knowledge.
Entropy measures distributional shape and predictability, so it is more closely associated with aleatoric uncertainty than epistemic uncertainty.
Uniform probability is commonly used for complete ignorance, but it cannot distinguish ignorance from a genuinely uniform random mechanism.
A uniform distribution is not invariant under reparametrization, creating another difficulty for representing ignorance.

A.3 Sets of distributions

Sets of probability distributions combine distributional and set-based representations to capture aleatoric and epistemic uncertainty. Credal sets and possibility measures provide increasingly structured ways to express uncertainty and ignorance.

Credal sets represent uncertainty as sets of probability distributions, combining the strengths of probabilistic and set-based approaches.
Credal sets are typically convex subsets of all probability distributions and may be specified through upper and lower event-probability bounds.
Possibility distributions can be interpreted both as graded sets and as representations of sets of dominated probability distributions.
Possibility measures are maxitive rather than additive, with Π(A ∪ B) = max(Π(A), Π(B)).
Complete ignorance is represented by making every nonempty event fully plausible while assigning zero necessity to all events.

A.4 Distributions of sets

Distributions of sets combine randomness and imprecision by assigning masses to subsets rather than individual hypotheses. Evidence theory derives belief and plausibility measures from these masses, while likelihood-based contours yield flexible constraints.

Random-set and Dempster–Shafer formalisms represent uncertainty through distributions over sets.
A basic belief assignment m distributes unit mass across subsets, with each positively weighted subset serving as a focal element and piece of evidence.
Assigning mass to subsets rather than single hypotheses allows evidence theory to combine randomness with imprecision.
Belief and plausibility functions generalize probability and possibility distributions; singleton focal sets recover probability, while nested focal sets recover possibility.
Normalized likelihood can define a plausibility contour whose focal sets are likelihood-based confidence intervals, forming a flexible set of constraints.

B Max-min versus sum-product aggregation

The paper replaces Bayesian sum-product aggregation with max-min aggregation, combining hypothesis plausibility and class support through a Sugeno-style operation. This avoids averaging information and can retain class plausibility supported by a sufficiently plausible alternative hypothesis.

Bayesian inference uses posterior probabilities and class probabilities, whereas this approach uses normalized hypothesis likelihoods and class plausibility given each hypothesis.
Max-min aggregation replaces Bayesian sum-product aggregation when computing the plausibility of an outcome.
The computation takes the maximum over pointwise minima of hypothesis plausibility and conditional class plausibility.
This aggregation corresponds to a Sugeno integral of outcome support with respect to a possibility measure induced by normalized likelihoods.
Unlike averaging, max-min aggregation avoids information loss and can deem a class plausible when a sufficiently plausible alternative hypothesis supports it.