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

Motivation

• Uncertainty is inherent to inductive modeling and prediction, while noisy or imprecise data and incorrect model assumptions introduce additional sources.Trustworthy uncertainty representation is especially important in safety-critical domains and supports uncertainty reduction in active learning and decision-tree induction.Source: §1 · pp. 1–2

Uncertainty distinction

• Aleatoric uncertainty reflects irreducible randomness, whereas epistemic uncertainty reflects reducible ignorance that can decrease with additional information.Conventional single-distribution probabilistic models often fail to distinguish these two sources (Hora, 1996; Der Kiureghian and Ditlevsen, 2009).Source: §1 · p. 2

Decision relevance

• Distinguishing uncertainty matters when decisions can be refused, delayed, or used to direct uncertainty-reducing actions such as active learning.When an agent must make an immediate prediction, the uncertainty source may instead be irrelevant, motivating purely probabilistic approaches.Source: §1 · p. 2

Related work

• Prior work applies the distinction to medical decisions, reliability maps, weather forecasting, and failures of autonomous systems (Senge et al., 2014; Kull and Flach, 2014; Varshney and Alemzadeh, 2016).Deep-learning studies likewise show confident failures and vulnerability to adversarial examples (Kendall and Gal, 2017; Papernot and McDaniel, 2018).Source: §1 · p. 3

Paper scope

• The paper surveys uncertainty-handling methods in supervised learning, emphasizing set-based and distributional approaches to modeling epistemic uncertainty about the optimal hypothesis.It covers epistemic-uncertainty modeling, concrete methods, and background on combining set-based with probabilistic representations.Source: §1 · pp. 3–4

2.1 Supervised learning and predictive uncertainty

• Supervised learning estimates a low-risk hypothesis from i.i.d. training examples, yet empirical-risk minimization leaves uncertainty about the hypothesis, its approximation quality, and its true risk.The induced hypothesis is applied to new query instances, motivating uncertainty assessment beyond average predictive accuracy.Source: §2.1 · pp. 5–6, Figure 3 · p. 6
• Predictive uncertainty concerns an individual query outcome and can be expressed through point predictions or predictive sets with marginal or conditional coverage guarantees.Conditional coverage fixes the query instance and averages over outcomes, whereas marginal coverage averages jointly over instances and outcomes.Source: §2.1 · pp. 6–7

2.2 Sources of uncertainty

• Aleatoric uncertainty comes from stochastic input–output dependencies, while model and approximation uncertainty are epistemic because they reflect missing knowledge about the predictor.Model uncertainty concerns choosing the hypothesis space; approximation uncertainty concerns how well the learned hypothesis estimates the best hypothesis within that space.Source: §2.2 · pp. 7–8, §2.3 · p. 8

2.3 Reducible versus irreducible uncertainty

• Aleatoric and epistemic uncertainty are context-dependent: changing the instance, output, hypothesis, or probability setting can transform one type into the other.Adding features may resolve apparent aleatoric overlap while increasing epistemic difficulty because the expanded representation requires more data.Source: §2.3 · p. 9

2.4 Approximation and model uncertainty

• With a fixed setting, more data can eliminate approximation uncertainty asymptotically, but model uncertainty remains difficult to quantify because the correct hypothesis space may be unknown.Assuming a correctly specified hypothesis space makes model misspecification risk easy to neglect but difficult to assess.Source: §2.4 · pp. 9–10
• Version space learning represents uncertainty about the optimal hypothesis with candidate sets, whereas Bayesian inference represents it with a probability distribution over the hypothesis space.Both approaches distinguish uncertainty about the hypothesis from uncertainty about the outcome at a query point.Source: §2.4 · p. 11

3.1 Version space learning

• Version space learning represents uncertainty as sets of candidate hypotheses and outcomes, making epistemic uncertainty correspond to the size of those sets.The approach assumes deterministic, noise-free data and that the true function belongs to the hypothesis space; uncertainty decreases as training data eliminate inconsistent hypotheses.Source: §3.1 · pp. 12–13, §3.3 · p. 17

3.2 Bayesian inference

• Bayesian model averaging produces predictive probabilities, but it can conceal epistemic uncertainty by yielding identical predictions under complete ignorance and complete certainty.For the coin example, both a uniform posterior and a point mass at a fair coin produce probability 1/2 for heads and tails; MAP inference also fails to capture epistemic uncertainty.Source: §3.2 · pp. 15–16
• Quantifying epistemic uncertainty from Bayesian predictive distributions remains nontrivial because spread, entropy, and variance each have important limitations.The paper notes that entropy can treat differently shaped distributions as equally uncertain, while variance is not maximized by the uniform distribution, a possible representation of minimal informedness.Source: §3.2 · p. 16

3.3 Representing a lack of knowledge

• Probability distributions are less suitable than candidate sets for representing ignorance because fixed total mass forces changes in one alternative to offset changes in another.Distributional uncertainty measures mainly capture peakedness or non-uniformity, which is closer to aleatoric uncertainty, whereas set size directly measures lack of knowledge.Source: §3.3 · p. 17
• Generalized probabilistic formalisms combine sets and distributions to represent uncertain knowledge, including imprecise probability, evidence theory, and possibility theory.These approaches respond to the argument that a single probability distribution cannot adequately represent ignorance.Source: §3.3 · p. 18

4 Machine learning methods for representing uncertainty

• The methods span frequentist and Bayesian traditions, distributional and set-valued predictions, and representations ranging from single distributions to sets or graded sets of distributions.The taxonomy is not unique: credal methods connect Bayesian and set-valued approaches, while reliable classification connects normalized likelihood with likelihood-based inference.Source: §4 · pp. 18–20, Figure 10 · p. 19

4.1 Probability estimation via scoring, calibration, and ensembling

• Ensembles estimate probabilities from multiple predictors, but vote-based probabilities can be biased and poorly calibrated; proper scoring rules, including log-loss, train probabilistic predictors.Examples include bagging, boosting, random forests, logistic regression, Bayesian networks, Gaussian processes, and neural-network methods.Source: §4.1 · p. 21
• Probability estimators usually represent aleatoric uncertainty but do not explicitly distinguish uncertainty types, because they often commit to a single learned hypothesis.Probabilistic predictors output distributions rather than point predictions, and calibration methods can transform scores into better-calibrated probabilities.Source: §4.1 · pp. 21–22

4.2 Maximum likelihood estimation and Fisher information

• Likelihood-based inference uses Fisher information and confidence regions to quantify epistemic uncertainty about parameters or hypotheses.Larger confidence regions and flatter likelihoods indicate greater uncertainty, while Fisher information also supports variance bounds and active-learning measures.Source: §4.2 · pp. 22–23

4.3 Generative models

• Generative models use input density to identify low-support regions as proxies for epistemic uncertainty, while class-conditional densities help represent both epistemic and aleatoric uncertainty.Density estimation underlies outlier detection and can support abstention or multi-label predictions when class densities overlap.Source: §4 · p. 19, §4.3 · pp. 23–24
• Generative uncertainty methods include kernel density estimation, Gaussian mixtures, isolation forests, auto-encoders, and radial basis function networks.Auto-encoder reconstruction error can measure distributional support, while one-class SVM variants separate outliers from normal data using hyperplanes or hyperspheres.Source: §4.3 · p. 24
• Set-valued generative prediction interprets a single-class output as low uncertainty, abstention as excessive epistemic uncertainty, and multi-class sets as excessive aleatoric uncertainty.Hechtlinger et al. (2019) fit class-conditional generative models and select prediction-set cardinality according to how many class densities sufficiently support the input.Source: §4.3 · p. 24
• Generative uncertainty methods remain limited by threshold selection, generative-model choice, data requirements, and high model uncertainty when samples are scarce.The conflict–ignorance distinction links overlapping class distributions with aleatoric uncertainty and sparse regions with epistemic uncertainty.Source: §4.3 · p. 25

4.4 Gaussian processes

• Gaussian processes generalize Bayesian inference from finite-dimensional random variables to infinite-dimensional functions, with finite function evaluations jointly following a multivariate normal distribution.A GP is specified by a mean function and covariance function, with the covariance function satisfying kernel properties.Source: §4.4 · pp. 25–26
• Given observed data, Gaussian processes replace a prior over functions with a posterior and provide posterior predictive distributions for new queries.For regression with Gaussian noise and a zero-mean prior, the predictive distribution remains Gaussian; general computations can otherwise be intractable.Source: §4.4 · p. 26
• For discrete outcomes, link functions connect observed labels to latent real-valued functions, but non-Gaussian likelihoods require approximate inference such as Laplace methods, expectation propagation, or MCMC.Binary classification is given as an example using a logistic link function.Source: §4.4 · pp. 26–27
• The confidence band around the predictive mean narrows as the number of observations increases, reflecting reduced predictive uncertainty.The distinction between aleatoric and epistemic sources can in principle be obtained by determining all GP hyperparameters, although estimation is difficult.Source: Figure 13 · p. 27, §4.4 · p. 27
• In regression, posterior predictive variance measures total uncertainty, while the observation-noise variance represents aleatoric uncertainty and their difference can represent epistemic uncertainty.The epistemic component is largely determined by kernel parameters, such as the characteristic length-scale of a squared exponential covariance function.Source: §4.4 · p. 27

4.5 Deep neural networks

• Bayesian neural networks represent weights probabilistically and approximate posterior predictive uncertainty, extending standard neural networks beyond aleatoric uncertainty to include epistemic uncertainty.Standard neural-network training produces probabilistic predictions without confidence information; Bayesian inference instead models p(w | D), typically using variational approximations and sampling.Source: §4.5 · pp. 27–28
• Deep-learning methods separate total uncertainty into aleatoric residual variation and epistemic parameter uncertainty, including heteroscedastic aleatoric uncertainty learned through loss attenuation.Loss attenuation penalizes errors less when predicted residual variance is high, while still penalizing predictions of high variance.Source: §4.5 · p. 28
• Predictive entropy can quantify total uncertainty, expected fixed-weight entropy quantifies aleatoric uncertainty, and their difference yields epistemic uncertainty as mutual information.Ensemble prediction variance likewise indicates epistemic uncertainty and motivates simple, parallelizable alternatives to Bayesian neural networks.Source: §4.5 · p. 29

4.6 Credal sets and classifiers

• Credal sets replace single probability distributions with sets of candidate priors, posteriors, and derived predictions, generalizing Bayesian inference under imprecise knowledge.In the Naïve Credal Classifier [Zaffalon, 2002a; Corani and Zaffalon, 2008a], local credal sets replace class priors and conditional probabilities, producing interval-valued posterior probabilities.Source: §4.6 · pp. 30–31

4.6.1 Uncertainty measures for credal sets

• Credal-set uncertainty distinguishes conflict, corresponding to aleatoric uncertainty, from non-specificity, corresponding to epistemic uncertainty.Generalized total uncertainty combines an aleatoric measure based on conflict or Shannon entropy with an epistemic measure based on non-specificity or generalized Hartley uncertainty.Source: §4.6.1 · p. 31
• Credal-set measures include generalized Hartley uncertainty with desirable axiomatic properties, while proposed entropy-based decompositions use upper entropy for total uncertainty and differences to quantify epistemic uncertainty.The generalized Hartley measure’s uniqueness was shown by Klir and Mariano (1987), whereas conflict-based Shannon-entropy generalizations are more difficult.Source: §4.6.1 · p. 32

4.6.2 Set-valued prediction

• Credal classifiers produce reliable set-valued predictions by retaining non-dominated outcomes, with uncertainty increasing when class-probability intervals widen or their midpoint approaches 1/2.Set-based modeling can impede inductive learning under complete ignorance, because extreme priors remain extreme after finite-sample updating; hierarchical Bayesian models are proposed as an alternative.Source: §4.6.2 · pp. 32–33

4.7 Reliable classification

• Senge et al. (2014) distinguish predictive uncertainty through two steps: deriving non-probabilistic plausibilities for candidate outcomes, then deriving aleatoric and epistemic uncertainty.The approach generalizes to multiclass classification through Nguyen et al. (2018).Source: §4.7 · p. 34

4.7.1 Modeling the plausibility of predictions

• Hypothesis plausibility is proportional to normalized likelihood, with the maximum-likelihood hypothesis assigned plausibility 1; outcome plausibility aggregates plausible hypotheses’ class support.Support for +1 is zero when h(xq) ≤ 1/2 and increases linearly to 1 at h(xq) = 1, with an analogous definition for −1.Source: §4.7.1 · pp. 34–35

4.7.2 From plausibility to aleatoric and epistemic uncertainty

• Epistemic uncertainty measures how plausible both classes are, whereas aleatoric uncertainty measures how implausible both are, with their sum bounded by 1.These quantities are bounds: initially both classes may be fully plausible, yielding ue = 1 and ua = 0 despite underlying aleatoric uncertainty; with more data, ue can vanish while ua approaches the true aleatoric uncertainty.Source: §4.7.2 · pp. 35–36
• Computing outcome plausibilities requires supremum optimization, whose complexity depends strongly on the hypothesis space H.Thus, the method’s computational cost can become substantial even though algorithmic aspects are not its focus.Source: §4.7.2 · p. 36

Construction

• Conformal prediction tests each candidate outcome and retains those not rejected, forming a prediction set guaranteed to cover the true outcome with probability 1−ϵ, such as 95%.For classification, the set contains classes; for regression, it is commonly an interval.Source: §4.8 · pp. 36–37

Validity

• Under exchangeability, the procedure’s p-values control rejection errors at level ϵ, making its coverage valid without parametric assumptions.Exchangeability requires the observation order to be irrelevant and is weaker than independence.Source: §4.8 · p. 37

Efficiency

• Validity and calibration hold regardless of the nonconformity function, but a better-suited function produces smaller prediction regions and therefore improves efficiency.The function assigns higher scores to less conforming input/output patterns.Source: §4.8 · p. 37

Uncertainty quantification

• Conformal prediction scores also quantify uncertainty: credibility is the largest p-value, while 1 minus the second-highest p-value measures how uniquely a candidate is supported.The highest-p-value candidate is selected when a point prediction is required.Source: §4.8 · pp. 37–38

Interpretation

• Unlike methods that predict a point and then attach uncertainty, conformal prediction fixes the confidence level first and adjusts the prediction region accordingly.The framework is primarily transductive, although inductive variants also exist (Papadopoulos, 2008).Source: §4.8 · pp. 37–38

• Unlike conformal prediction, these methods seek utility-maximizing sets rather than sets containing the true class with high probability, and they primarily address multi-class classification.Most methods derive decisions from conditional class probabilities and therefore capture conditional, chiefly aleatoric, uncertainty.Source: §4.9 · pp. 38, 40
• Bayes-optimal set-valued prediction maximizes expected utility u(y, bY ) under p(y | xq), although brute-force optimization over all label subsets is exponentially costly.For many utility functions, more efficient solutions exist.Source: §4.9 · p. 38
• The common utility family rewards singleton true-label predictions, decreases utility as predicted-set size grows, and requires risk-averse sets to outperform random guessing.The condition g(s) ≥1/s formalizes preferring abstention to random guessing when uncertainty is substantial.Source: §4.9 · p. 39
• Precision, recall, and F1 are special cases of the utility family, while credal-classification variants use alternative g functions to encode different degrees of risk aversion.Precision uses gP (s) = 1/s, recall uses gR(s) = 1, and F1 uses gF1(s) = 2/(1+s).Source: §4.9 · p. 39
• Reject-option classifiers restrict predictions to a singleton or the full label set, assigning abstention a lower reward 1 −α with 0 < α < 1 −1/K.The full-set prediction represents abstention, while the singleton receives reward one when correct.Source: §4.9 · pp. 39–40
• Hierarchical classifiers restrict predictions to sets represented by hierarchy nodes, but tree-distance loss often favors broad near-root sets and therefore weak abstention behavior.Some hierarchical methods optimize members of the utility family or fix predicted-set size instead.Source: §4.9 · p. 40
• Most approaches use frequentist conditional probabilities and capture aleatoric uncertainty, whereas Yang et al. (2017a) and Nguyen et al. (2018) pursue sets capturing both uncertainty types.These exceptions draw on imprecise probability theory and reliable classification.Source: §4.9 · p. 40

• Epistemic uncertainty reflects limited knowledge about the Bayes-optimal hypothesis and can, in principle, be reduced with additional data, whereas aleatoric uncertainty arises from irreducible stochasticity in outcomes.Source: §5 · p. 40
• Bayesian posterior spread represents epistemic uncertainty, but Bayesian model averaging may lose information that graded set-based representations, such as version spaces, can preserve.The Senge et al. (2014) approach combines generalized version-space learning with Bayesian inference.Source: §5 · pp. 40–41
• Uncertainty modeling remains unsettled: methods including conformal prediction do not have immediately clear aleatoric–epistemic interpretations, motivating an axiomatic derivation of total uncertainty and its decomposition.The paper calls for a mathematically rigorous basis comparable to axiomatic justifications of entropy (Csiszár, 2008).Source: §5 · p. 41
• Empirical evaluation lacks ground-truth uncertainty labels, so methods are often assessed indirectly through improved prediction or decisions, including utility maximization and monotone accuracy-rejection curves under abstention.A well-calibrated uncertainty estimate should produce increasing accuracy as the allowed rejection percentage increases, unlike random abstention.Source: §5 · p. 41
• Most approaches assume a correctly specified hypothesis space, but model misspecification can create conflict or inconsistency that should be represented as model uncertainty.A non-normalized plausibility function can quantify this through the gap 1−sup_h π_H(h) between one and the maximum plausibility.Source: §5 · pp. 41–42
• Uncertainty methods should express ignorance when new classes emerge or inputs are out of distribution, because the commonly used closed-world assumption is often violated.Initial proposals include DeVries and Taylor (2018), Lee et al. (2018b), Malinin and Gales (2018), and Sensoy et al. (2018).Source: §5 · p. 42
• Quantified uncertainty can inform active learning and set-valued prediction, with epistemic uncertainty argued to be more relevant than aleatoric uncertainty for uncertainty sampling.Set-valued prediction enables partial abstention by returning subsets of possible outcomes.Source: §5 · p. 42

A Background on uncertainty modeling

• Uncertainty has long been important across economics, psychology, social sciences, artificial intelligence, and engineering, where it is also studied as uncertainty quantification.In AI, uncertainty modeling addresses inconsistency, incompleteness, imprecision, and vagueness in knowledge representation.Source: A Background on uncertainty modeling · p. 42
• Contemporary uncertainty literature includes diverse calculi and frameworks, many of which generalize standard probability theory.Figure 14 surveys these formalisms and their relationships.Source: A Background on uncertainty modeling · p. 43, Figure 14 · p. 43
• Probabilistic representations are suited to aleatoric uncertainty, whereas set-based approaches are argued to better capture epistemic uncertainty arising from ignorance or limited knowledge.The distinction between distributional and set-based modeling is therefore central to discussions of aleatoric and epistemic uncertainty.Source: A Background on uncertainty modeling · p. 43

Sets versus distributions

• The exhaustiveness of Ω distinguishes closed-world from open-world settings, where the empty set can indicate that the true state lies outside Ω.Open-world classification allows unknown or emerging classes, unlike the closed-world assumption that excludes the empty set as valid information.Source: §A.1 · pp. 43–44
• A subset C ⊆ Ω constrains the truth ω* to C, distinguishing possible values from excluded ones and representing imprecision such as an interval.The reference set Ω contains exhaustive, mutually exclusive hypotheses, with one corresponding to the truth; the exposition assumes Ω is finite or countable.Source: §A.1 · pp. 43–44
• A distribution assigns each hypothesis a degree-of-belief weight, but probability distributions are not simply set representations because their nonnegative weights must sum to 1.Probability additionally requires P(∅) = 0 and P(Ω) = 1, whereas set-based information need not satisfy normalization.Source: §A.1 · p. 44
• Set-based and probabilistic information use fundamentally different calculi: probability is additive, whereas constraints evaluate events by logical consistency and yield a maxitive set-function.An event is possible when it intersects the constraint C and impossible otherwise, while probability compares the mass of competing events.Source: §A.1 · pp. 44–45

Set-based representation

• Set-based uncertainty represents increasing ignorance through larger candidate sets, whereas probability measures cannot weaken knowledge straightforwardly because total belief remains fixed.The set-based correspondence is between imprecision and |C|; adding elements to C weakens knowledge, while increasing one probability weight requires decreasing another.Source: §A.2 · p. 45

Probability-based measures

• Entropy measures distributional shape and predictability, so they are more akin to aleatoric uncertainty than epistemic uncertainty.The passage contrasts probability-distribution shape measures with set-based approaches for capturing epistemic uncertainty.Source: §A.2 · p. 45

Uniform distributions and ignorance

• Uniform probability distributions cannot distinguish precise probabilistic knowledge, such as a fair coin, from complete ignorance about an incompletely described experiment.Both states may receive the same uniform distribution under the principle of indifference or maximum entropy, despite representing different knowledge states.Source: §A.2 · pp. 45–46

Reparametrization

• Uniform priors are not invariant under reparametrization, so modeling ignorance uniformly for one quantity can imply informedness about an equivalent transformation.A uniform distribution on ω does not remain uniform on 1/ω, and uniform cube length x does not yield uniform volume x3, making uniform priors for epistemic uncertainty debatable.Source: §A.2 · p. 46

Sets of distributions

• Credal sets represent incomplete knowledge about the true probability distribution by restricting it to a set of possible candidate distributions (Cozman, 2000; Zaffalon, 2002b).They are typically convex subsets of all distributions and may be specified using upper and lower event-probability bounds.Source: §A.3 · p. 46

Possibility theory

• Possibility distributions offer a simpler, less expressive credal-set construction by assigning graded values π: Ω→[0, 1] and inducing upper bounds on probabilities.Their associated possibility measure is maxitive rather than additive, with Π(∅)=0, Π(Ω)=1, and Π(A∪B)=max(Π(A),Π(B)).Source: §A.3 · pp. 46–47
• Possibility theory combines set-based and distributional approaches by treating possibility distributions as graded generalized sets and non-additive measures that express degrees of belief.Probability and possibility distributions can be viewed as two extremes among uncertainty representations, with possibility calculus based on maxima rather than sums.Source: §A.3 · p. 47
• The dual measures (N, Π) represent ignorance because an event can remain plausible without its complement being implausible, unlike self-dual probability measures.Complete ignorance is modeled when Π(A)=1 for every nonempty A, yielding N(A)=0 for all A.Source: §A.3 · p. 47

Distributions of sets

• The distributions-of-sets perspective underlies random-set calculus and Dempster-Shafer theory, combining sets with distributions in the reverse direction from distributions of sets.These formalisms treat uncertainty through collections of sets rather than a single set or constraint.Source: §A.4 · p. 48
• Evidence theory assigns masses to subsets of Ω, whose positive-mass focal elements represent evidence and allow randomness and imprecision to be combined.The mass function induces belief and plausibility measures over Ω, with each focal element acting as a weighted constraint.Source: §A.4 · p. 48
• Plausibility and belief generalize probability and possibility distributions: singleton focal elements yield probability, while nested focal sets yield possibility and necessity measures.The associated possibility distribution is the plausibility contour π(ω) = Pl({ω}).Source: §A.4 · p. 48
• Normalized likelihood defines a consonant plausibility function whose focal sets are likelihood-based confidence intervals, replacing one rigid constraint with a flexible set of constraints.For observed data D, the contour is π(ω) = L_n(ω; D), with focal sets C_α = {ω | L_n(ω; D) ≥ α}.Source: §A.4 · p. 49

B Max-min versus sum-product aggregation

• Max-min aggregation replaces Bayesian sum-product aggregation, using normalized hypothesis likelihoods and class plausibilities rather than posterior and probability representations.The resulting plausibility is formally expressed through a Sugeno integral with respect to the possibility measure induced on the hypothesis space.Source: B Max-min versus sum-product aggregation · pp. 49–50
• Max-min inference avoids information loss from averaging and generalizes existential aggregation, but requires commensurable normalized measures πH(h) and π(+1 | h, xq).This requirement motivates normalizing the likelihood before aggregation.Source: B Max-min versus sum-product aggregation · p. 50
• Unlike MAP inference, max-min inference incorporates class support and can deem a class plausible when a nearly most-likely alternative hypothesis supports it sufficiently.MAP considers only hypothesis probabilities and ignores the probabilities assigned to classes.Source: B Max-min versus sum-product aggregation · p. 50