Epistemic Uncertainty Modeling

Updated 28 October 2025

Epistemic uncertainty modeling is the formal quantification of knowledge gaps in predictive systems, distinguishing it from inherent aleatoric randomness.
It employs methodologies such as Dempster–Shafer theory, possibility theory, and Bayesian frameworks to isolate, propagate, and assess uncertainty sources.
Robust quantification of epistemic uncertainty underpins improved decision-making in high-stakes domains by addressing data limitations, model misspecification, and procedural biases.

Epistemic uncertainty modeling is the formal quantification and propagation of “uncertainty due to lack of knowledge” within mathematical models and predictive systems. Distinguished from aleatoric uncertainty—which reflects stochastic variability or inherent randomness—epistemic uncertainty originates from limitations in data, unknown parameters, model misspecification, or incomplete causal understanding. In modern AI, engineering, and scientific workflows, rigorous epistemic uncertainty modeling is crucial for robust design, safety assurance, high-confidence forecasting, and principled decision-making.

1. Core Definitions, Principles, and Taxonomies

Epistemic uncertainty is formally defined as the uncertainty arising from ignorance or incomplete information, as opposed to the irreducible indeterminacy of aleatoric (stochastic) uncertainty. In uncertainty quantification, epistemic and aleatoric uncertainty are strictly complementary but not reducible to one another.

A recurring taxonomy (Jiménez et al., 29 May 2025) organizes epistemic uncertainty in machine learning into several components:

Component	Description	Typical Source
Model uncertainty	Hypothesis space lacks the true model (misspecification bias)	Model class/design limitations
Data (estimation) uncertainty	Uncertainty in parameters/outputs due to limited or unrepresentative data	Data scarcity/sampling artifact
Procedural uncertainty	Randomness in training process (random init., optimization, data order)	Procedural randomness in training pipelines
Distributional uncertainty	Uncertainty due to train/test distribution drift	Shifted environment or covariate shift

The bias–variance decomposition is foundational. The total expected squared error at $x$ decomposes as: $\mathbb{E}_{y|x}\Big[\mathbb{E}_{\mathcal{D}_N, \gamma}\big((y - \hat{y})^2\mid x\big)\Big] = \sigma^2(x) + \mathrm{Var}_{\mathcal{D}_N, \gamma}(\hat{y}\mid x) + \mathrm{bias}^2\big[\hat{y}\mid x\big]$ Here, $\sigma^2(x)$ is the aleatoric (intrinsic) noise, while $\mathrm{Var}_{\mathcal{D}_N, \gamma}(\hat{y}\mid x)$ and $\mathrm{bias}^2$ stem from epistemic contributions (Jiménez et al., 29 May 2025).

In dynamic systems, the segregation and separate propagation of stochastic (aleatory) variability and epistemic ignorance is critical (Terejanu et al., 2011, Li et al., 2012).

2. Theoretical Frameworks for Epistemic Uncertainty Quantification

Several mathematical frameworks underpin epistemic uncertainty modeling:

Dempster–Shafer Theory: Models epistemic uncertainty via belief functions and random sets, assigning “mass” to sets of possible outcomes (focal elements) rather than singleton events. DS structures on closed intervals form belief–plausibility envelopes, representing ignorance directly (Terejanu et al., 2011, Manchingal et al., 2022, Sultana et al., 4 May 2025). Output is typically a set-valued or “credal” structure (e.g., p‐box).
Possibility Theory and Evidence Theory: Possibility distributions $T_\Theta(x)$ , with associated possibility ( $\mathrm{Pos}(A) = \sup_{x\in A} T_\Theta(x)$ ) and necessity ( $\mathrm{Nec}(A) = 1 - \sup_{x \notin A} T_\Theta(x)$ ) measures, bound the “probability” of events in the face of imprecision or ambiguous information (Li et al., 2012, Kimchaiwong et al., 28 Nov 2024). Evidence theory merges probabilistic and possibilistic variables for hybrid uncertainty propagation.
Bayesian Frameworks: Treats parameters or model weights as random variables with prior/posterior distributions. Epistemic uncertainty is quantified via the posterior variance in predictions or, more formally, via mutual information terms (Jose et al., 2021). Imprecise Bayesian methods (e.g., imprecise Dirichlet or Gamma models) allow for hyperparameter intervals as robust guards against overconfident inference (Troffaes et al., 2013).
Frequentist and Simulation-Based Evaluation: Model “double-guessing” and estimates epistemic uncertainty by training models to produce two conditionally related predictions, then using their covariance as the epistemic variance contribution (Foglia et al., 17 Mar 2025). Simulation-based protocols (resampling, procedural randomization) enable empirical quantification of epistemic uncertainty due to both data and algorithmic randomness (Jiménez et al., 29 May 2025).
Information-Theoretic Measures: Predictive uncertainty (entropy) decomposed into aleatoric (irreducible, $H(Y|X,W)$ ) and epistemic (parameter, $I(Y; W \mid X,Z)$ ) contributions. In meta-learning, the decomposition extends to latent hyperparameters (Jose et al., 2021). Epistemic uncertainty reflects "excess risk" due to parameter or model inadequacy.

3. Practical Modeling and Propagation Strategies

Separation and Joint Propagation

Hybrid approaches strictly stratify aleatoric and epistemic sources (Terejanu et al., 2011, Li et al., 2012, Katwyk et al., 6 Oct 2025):

Aleatoric uncertainty addressed via representative parametric models (e.g., Gaussian, normalizing flows, conditional normalizing flows), with canonical finite parametrization (means, variances).
Epistemic uncertainty propagated through hierarchical mechanisms: DS or possibility structures over moments/parameters, with moment evolution equations or polynomial chaos expansions moving the belief–plausibility envelopes forward (Terejanu et al., 2011).
In ensemble-based ML or deep learning, epistemic uncertainty is estimated from the variance across ensemble outputs (or via MC-dropout, variational inference) (Zhou et al., 2021, Berry et al., 2023).

Model Wrapping, Lifting, and Advanced Representations

Epistemic Wrapping: Transforms BNN parameter posteriors (weight distributions) into belief function posteriors, “wrapping” them into the random set domain for second‐order uncertainty quantification. Dirichlet distributions fit to the mass values provide tractable credal representations, improving robustness and sensitivity to out-of-distribution samples (Sultana et al., 4 May 2025).
Random-Set/Belief Function Neural Networks: Replace classic probability-based outputs with set-valued outputs, augmenting predictive expressivity to model epistemic ambiguity (Manchingal et al., 2022).
HybridFlow: Separates modeling components—aleatoric uncertainty with conditional masked autoregressive flows, epistemic with a Bayesian/ensemble probabilistic predictor—enabling modular calibration and diagnosis (Katwyk et al., 6 Oct 2025).
Frequentist Double-Scoring: Single-model approaches, where model outputs are conditioned on initial predictions, have shown effective practical discrimination between epistemic and aleatoric sources (Foglia et al., 17 Mar 2025).

Conformal and Distribution-Free Methods

EPICSCORE extends classical conformal prediction by augmenting any conformal score with a Bayesian predictive distribution over the score, widening intervals adaptively in data-sparse regions while retaining distribution-free marginal coverage (Cabezas et al., 10 Feb 2025). Asymptotic conditional coverage is theoretically achieved under uniform posterior convergence assumptions.

4. Evaluation, Calibration, and Limitations

Performance metrics for epistemic uncertainty quantification span predictive accuracy, calibration scores (ECE, PICP, MPIW, Winkler, CRPS), sharpness of credible intervals, and explicit measures of “ignorance” (e.g., belief–plausibility gap, NIDI/NIigF indices) (Terejanu et al., 2011, Katwyk et al., 6 Oct 2025).

Key limitations and challenges highlighted include:

Epistemic Uncertainty Hole: Unexpected collapse of epistemic uncertainty in overparameterized or undertrained BNNs (especially with MC-Dropout or ensembles), resulting in unreliable OOD detection and overconfident predictions in poorly characterized regions (Fellaji et al., 2 Jul 2024).
Bias Underestimation: Standard second-order or Bayesian uncertainty estimates often fail to represent model bias, instead conflating it with aleatoric uncertainty, and yield misleadingly low epistemic estimates in high-bias regions (Jiménez et al., 29 May 2025).
Adequate conditional coverage is difficult; most methods guarantee only marginal coverage except for EPICSCORE, which achieves both distribution-free marginal and asymptotic conditional coverage (Cabezas et al., 10 Feb 2025).
Assessment protocols must account for data, procedural, and modeling uncertainties, not just sample variance. Simulation-based or resampling methods enable fuller characterization (Jiménez et al., 29 May 2025).

5. Applications in Engineering, Science, and Safety Assurance

Epistemic uncertainty modeling is foundational in domains requiring robust decisions under incomplete knowledge:

Dynamic Systems: Propagating DS structures or possibility distributions through nonlinear stochastic systems permits hazard/risk assessment in engineering and safety-critical infrastructure (Terejanu et al., 2011, Li et al., 2012).
Power Systems and Renewable Generation: Hybrid evidence/possibility theory frameworks capture imprecision due to poorly characterized generator parameters, yielding more faithful adequacy and reliability measures as renewable penetration increases (Li et al., 2012).
Reliability and Risk Analysis: Robust Bayesian and imprecise probability models provide interval-valued risk assessments vital for high-consequence engineering domains (Troffaes et al., 2013).
Socio-Technical and Safety Analysis: Systematic identification and tracking of epistemic uncertainty in causal modeling enhances completeness and dynamic updating of safety cases (HOT-PIE diagram, STPA integration) (Leong et al., 2017).
Machine Learning and AI: Improved OOD detection, active learning efficiency, and interpretability—via ensemble diversity, epistemic deep learning, and feature‐gap approaches—drive advancements in automated medical diagnosis, autonomous vehicles, and scientific emulation (Manchingal et al., 2022, Berry et al., 2023, Bakman et al., 3 Oct 2025).
LLMs: Techniques leveraging internal representations, linear probes, or semantic feature gaps enable token-level attribution of epistemic versus aleatoric uncertainty, supporting hallucination reduction and scalable oversight (Ahdritz et al., 5 Feb 2024, Bakman et al., 3 Oct 2025).

6. Interpretation, Explanation, and Future Directions

Recent work emphasizes interpretability and actionable reduction of epistemic uncertainty:

Ensured Explanations: Explanations are extended to account for not just predicted label but for actionable feature modifications that provably reduce epistemic uncertainty (quantified as interval tightness), categorized as counter-, semi-, or super-potential depending on their effect (Löfström et al., 7 Oct 2024).
Feature-Gap Theories: In large models, epistemic uncertainty is interpreted as a “semantic feature gap” in the hidden space relative to an ideal or more knowledgeable reference, estimated via linear probes, contrastive prompts, and low-dimensional summaries (Bakman et al., 3 Oct 2025).
Calibration and Ranking: Metrics such as ensured ranking balance the tradeoff between minimized epistemic interval width and maximized class probability, enabling more trustworthy, user-centred explanations.

Open research directions include: better integration of possibility/evidence with probabilistic reasoning for hybrid uncertainties (Kimchaiwong et al., 28 Nov 2024); more faithful representation and propagation of model bias (Jiménez et al., 29 May 2025); scalable, computationally-efficient ensembling and hybridization (Beachy et al., 2023); principled evaluation protocols beyond marginal coverage; and automated, interpretable approaches for high-stakes autonomy and safety assurance.

7. Summary Table: Frameworks and Key Methods

Framework/Method	Epistemic Modeling Approach	Key References
Dempster–Shafer belief functions, p-boxes	Set-valued evidence, interval mass	(Terejanu et al., 2011, Manchingal et al., 2022)
Possibility theory, evidence theory	Possibility/necessity, α-cut	(Li et al., 2012, Kimchaiwong et al., 28 Nov 2024)
Bayesian imprecise models	Interval-valued priors/posteriors	(Troffaes et al., 2013, Zhou et al., 2021)
Ensemble/MC-dropout/HybridFlow	Variance across predictions, modularity	(Berry et al., 2023, Katwyk et al., 6 Oct 2025)
Frequentist double-scoring	Model covariance via feedback	(Foglia et al., 17 Mar 2025)
EPICSCORE (conformal)	Bayesian-augmented conformal score	(Cabezas et al., 10 Feb 2025)
Epistemic Wrapping (belief-wrapped BNNs)	Dirichlet/wrapped credal parameter	(Sultana et al., 4 May 2025)
HOT-PIE/systemic hazard models	Causal checklist, traceable factors	(Leong et al., 2017)
Feature-gap/LLM linear probe	Hidden-representation gap analysis	(Ahdritz et al., 5 Feb 2024, Bakman et al., 3 Oct 2025)
Ensured Explanations	Feature actions for uncertainty reduction	(Löfström et al., 7 Oct 2024)

In sum, epistemic uncertainty modeling is a multifaceted discipline encompassing advanced mathematical formalisms, computational strategies for uncertainty propagation, and application protocols designed to foster interpretability, safety, and robust inference. The field continues to advance in response to the practical demands of high-stakes scientific, engineering, and AI domains.