Aleatoric and Epistemic Uncertainties

• Aleatoric and epistemic uncertainties are distinct concepts that model inherent randomness and knowledge gaps in complex systems.
• They are represented using probabilistic methods and Dempster–Shafer structures to separate and propagate randomness and ignorance.
• Applications in risk assessment, decision analysis, and engineering design yield actionable insights for improving model reliability.

Aleatoric and Epistemic Uncertainties are foundational constructs in the quantification and propagation of uncertainty for dynamic systems, probabilistic machine learning, decision analysis, and engineering risk. These uncertainties, corresponding respectively to inherent randomness and ignorance or lack of knowledge, inform the modeling, analysis, and interpretation of complex phenomena across scientific disciplines. Rigorous treatment of both forms of uncertainty necessitates hierarchical mathematical representations, careful separation during propagation, and principled aggregation for inference and decision-making.

1. Conceptual Definitions and Hierarchical Representation

Aleatoric uncertainty is defined as the irreducible randomness or variability inherent in a system or process. It captures stochasticity that persists even under perfect knowledge of the governing mechanisms—e.g., thermal noise, quantum fluctuations, or white noise excitation in differential systems. In mathematical terms, aleatoric uncertainty is typically modeled through probability distributions whose parameters may themselves be exactly known or unknown.

Epistemic uncertainty, in contrast, encodes incomplete knowledge or ignorance regarding system parameters, initial or boundary conditions, model structure, or measurement processes. Epistemic uncertainty is, in principle, reducible by acquiring additional information, refining the model, or performing more experiments. In advanced frameworks, epistemic uncertainty is represented as sets or distributions over possible probability distributions; the Dempster–Shafer (DS) structure and credal sets are canonical examples, assigning interval-valued or set-valued belief masses to parameter ranges or alternative hypotheses.

The hierarchical combination of these uncertainties yields a second-order uncertainty structure: aleatoric variability is represented by a conditional probability density (or distribution) parameterized by epistemic variables, themselves characterized by set-based or interval-based ignorance structures (Terejanu et al., 2011).

2. Mathematical Frameworks for Modeling and Propagation

In the context of stochastic dynamic systems, a canonical propagation scheme involves two stages:

Aleatoric Uncertainty Modeling

Conditioning on a set of (possibly epistemically uncertain) model parameters, the state’s time-evolution is represented via a conditional probability density function (pdf), e.g., $$p(t, x \mid e_D) \approx \mathcal{N}(x; \mu, \sigma^2)$$ where $\mu$ and $\sigma^2$ are the conditional mean and variance, evolving according to the system’s stochastic dynamics (e.g., Fokker–Planck–Kolmogorov formalism). Finite-dimensional parameterization, such as assuming (approximate) Gaussianity, is introduced to tractably capture evolving uncertainty under white-noise excitation.

Epistemic Uncertainty Propagation

Model parameters—including coefficients, noise intensity, and initial statistical moments—are represented as Dempster–Shafer structures on closed intervals, i.e., $$x \sim \{\, ([\underline{x}_i, \overline{x}_i], p_i) \}_{i=1}^n$$ The time-propagation of uncertainty is performed through moment evolution equations derived, e.g., via Itô’s lemma for polynomial systems, $$\dot{M}_k|_{e_0} = -k \sum_i \alpha_i m_{i+k-1}|_{e_0} + \frac{1}{2} k(k-1) q^2 m_{k-2}|_{e_0}$$ where all parameters and moments are interval-valued and propagated according to their DS masses through the corresponding ordinary differential equations (ODEs). To mitigate dependency-induced conservatism in interval arithmetic, advanced bounding approaches—including polynomial chaos expansion and Bernstein polynomial transformations—are employed.

3. Output Representations and Decision Aggregation

After sequential propagation, the resulting uncertainty in system response is expressed by a DS structure (with assigned masses) over sets of cumulative distribution functions (p-boxes): $\{\, ([F_l(x), F_u(x)], p_i) \}$ where each $F_l, F_u$ bounds the cdf envelopes induced by propagated moment intervals for each focal element of the DS structure.

Prior to or during decision-making, second-order uncertainty can be "crunched" into a singleton cdf or pdf—often via the pignistic transformation, $$P_{\text{Bet}}(X \leq x) = \frac{1}{2} \sum_i \left( \underline{N}_i(x) + \overline{N}_i(x) \right) p_{D,i}$$ turning a set-valued (ambiguous) belief structure into a single actionable distribution for expected utility calculation and risk analysis. Quantitative indices (such as NIDI or the ignorance function IgF) are provided to assess the residual ambiguity and guide confidence-aware decision policies.

4. Key Mathematical Constructs

The propagation and aggregation workflow is summarized by the following formulations:

Mathematical Expression	Interpretation
$$\dot{M}_k|_{e_0} = - k \sum_i \alpha_i m_{i+k-1}|_{e_0} + \frac{1}{2} k(k-1) q^2 m_{k-2}|_{e_0}$$	Moment evolution with epistemic intervals
$$x \sim \{([\underline{x}_i, \overline{x}_i], p_i)\}$$	DS structure for epistemic parameter uncertainty
$$p_{\text{Bet}}(x) = \sum_i \dfrac{p_i}{\overline{x}_i - \underline{x}_i} I(x; [\underline{x}_i, \overline{x}_i])$$	Pignistic transformation to singleton pdf
$$N_l(x_f) = \mathcal{N}(x_f; \mu_l, \sigma^2_l),\ N_u(x_f) = \mathcal{N}(x_f; \mu_u, \sigma^2_u)$$	Envelope bounds for Gaussian cdf with interval moments
$$P_{\text{Bet}}(X \leq x_f) = \frac{1}{2} \sum_i ( N_l^i(x_f) + N_u^i(x_f) ) p_{D,i}$$	Singleton cdf for decision analysis

5. Applications and Implications in Engineering and Risk Analysis

This approach yields substantial utility in scenarios where system parameters and initial/boundary conditions are characterized by epistemic ignorance, possibly due to sparse, conflicting, or expert-elicited data. Typical application domains include:

• Risk and hazard assessment: Quantifies how both stochastic variability and lack of knowledge about, e.g., rare event rates, propagate to system-level failure probabilities.
• Decision-making under uncertainty: Enables rational choices under ambiguity, deferring action or selecting robust alternatives if propagated epistemic uncertainty is unacceptably high.
• Engineering design and safety: Supports robust control, reliability assessment, and model-based design by identifying if uncertainty bounds are dominated by intrinsic noise or by lack of knowledge, guiding data acquisition or model refinement priorities.

An explicit separation between aleatoric and epistemic components throughout the propagation process provides actionable diagnostic information. Only when ignorance is compressed (via pignistic transformation) is a single expected utility recommendation produced; before this step, residual ignorance is explicitly quantifiable.

6. Advantages and Limitations of the DS-based Dual Uncertainty Framework

Advantages:

• Maintains interpretability by quantifying the impact of incomplete knowledge separately from irreducible randomness.
• The DS structure over p-boxes provides a systematic route to combine evidence and update beliefs with new information.
• Enables calibration of decision rules according to the degree of residual ignorance, reducing the risk of overconfident conclusions.

Limitations:

• Interval arithmetic and DS combination rules may be computationally demanding for high-dimensional systems or numerous focal elements.
• The approach relies on justifiable interval representations and focal element partitioning for epistemic uncertainty, which can be nontrivial in some domains.
• Conservativeness in the propagation of epistemic uncertainty may yield overly wide p-boxes if dependence among intervals is not appropriately handled.

Overall, the framework for dual propagation of aleatoric and epistemic uncertainties as presented in (Terejanu et al., 2011) delivers a rigorously structured mechanism to represent the full spectrum of uncertainty in dynamic models, facilitating informed risk assessment and robust decision-making where both randomness and ignorance are present in critical parameters and system behaviors.