Epistemic Uncertainty and Aleatory Uncertainty

Good question. Roughly speaking, aleatory uncertainty is the inherent variability of nature itself; it cannot be eliminated no matter how much data you collect. On the other hand, epistemic uncertainty arises from a lack of knowledge and can be reduced by improving models or adding more data.

Let's consider a car crash test. Even if you crash the same car under the same conditions, the yield stress of the steel sheet will vary slightly from lot to lot. This is natural variability—in other words, aleatory uncertainty. Even if you run the test a million times, this variability will never become zero.

On the other hand, the friction coefficient used in a crash simulation might be taken directly from literature values. We simply assume $\mu = 0.15$ because we don't know the true value, but we could get closer to it with precise experiments. This is epistemic uncertainty.

Exactly. Another important point is that the international standard for V&V (Verification and Validation), ASME V&V 20, requires that the two be clearly distinguished and quantified separately. If you mix them up, you won't know where there's room for improvement.

Aleatory uncertainty is handled using probability theory. You assign a probability density function $f_X(x)$ to the input variable $X$ and then find the statistical quantities of the output $Y = g(X)$.

The expected value and variance of the output $Y$ can be expressed as:

Exactly. However, in practice, $g(\cdot)$ is a huge CAE model, so analytical integration is impossible. That's why we use Monte Carlo sampling or Polynomial Chaos Expansion (PCE).

That's a sharp question. Actually, there are two schools of thought. The Bayesian subjective probability school believes it's acceptable to assign probability distributions to epistemic uncertainty as well. On the other hand, the non-probabilistic approach school argues that "assuming a probability distribution when data is insufficient is dangerous" and represents it using intervals or fuzzy sets.

In the interval representation, we only know the possible range of the variable $\theta$:

For example, the constant $C_\mu$ in a turbulence model has a range like $0.07 \leq C_\mu \leq 0.11$ depending on the literature. At present, we don't know which value within this range is the "true" one. This is epistemic uncertainty represented by an interval.

Exactly. That's why frameworks like P-boxes and Dempster-Shafer theory are needed to handle the two types together.

A P-box (Probability box) is a method of representing uncertainty using the upper and lower bounds of a cumulative distribution function (CDF). Its biggest advantage is that it can handle both aleatory and epistemic uncertainty simultaneously.

Let's think of a concrete example. Suppose the yield stress of a steel material follows a normal distribution $N(\mu, \sigma^2)$ (this is the aleatory uncertainty). However, the mean value $\mu$ is not known precisely due to insufficient lot data, and we only know $\mu \in [340, 360]$ MPa (this is the epistemic uncertainty). In this case, the CDF is not a single curve but a band-shaped region. This band is the P-box.

Exactly. P-boxes can also be used in cases with even less information (where even the distribution shape is unknown). For distribution-free P-boxes, upper and lower bounds are constructed using Chebyshev's inequality based only on the range of the mean and variance.

Dempster-Shafer (DS) theory is a framework for representing degrees of belief based on incomplete evidence. While probability theory assigns a single probability to each event, DS theory uses two measures—the belief function Bel and the plausibility function Pl—to represent the lower and upper bounds of probability.

Here's how it works. First, define a Basic Probability Assignment (BPA) $m$. This is a function representing the "mass" of each piece of evidence, assigned to subsets (excluding the empty set):

Then the belief and plausibility functions are defined as follows:

Perfect understanding. In practice, it's used, for example, when multiple experts hold different opinions. Expert A says, "The parameter is in the range $[2, 5]$," and Expert B says, "It's in the range $[4, 7]$." You assign a degree of confidence (BPA) to each opinion and combine them using Dempster's rule of combination. This allows you to quantify how much evidence supports each interval.

ASME V&V 20-2009 "Standard for Verification and Validation in Computational Fluid Dynamics and Heat Transfer" defines the model validation metric $E$ as follows:

Here, $S$ is the simulation result and $D$ is the experimental data. Then, the uncertainty constituting this $E$ is decomposed into the following three types:

• $u_{\text{num}}$ — Numerical uncertainty (discretization error, iteration error, etc.): Epistemic
• $u_{\text{input}}$ — Input uncertainty (variation in material constants, boundary conditions): Mixed aleatory and epistemic
• $u_D$ — Experimental uncertainty (measurement error, reproducibility): Mixed aleatory and epistemic

The key point is that if $|E| \leq u_{\text{val}}$, you cannot yet say "the model is validated"; you must state that it is "consistent within the validation uncertainty bounds." Furthermore, epistemic uncertainty is required to be explicitly identified as a part that can be reduced by future research.

The most standard method is Monte Carlo Sampling (MCS). You generate $N$ samples from the probability distribution of the input variables and execute the CAE model for each sample. A histogram of the outputs is naturally created.

The convergence rate is $O(1/\sqrt{N})$, which is independent of input dimension—that's its strength. However, to accurately capture the tails of the distribution, such as the 99th percentile, you'd want $N \geq 10{,}000$. In practice, it's common to use Latin Hypercube Sampling (LHS) to significantly reduce the required number of samples.

Correct. When epistemic uncertainty is represented by an interval, you use interval analysis to find the maximum and minimum values of the output:

For simple cases, you can just evaluate the endpoints if $g$ is monotonic. But if there are multiple epistemic parameters, you need to solve an optimization problem. Global optimization algorithms (like genetic algorithms or particle swarm optimization) are often used for this.

In that case, you evaluate the model for each focal element and construct the output-side Bel and Pl. If there are $m$ focal elements, you need at least $2m$ model evaluations (the upper and lower bounds of each focal element).