Modeling Uncertainty in Complex Systems
An Introduction to Probability Theory and Possibility Theory Frameworks
November 11, 2025
Introduction
In our previous post, we explored complex systems theory, with a focus on complex adaptive systems: systems made up of many interacting parts that evolve and adapt over time.
One concept we touched on briefly was uncertainty, including aleatory vs epistemic uncertainty. In this post, we expand on uncertainty. Specifically, we explore how uncertainty can be modeled, distinguishing between probabilistic and possibilistic approaches.
Recap of Uncertainty (if you’ve read our most recent post, skip to the next section)
Complex systems are inherently uncertain. This raises an interesting question: What does “uncertainty” mean?
There are two primary types of uncertainty – one tied to randomness, and the other to incomplete knowledge.
Aleatory uncertainty refers to inherent variability or noise – randomness that remains even when the governing rules of a system are known. For example, weather models include stochastic fluctuations in atmospheric motion that cannot be predicted precisely, even with perfect equations. This kind of uncertainty is irreducible: it reflects the system’s intrinsic randomness.
Epistemic uncertainty, on the other hand, arises from limits to what we know about the system itself. We may not know all the relevant variables, parameters, or feedback structures; our models may simplify or omit mechanisms that matter. This type of uncertainty is reducible in principle, because it stems from incomplete knowledge rather than randomness. But in practice, it can be difficult to model.
A related distinction, often attributed to economist Frank Knight, is between risk and uncertainty:
Risk corresponds to known unknowns – situations where possible outcomes are known and probabilities can be assigned (for example, rolling a die or estimating rainfall variability).
Uncertainty, in the stricter sense, refers to unknown unknowns – situations where we do not even know all the possible outcomes or mechanisms, much less their probabilities.
Aleatory uncertainty aligns with risk: we can describe inherent variability statistically.
Epistemic uncertainty aligns with true uncertainty: our ignorance prevents probabilistic specification.
Modeling Uncertainty: Probability and Possibility
This brings us to a discussion of techniques for modeling various forms of uncertainty.
Probability Theory
Probability theory quantifies uncertainty by assigning probabilities to possible outcomes or states of knowledge, whether based on observed frequencies (as in frequentist probability) or on degrees of belief about model parameters (as in Bayesian probability).
Frequentist probability is data-based and treats probability as the long-run frequency of outcomes under repeated trials or experiments. It deals with aleatory uncertainty – randomness intrinsic to the system – and there is an underlying assumption that the processes are assumed to be governed by known or estimable distributions. These methods are powerful when historical data or repeated observations allow for reliable estimation of distributions.
However, in the context of complex adaptive systems, many forms of uncertainty are epistemic: they stem not from randomness but from limited knowledge about structure, parameters, or causal mechanisms. Frequentist models capture randomness, but they do not capture ignorance. When we cannot specify a complete probability distribution (for instance, when future system states are only partially understood or data are sparse), there are alternatives to frequentist methods.
For example, take Bayesian probability, which extends probabilistic reasoning to situations of partial knowledge. Rather than relying solely on observed frequencies, Bayesian methods represent uncertainty about model parameters or hypotheses through prior probability distributions (formal expressions of what is believed or assumed about those parameters before observing data), which are then updated as new data becomes available. Through these priors, Bayesian models can incorporate expert judgment, theory-based expectations, or information from analogous systems, adjusting those beliefs iteratively as the system evolves.
However, a major criticism of Bayesian probability is that the priors are rather subjective, as they require assigning precise numerical values even when that prior knowledge is vague (e.g., when the information itself or the degree of confidence in it is unclear). They aren’t based on any rigorous empirical grounding or universally accepted method of selection.
Probabilistic models are a widely used tool for complex systems analysis. However, for the reasons outlined above (1. Frequentist probability accounts for aleatory uncertainty but not epistemic uncertainty, and 2, Bayesian probability introduces subjectivity through priors that often lack a rigorous empirical or mechanistic basis), they are limited in modeling the full scope of uncertainty present in complex adaptive systems.
Possibility Theory
This is where possibility theory enters the picture. Developed in the late twentieth century as a complement to probability theory, it offers a framework for representing uncertainty when probabilities cannot be meaningfully specified; that is, when our knowledge is too incomplete, imprecise, or ambiguous to define a full probability distribution. Possibility theory measures plausibility, the degree to which an event or state is consistent with available information. Rooted in fuzzy set theory, it represents uncertainty in terms of set membership rather than frequency: each state of the world is assigned a degree of membership between 0 and 1, indicating how compatible that state is with what is known. Possibility theory tolerates imprecision and allows incomplete knowledge to be expressed without forcing arbitrary numerical degrees of belief. Read more about the formalization of possibility theory here.
In practice, both probability theory and possibility are useful. It all depends on the context, and on the type of uncertainty being modeled.
~The GaiaVerse Team
Up Next
In our next blog post, we’ll explore how the word “sustainability” has been stretched across so many contexts that its precision has faded. What does sustainability really mean when treated with the precision that honors its true scope and rigor.