How Chaos Theory Better Captures the Dynamics of Economic Uncertainty Compared to Bayesian Approaches

Economic uncertainty can be defined as “a situation in which the future economic environment is challenging to predict, and there is a high degree of risk or unknowns involved” (Mohamed 2024). Currently, 2 key approaches in modelling this uncertainty are Chaos Theory and Bayesian Inference. This paper argues that Chaos is better suited to understanding real-life economic uncertainty, whilst Bayesian Inference can still be supportive rather than an alternative.

Logistics map equation (Pierre François Verhulst)

The equation and its corresponding bifurcation diagram illustrates how simple equations can yield complex behaviors as the control parameter increases. This can cause the system the shift from stable equilibrium to periodic cycles and eventually chaos. For example, when r = 2.5, the system corresponds to 1 singular point for the population whereas when r = 3.6, the system exhibits chaos that arises purely internally. This supports the idea of real economies behaving more like nonlinear dynamical systems rather than simple probabilistic models. Hence demonstrating that unpredictable behaviours can emerge from within the system rather than external triggers or randomness. Therefore suggesting that economic uncertainty may arise endogenously due to speculations, herd behaviour, or fallen confidence ; not from external shocks.

On the other hand, the Bayesian inference assumes that the uncertainty that arises is epistemic, stemming from a lack of knowledge. This makes Bayesian models prone to oversimplifying economic systems that are non-linear. According to De Finetti (1974), “it is senseless to speak of the probability of an event unless we do so in relation to the body of knowledge possessed by a given person.” implying that Bayesian probabilities are subjective and based on a limited perception rather than the objective structure of the system.

This is the Bayes’ formula, while it is useful for updating probability as new events occur, a fixed prior probability of event A is required. This theorem also requires a complete set of possible outcomes, as well as assuming that all outcomes are known. Therefore in nonlinear economic systems or during black swan events,  these prior become invalid as the crises often arise outside of known past experiences; such as the 2008 housing crisis. Since the type of crash had never happened before, the probability of such an event B from a Bayesian view should be 0 as it was never part of the sample space. As a result, this will leave the entire formula undefined as the denominator P(B) is zero, leading to a meaningless prediction. This suggests that Bayesian inference must be based on stable data generation alongside a consistent environment over time. Therefore may not predict economic

systems accurately as they are driven by nonlinear feedback.

Furthermore, in real economic systems, small internal changes can trigger significant changes that are disproportionately large. This is due to a feature of chaos called “sensitivity to initial conditions.”. The Lyapunov exponent expresses this principle by measuring the rate at which 2 nearly identical trajectories separate over time. This mathematical insight implies that uncertainty is structural since when the lambda is larger than 0, the divergence can grow exponentially even if the initial state is known almost perfectly at time t=0. This

challenges the Bayesian assumption that uncertainty is epistemic and is caused by incomplete or imperfect knowledge.

Hsieh(1991) supports his view by arguing that financial markets may behave like nonlinear deterministic systems that “look random”. This is not because they are stochastic, but because they exhibit chaotic dynamics.  Therefore, even if a model is mathematically precise with extremely accurate data, a positive Lyapunov exponent can still ensure the predictions will diverge inevitably over time. Hence, making Chaos better suited for modelling economic uncertainty as it accounts for all events that emerge naturally within the economy.

But this is not all. While Bayesian models struggle with structural uncertainty, they remain valuable in modelling epistemic uncertainties and when modelling individuals’ learning.

As Farmer & Foley (2009) explain, “ ABMs potentially present a way to model the financial economy as a complex system…while taking human adaptation and learning into account.”. This supports the view that Bayesians shouldn’t be scared, but used within a chaotic framework. Hence not as an alternative to Chaos but as a complement.

This tree shows an ABM of a complex system called Watershed World, where there are chaotic agents ( climate and hydrology), interacting with Bayesian agents (farmers and markets). In which the Bayesian agents make decisions based on available data while chaotic elements introduce structural uncertainty. These interactions can lead to emergent outcomes, like floods or market shifts. Showing how minor decisions can trigger large macroeconomic effects.

In conclusion, Chaos theory is better suited for modelling economic uncertainty as it recognizes unpredictability as a built-in feature within the system rather than an unknown. Bayesian inference is still valuable, especially for modelling how individuals update their beliefs with new information. However, it is not enough to stand alone when modelling systems with a nonlinear dynamic due to its assumption of a stable and known system. Additionally, ABMs offer a hybrid approach, allowing it to be more capable of representing realistic economies where small decisions can lead to complex, unpredictable behaviours.

Citations:

Mohades, S. (2024, May 14). Economic uncertainty and business decisions – what is the link? Maastricht University.

Fernández-Díaz, A. (2023). Overview and perspectives of chaos theory and its applications in economics. Mathematics, 12(1), 92.

De Finetti, B. (1974). Theory of probability (Vol. 1, A. Machì & A. Smith, Trans.). Wiley.

Hsieh, D. A. (1991). Chaos and nonlinear dynamics: Application to financial markets. The Journal of Finance, 46(5), 1839–1877.

Farmer, J. D., & Foley, D. (2009). The economy needs agent-based modelling. Nature, 460(7256),