Briefing: Chaotic Bayesian Inference for Black Swan Events

Source: "Chaotic Bayesian Inference: Strange Attractors as Risk Models for Black Swan Events" by Crystal Rust (September 11, 2025)

I. Executive Summary

This paper introduces a novel dual‑model framework for risk modeling that addresses the challenge of anticipating rare, high‑impact events, often termed "Black Swans." Traditional risk models, which often assume Gaussian behavior, systematically underestimate these extreme events. The proposed framework embeds chaotic dynamics (specifically Lorenz and Rössler attractors) directly into Bayesian inference, utilizing fat‑tailed priors. This approach allows for the natural emergence of "endogenous volatility clustering, power‑law tails, and extreme events." The framework offers "complementary views": Model A focuses on geometric stability and baseline dynamics, while Model B emphasizes statistical recurrence and volatile bursts, employing Fibonacci‑based diagnostics to detect rare‑event patterns across multiple timescales. This work provides a "constructive mathematical link to Taleb's Black Swan and antifragility framework" and has broad implications beyond finance, including climate, epidemiology, and infrastructure networks.

II. Main Themes and Key Ideas

A. The Problem with Traditional Risk Models and the Black Swan Concept:

• Traditional financial risk models, such as GARCH‑type models, assume "Gaussian or near‑Gaussian returns" and "systematically underestimate the likelihood and impact of extreme events."
• Nassim Nicholas Taleb's Black Swan framework argues that "rare, high‑impact shocks are not outliers but intrinsic to complex systems." The paper aims to "operationalize these ideas within a Bayesian inference framework."

B. Chaotic Attractors and Bayesian Inference for Black Swans:

• The core innovation is to "embed chaotic dynamics into Bayesian inference," where "posterior distributions live on Lorenz- or Rössler‑type attractors."
• This construction, combined with "heavy‑tailed priors," generates "probability structures that exhibit endogenous volatility clustering, bursts, and fat tails."
• Chaotic attractors provide a geometric and dynamical basis for understanding how extreme events "emerge naturally from the dynamics."

C. Dual‑Model Framework (Model A and Model B): The framework consists of two complementary models:

• Model A: Poincaré–Mahalanobis (Geometric Stability)
• Focus: "Anchors inference in the attractor's geometry by evaluating how well simulated trajectories reproduce the observed Poincaré section." It recovers "stable system structure, serving as a baseline reference."
• Methodology: Uses Poincaré sections (intersections of trajectories with a hyperplane) and Mahalanobis distance to compare simulated and observed attractor geometry.
• Role: Provides a "stable geometric anchor" and recovers "canonical parameter regimes" for "baseline system stability."
• Model B: Correlation–Integral with Fibonacci Diagnostics (Volatility Clustering and Rare Events)
• Focus: "Emphasizes recurrence statistics and volatility bursts, using correlation integrals and recursive diagnostic windows to capture rare‑event structure." It "highlights tail risks."
• Methodology: Employs the correlation integral to characterize attractor dimension and integrates "volatility diagnostics based on recursive Fibonacci windows (21, 34, 55, 89)" to detect bursts across multiple timescales.
• Role: "Re‑weights inference toward tail‑sensitive statistics," capturing "rare‑event clustering" and providing a fundamentally different view of the system's dynamics compared to Model A.

D. Fat‑Tailed Priors and Rare‑Event Sensitivity:

• A "central feature of rare‑event modeling is the presence of fat‑tailed (heavy‑tailed) probability distributions."
• The study "adopt[s] fat‑tailed priors (e.g., Student‑t or power‑law families) to represent the underlying uncertainty in parameters subject to rare shocks."
• This directly links "the statistical signature of Black Swans in probability space (fat tails) with their dynamical expression in phase space (chaotic attractors)."

E. Experimental Validation (Lorenz‑Lorenz and Lorenz‑Rössler):

• Lorenz‑Lorenz Experiment:
• Applied both models to the Lorenz attractor to "isolate the methodological difference."
• Model A "reproduced the butterfly geometry" and concentrated posteriors around "canonical values," confirming its role in "baseline attractor structure."
• Model B "emphasized volatility bursts," identifying subsequences exceeding multi‑scale diagnostic thresholds and aligning accepted proposals with burst clustering.
• Key takeaway: Diagnostic weighting fundamentally shifts posterior emphasis from "geometric stability (Model A) to rare‑event clustering (Model B)."
• Lorenz‑Rössler Experiment:
• Model A applied to Lorenz (as a "stable geometric anchor") and Model B applied to the Rössler attractor.
• Demonstrated "generality": the Correlation–Integral framework with Fibonacci diagnostics "transfers across distinct chaotic systems," capturing "volatility structure in the Rössler attractor" with its characteristic spiral geometry and burst dynamics.

F. Superiority of Fibonacci Diagnostics:

• Compared to "conventional diagnostics" like rolling volatility estimates and standardized returns using fixed windows, Fibonacci diagnostics "capture bursts across multiple scales where fixed windows failed." Fixed windows "smoothed over short‑lived bursts, limiting sensitivity to clustering."

III. Broader Impacts and Future Directions

A. Wider Implications Beyond Finance:

• The framework has "wider implications" for other complex systems vulnerable to "rare but catastrophic shocks," including:
• Climate: "tipping points in ice sheets and ocean circulation."
• Epidemiology: "sudden outbreak accelerations."
• Infrastructure networks: "cascading breakdowns" in power grids or supply chains.
• It offers a "more realistic way to model cascading failures, sudden regime shifts, and systemic vulnerabilities" by making "extreme events an intrinsic part of the system's probability structure."

B. Implications for Risk Analysis:

• Complementary insights: Model A provides "interpretability for conventional practice" by recovering stable geometry, while Model B "foregrounds low‑probability but high‑impact deviations."
• Enhanced risk management: The dual perspective offers a "richer basis for anticipating rare events and Black Swan dynamics." The D‑trace in Model B provides a "volatility‑sensitive diagnostic stream."

C. Future Work:

• Calibration to "real financial data."
• Extension to "higher‑dimensional attractors" (e.g., coupled Lorenz–Rössler systems).
• Integration with "agent‑based models for systemic risk."
• Aim to determine if the dual approach can provide "early‑warning signals of rare, destabilizing events in practice."

IV. Key Quotes

• "We introduce a novel risk modeling framework in which chaotic attractors define the geometry of posterior distributions in Bayesian inference. By combining fat‑tailed priors with Lorenz and Rössler attractor dynamics, we show how endogenous volatility clustering, power‑law tails, and extreme events emerge naturally from the dynamics."
• "This construction provides a constructive mathematical link to Taleb's Black Swan and antifragility framework."
• "Traditional models often underestimate these risks by assuming smooth or Gaussian behavior. By contrast, our framework makes extreme events an intrinsic part of the system's probability structure, offering a more realistic way to model cascading failures, sudden regime shifts, and systemic vulnerabilities."
• "Taleb's Black Swan framework emphasizes that such rare, high‑impact shocks are not outliers but intrinsic to complex systems."
• "Model A (Poincaré–Mahalanobis) anchors inference in the attractor's geometry by evaluating how well simulated trajectories reproduce the observed Poincaré section."
• "Model B (Correlation–Integral with Fibonacci diagnostics) emphasizes recurrence statistics and volatility bursts, using correlation integrals and recursive diagnostic windows to capture rare‑event structure."
• "This construction directly links the statistical signature of Black Swans in probability space (fat tails) with their dynamical expression in phase space (chaotic attractors)."
• "The Lorenz–Lorenz comparison shows how diagnostic weighting shifts posterior emphasis from geometric stability (Model A) to rare‑event clustering (Model B)."
• "Fibonacci diagnostics detect bursts across multiple scales where fixed windows failed."