A Bayesian Adaptive Mixture Framework with Convergence Guarantees for Detecting Causal Emergent Features in Multi-Regime Time Series

Introduction: Real-world time series often arise from mixtures of multiple causal generating processes, each subject to sudden emergence or parameter changes—phenomena that challenge classical models assuming a single dynamic process. While recent work introduces the Adaptive Logistic Model (ALM) to address this using nonlinear least squares, it lacks uncertainty quantification and formal convergence guarantees. This paper develops a Bayesian adaptive mixture framework that detects emergent dynamics, quantifies uncertainty, and provides theoretical convergence guarantees.

Methods: We propose a hierarchical Bayesian model where observations are generated by a mixture of nonlinear dynamic processes (e.g., logistic growth/decay) with unknown regime allocations and change points. Dirichlet process priors automatically infer the number of regimes, while Gaussian process priors capture parameter evolution. We derive a Markov chain Monte Carlo sampler and prove its geometric ergodicity under regularity conditions, establishing rates of posterior convergence. The framework outputs posterior probabilities for change-point locations and regime parameters, enabling uncertainty-aware forecasting and causal interpretation.

Results: Simulation studies demonstrate that the Bayesian framework accurately recovers true change points and regime parameters, with coverage rates matching nominal levels—an improvement over point-estimate methods. In empirical applications, the model detects known regime shifts in U.S. GDP growth (recessions), river flow data (flood events), and COVID-19 case counts (new variants), with change points aligning closely with documented external events. Forecast accuracy matches or exceeds the original ALM while providing uncertainty intervals.

Conclusions: This paper provides the first Bayesian extension of the Adaptive Logistic Model with formal convergence guarantees, enabling reliable detection of causal emergent features in multi-regime time series. The framework offers a mathematically rigorous tool for economists, hydrologists, and epidemiologists to understand when and why dynamics change.