Resolving Structural Uncertainty in Stochastic Dynamical Systems via Ensemble-Based Bayesian Inference

Abstract

Recovering the interaction structure of complex stochastic dynamical systems from sparse time-series observations is an inherently ill-posed inverse problem. The posterior over network configurations is high-dimensional, highly degenerate, and shaped by the non-equilibrium nature of the available data. To address this, we propose a fully Bayesian, ensemble-based inference framework. Our approach grounds a continuous-time Stochastic Differential Equation (SDE) likelihood formulation---acting mathematically as an Onsager-Machlup quasi-potential for non-equilibrium paths---within a non-parametric Minimum Description Length (MDL) structural prior. This framework rigorously penalizes model complexity and avoids kinetic overfitting by forcing continuous parameters into a finite vocabulary of discrete categories. The joint posterior is optimized using a block-coordinate scheme combining sub-quadratic Markov Chain Monte Carlo (MCMC) topological sampling with deterministic Nelder-Mead parameter optimization. Using highly non-linear macroscopic population dynamics as a benchmark, we demonstrate that inference requires non-stationary transient data to resolve causal information. Furthermore, we prove that propagating structural uncertainty via ensemble-based posterior analysis produces significantly more robust and reliable predictions for downstream perturbation outcomes than traditional single-point (MAP) estimates. The framework is highly generalizable, capturing fundamental topological constraints even under severe generative model mismatches, offering a robust tool for uncertainty-aware inference in complex stochastic systems.