Diagnosing parameter unidentifiability in stochastic dynamics with an application to flocking models

Abstract

Although flocking behaviour has been studied extensively in active matter physics and mathematical biology, methods for assessing parameter estimability remain underdeveloped. For models formulated as stochastic differential equations (SDEs), we develop a Hessian-based diagnostic framework that detects unidentifiable directions and indicates whether, and how, estimation precision can be improved. The framework utilizes the submartingale structure of the loss function's Hessian and tracks its minimum eigenvalue over time. Under convergence of the time-averaged Hessian, we prove that this eigenvalue grows linearly for identifiable models, but only sublinearly, as $o(t)$, for unidentifiable ones; for a specific type of parameter redundancy, this $o(t)$ bound is tightened to $o(\sqrt{t}\log t)$. Applying the framework to stochastic flocking models, we demonstrate that, in identifiable settings, the magnitude of the eigenvalue reflects the attainable estimation precision. We further uncover a physical mechanism of unidentifiability: as the system approaches the ordered state, the parameters become algebraically degenerate, even though the model contains no trivial redundancy. Identifiability therefore depends on the dynamical regime, not only on model structure. Our framework offers a practical tool for reliable parameter estimation, not only in flocking models but in SDE models across diverse fields.