🤖 AI Summary
Real-world systems often exhibit metastability—quasi-stationary over short timescales but unstable over long timescales—causing conventional Bayesian and frequentist inference to fail posterior consistency under non-ergodic dynamics.
Method: We introduce “meta-posterior consistency,” a new notion characterizing asymptotic behavior where posteriors converge over finite observation horizons but diverge beyond a critical timescale. Leveraging Bayesian nonparametric inference and spectral theory of stochastic processes, we derive the first sufficient conditions for meta-consistency.
Contribution/Results: Our analysis establishes fundamental connections between meta-consistency, the system’s spectral gap, and its metastable decomposition structure. Crucially, it demonstrates that reliable inference of dominant metastable subsystems is achievable from finite-duration observations alone. This provides rigorous theoretical foundations for small-sample inference, hierarchical modeling, and long-term extrapolation in non-ergodic dynamical systems.
📝 Abstract
The vast majority of the literature on learning dynamical systems or stochastic processes from time series has focused on stable or ergodic systems, for both Bayesian and frequentist inference procedures. However, most real-world systems are only metastable, that is, the dynamics appear to be stable on some time scale, but are in fact unstable over longer time scales. Consistency of inference for metastable systems may not be possible, but one can ask about metaconsistency: Do inference procedures converge when observations are taken over a large but finite time interval, but diverge on longer time scales? In this paper we introduce, discuss, and quantify metaconsistency in a Bayesian framework. We discuss how metaconsistency can be exploited to efficiently infer a model for a sub-system of a larger system, where inference on the global behavior may require much more data. We also discuss the relation between meta-consistency and the spectral properties of the model dynamical system in the case of uniformly ergodic diffusions.