Variational analysis of inference from dynamical systems

Understand

We introduce and study a variational framework for the analysis of empirical risk based inference for dynamical systems and ergodic processes.

• The analysis applies to a two-stage estimation procedure in which (i) the trajectory of an observed (but unknown) system is fit to a trajectory from a known reference system by minimizing cumulative per-state loss, and (ii) a parameter estimate is obtained from the initial state of the best fit reference trajectory.
• We show that the empirical risk of the best fit trajectory converges almost surely to a constant that can be expressed in a variational form as the minimal expected loss over dynamically invariant couplings (joinings) of the observed and reference systems.
• Moreover, we establish that the family of joinings minimizing the expected loss is convex and compact, and that it fully characterizes the asymptotic behavior of the estimated parameters, addressing both identifiability and misspecification.