Variational analysis of inference from dynamical systems

Abstract

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. The two-stage estimation framework and associated variational analysis apply to a broad family of empirical risk miminization procedures for dependent observations. To illustrate this, we apply variational analysis to the well studied problems of maximum likelihood and non-linear regression, and then undertake an extended analysis of system identification from quantized trajectories subject to noise, a problem of interest in dynamics, where the models themselves exhibit dynamical behavior across time.

Gibbs posterior convergence and the thermodynamic formalism

“…The setting and results of [36] and [37] are worthy of some discussion, as they may be considered frequentist analogues of the present work. Indeed, the setting of this previous work involves observations from an unknown ergodic system, a model family consisting of topological dynamical systems, and a loss function connecting the models to the observations, as in the present work.…”

“…Interestingly, this variational expression has been studied recently as part of an asymptotic analysis of estimators based on empirical risk minimization for dynamical systems [36,37]. Indeed, the solution set Θ ∞ of this ground state variational problem exactly characterizes the set of possible limits of parameter estimates that asymptotically minimize average empirical risk.…”

“…setting and considering statistical inference for dependent processes, including processes arising from dynamical systems. Statistical problems receiving recent attention in the context of dynamical systems include denoising (or filtering) [28,29], consistency of maximum likelihood estimation [34], forecasting and density estimation [21,44], empirical risk minimization [36,37], and data assimilation and uncertainty quantification [30]. For a survey of this area, see [35].…”

“…In this setting, one would like to understand what happens as β tends to infinity. In Section 3.7, we identify the limit of both V and Θ min as β tends to infinity in terms of variational problems considered in previous work [36].…”