Eigenfunction martingale estimators for interacting particle systems and their mean field limit

Abstract

We study the problem of parameter estimation for large exchangeable interacting particle systems when a sample of discrete observations from a single particle is known. We propose a novel method based on martingale estimating functions constructed by employing the eigenvalues and eigenfunctions of the generator of the mean field limit, linearized around the (unique) invariant measure of the mean field dynamics. We then prove that our estimator is asymptotically unbiased and asymptotically normal when the number of observations and the number of particles tend to infinity, and we provide a rate of convergence towards the exact value of the parameters. Finally, we present several numerical experiments which show the accuracy of our estimator and corroborate our theoretical findings, even in the case the mean field dynamics exhibit more than one steady states.

Neural parameter calibration for large-scale multiagent models

“…In the second approach, a model is realized as a discretized version of equations in continuous time and space (21,22); this approach opens models up to analysis using mean-field theory, stochastic analysis, statistical physics, and kinetic theory (23)(24)(25)(26). A large and sophisticated toolset of statistical methods exists to estimate parameters, interaction kernels, or network structures from data, such as maximum likelihood estimators (27)(28)(29)(30)(31), Markov chain Monte Carlo methods based on a Bayesian paradigm (32)(33)(34)(35), martingale estimators (36), estimation of active terms in ODE and PDE systems (see ref. 37 for a review), entropy maximization (38), and regression-based learning methods (39,40).…”