A method of moments estimator for interacting particle systems and their mean field limit

TL;DR

This paper introduces a simple, moment-based estimator for learning parameters in stochastic interacting particle systems, validated through theoretical analysis and numerical experiments, effective in mean field and large data limits.

Contribution

It proposes a novel, easy-to-implement method for parameter estimation in interacting particle systems using moments and linear systems, with proven asymptotic unbiasedness.

Findings

Estimator is asymptotically unbiased with infinite data and particles.

Method accurately infers parameters in numerical experiments.

Approach is computationally simple, relying on moments and linear systems.

Abstract

We study the problem of learning unknown parameters in stochastic interacting particle systems with polynomial drift, interaction and diffusion functions from the path of one single particle in the system. Our estimator is obtained by solving a linear system which is constructed by imposing appropriate conditions on the moments of the invariant distribution of the mean field limit and on the quadratic variation of the process. Our approach is easy to implement as it only requires the approximation of the moments via the ergodic theorem and the solution of a low-dimensional linear system. Moreover, we prove that our estimator is asymptotically unbiased in the limits of infinite data and infinite number of particles (mean field limit). In addition, we present several numerical experiments that validate the theoretical analysis and show the effectiveness of our methodology to accurately infer parameters in systems of interacting particles.