Recursive Maximum Likelihood Estimation for Interacting Particle Systems using Virtual Particles

TL;DR

This paper introduces a recursive maximum likelihood estimation method for interacting particle systems using virtual particles, enabling convergence to the mean-field system's stationary points through stochastic gradient techniques.

Contribution

It develops a novel stochastic gradient algorithm employing virtual particles for parameter estimation in mean-field interacting particle systems, with proven convergence properties.

Findings

Algorithms drive the surrogate gradient to zero as time approaches infinity.

Surrogate gradients converge uniformly to the true gradient of the mean-field system.

Numerical examples demonstrate the method's effectiveness on various models.

Abstract

We study recursive maximum likelihood estimation for stochastic interacting particle systems based on continuous observation of a single particle. In this regime, consistent estimation of the finite-particle log-likelihood is not possible, even in the limit as the number of particles $N\rightarrow\infty$ and the time horizon $t\rightarrow\infty$. We thus seek to optimise the stationary log-likelihood of the limiting mean-field system. We achieve this via a form of stochastic gradient estimate in continuous time, with stochastic gradient estimates computed using the continuous trajectory of the single observed particle, alongside a virtual interacting particle system and a virtual tangent interacting particle system, which are integrated with the online parameter estimate. For fixed numbers of real and virtual particles, we show that the resulting algorithms drive the gradient of a finite-particle surrogate objective to zero as $t\to\infty$. We then prove that, in the iterated limit $t\to\infty$ followed by $N,M\to\infty$, these surrogate gradients converge uniformly to the gradient of the stationary log-likelihood of the limiting mean-field system, yielding convergence to its stationary points. We illustrate the method on several numerical examples, including a model with quadratic confinement and interaction potentials, a model of interacting FitzHugh--Nagumo neurons, and a stochastic Kuramoto model.