Recursive Maximum Likelihood Estimation for Interacting Particle Systems using Virtual Particles

2026-05-01

AI Summary

This work addresses the challenge of consistently estimating the log-likelihood of finite-particle systems from continuous single-particle observations, a task that becomes intractable when directly optimizing as both the number of particles and time horizon grow large. To overcome this, the authors propose constructing an auxiliary particle system along with its tangent process and develop a continuous-time stochastic gradient algorithm that optimizes a surrogate objective approximating the stationary log-likelihood under the mean-field limit. Theoretical analysis establishes that the surrogate gradient converges uniformly to the true gradient in the joint limit of infinite time and particle count, thereby ensuring consistent parameter estimation. The algorithm drives the gradient to zero over long time horizons and demonstrates empirical efficacy on benchmark models including quadratic potentials, FitzHugh–Nagumo neurons, and the stochastic Kuramoto model.

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.

Problem

• interacting particle systems
• maximum likelihood estimation
• mean-field limit
• continuous observation
• parameter estimation

Innovation

• recursive maximum likelihood
• interacting particle systems
• virtual particles
• mean-field limit
• stochastic gradient estimation