# Recursive Maximum Likelihood Estimation for Interacting Particle Systems using Virtual Particles

arXiv: https://arxiv.org/abs/2605.00786 · HTML: https://mlsanity.com/paper/2605.00786/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 91 references that are in the map.
- First posted on arXiv 1 May 2026.

## Builds on (most cited first, 20 of 20)

- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [Trainability and Accuracy of Neural Networks: An Interacting Particle System Approach](https://mlsanity.com/paper/1805.00915/) (2018)
- [Mean-Field Langevin Dynamics and Energy Landscape of Neural Networks](https://mlsanity.com/paper/1905.07769/) (2019)
- [Nonparametric inference of interaction laws in systems of agents from trajectory data](https://mlsanity.com/paper/1812.06003/) (2018)
- [Learning interaction kernels in heterogeneous systems of agents from multiple trajectories](https://mlsanity.com/paper/1910.04832/) (2019)
- [Mean Field Game of Controls and An Application To Trade Crowding](https://mlsanity.com/paper/1610.09904/) (2016)
- [Stochastic Gradient Descent in Continuous Time](https://mlsanity.com/paper/1611.05545/) (2016)
- [Identifiability of interaction kernels in mean-field equations of interacting particles](https://mlsanity.com/paper/2106.05565/) (2021)
- [Stochastic Gradient Descent in Continuous Time: A Central Limit Theorem](https://mlsanity.com/paper/1710.04273/) (2017)
- [Maximum likelihood estimation of potential energy in interacting particle systems from single-trajectory data](https://mlsanity.com/paper/2007.11048/) (2020)
- [Parameter Estimation for the McKean-Vlasov Stochastic Differential Equation](https://mlsanity.com/paper/2106.13751/) (2021)
- [Online Maximum Likelihood Estimation of the Parameters of Partially Observed Diffusion Processes](https://mlsanity.com/paper/1611.00170/) (2016)
- [Two-Timescale Stochastic Gradient Descent in Continuous Time with Applications to Joint Online Parameter Estimation and Optimal Sensor Placement](https://mlsanity.com/paper/2007.15998/) (2020)
- [Continuous-time stochastic gradient descent for optimizing over the stationary distribution of stochastic differential equations](https://mlsanity.com/paper/2202.06637/) (2022)
- [A Forward Propagation Algorithm for Online Optimization of Nonlinear Stochastic Differential Equations](https://mlsanity.com/paper/2207.04496/) (2022)
- [Joint Online Parameter Estimation and Optimal Sensor Placement for the Partially Observed Stochastic Advection-Diffusion Equation](https://mlsanity.com/paper/2009.08693/) (2020)
- [Two-Timescale Stochastic Approximation for Bilevel Optimisation Problems in Continuous-Time Models](https://mlsanity.com/paper/2206.06995/) (2022)
- [Parameter Estimation for Partially Observed McKean-Vlasov Diffusions](https://mlsanity.com/paper/2411.06716/) (2024)
- [Efficient Online Learning in Interacting Particle Systems](https://mlsanity.com/paper/2602.20875/) (2026)

## Abstract (arXiv)

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 Nrightarrow∞ and the time horizon trightarrow∞. 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→∞. We then prove that, in the iterated limit t→∞ followed by N,M→∞, 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.