Bayesian Inference for Partially Observed McKean-Vlasov SDEs with Full Distribution Dependence

2026-01-18

AI Summary

This work proposes the first Bayesian inference framework for partially observed McKean–Vlasov stochastic differential equations (MVSDEs) whose drift and diffusion coefficients depend on the full law of the process, enabling joint estimation of latent states and parameters. By combining time discretization with particle filtering, a computable likelihood estimator is constructed, and both single-level and multi-level particle Markov chain Monte Carlo (PMCMC/MLPMCMC) algorithms are developed. The approach innovatively introduces a multi-level particle system with coupling across discretization levels, and under standard regularity conditions, it is shown to converge while reducing computational complexity from \(O(\varepsilon^{-7})\) to \(O(\varepsilon^{-6})\). Numerical experiments demonstrate the method’s significant gains in both accuracy and efficiency.

Technology Category

• Machine Learning: Bayesian Learning
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

McKean-Vlasov stochastic differential equations (MVSDEs) describe systems whose dynamics depend on both individual states and the population distribution, and they arise widely in neuroscience, finance, and epidemiology. In many applications the system is only partially observed, making inference very challenging when both drift and diffusion coefficients depend on the evolving empirical law. This paper develops a Bayesian framework for latent state inference and parameter estimation in such partially observed MVSDEs. We combine time-discretization with particle-based approximations to construct tractable likelihood estimators, and we design two particle Markov chain Monte Carlo (PMCMC) algorithms: a single-level PMCMC method and a multilevel PMCMC (MLPMCMC) method that couples particle systems across discretization levels. The multilevel construction yields correlated likelihood estimates and achieves mean square error $(O(\varepsilon^2))$ at computational cost $(O(\varepsilon^{-6}))$, improving on the $(O(\varepsilon^{-7}))$ complexity of single-level schemes. We address the fully law-dependent diffusion setting which is the most general formulation of MVSDEs, and provide theoretical guarantees under standard regularity assumptions. Numerical experiments confirm the efficiency and accuracy of the proposed methodology.

Problem

• McKean-Vlasov SDEs
• partial observation
• distribution-dependent diffusion
• Bayesian inference
• latent state estimation

Innovation

• McKean-Vlasov SDEs
• Bayesian inference
• particle MCMC
• multilevel Monte Carlo
• fully distribution-dependent diffusion