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

Ning Ning, Amin Wu

TL;DR

This work tackles Bayesian latent-state inference and parameter estimation for partially observed McKean–Vlasov SDEs where both drift and diffusion depend on the evolving population distribution. It introduces two particle MCMC frameworks: a single-level PMCMC and a multilevel PMCMC (MLPMCMC) that couples particle systems across discretization levels to reduce variance. Theoretical results establish regularity conditions and provide MSE and computational-cost bounds, showing MLPMCMC achieves the optimal $O(\varepsilon^{-6})$ cost for $O(\varepsilon^{2})$ accuracy, a substantial improvement over single-level schemes. Numerical experiments on a 3D neuron mean-field model with distribution-dependent volatility corroborate the theory, demonstrating improved scalability and accurate inference for complex, law-dependent dynamics.

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.

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

TL;DR

This work tackles Bayesian latent-state inference and parameter estimation for partially observed McKean–Vlasov SDEs where both drift and diffusion depend on the evolving population distribution. It introduces two particle MCMC frameworks: a single-level PMCMC and a multilevel PMCMC (MLPMCMC) that couples particle systems across discretization levels to reduce variance. Theoretical results establish regularity conditions and provide MSE and computational-cost bounds, showing MLPMCMC achieves the optimal cost for accuracy, a substantial improvement over single-level schemes. Numerical experiments on a 3D neuron mean-field model with distribution-dependent volatility corroborate the theory, demonstrating improved scalability and accurate inference for complex, law-dependent dynamics.

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 at computational cost , improving on the 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.
Paper Structure (29 sections, 6 theorems, 139 equations, 3 figures, 2 tables, 3 algorithms)

This paper contains 29 sections, 6 theorems, 139 equations, 3 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Our Contributions
  4. Organization of the Paper
  5. Model Formulation
  6. Partially Observed McKean-Vlasov SDEs
  7. Time Discretization
  8. Methodology
  9. Single-level PMCMC Algorithm
  10. Single-level PMCMC Sampling Algorithm
  11. Multilevel PMCMC algorithm
  12. Bi-Level Telescoping Difference
  13. Coupled Posterior Distribution Framework
  14. Multilevel PMCMC Sampling Algorithm
  15. Theoretical Results
  16. ...and 14 more sections

Key Result

Theorem 4.1

Under Assumptions (Aass:1)-(Aass:3), let $\varphi\in\mathcal{C}_b^2(\Theta\times\mathbb{R}^{dT})\cap\mathcal{B}_b(\Theta\times\mathbb{R}^{dT})$ be a test function, there exists a constant $C<\infty$ such that for any multilevel configuration specified by integers $l_{\star}<L$ and particle/iteration where the error components are defined as follows:

Figures (3)

  • Figure F.1: Trace plots for the estimated parameters of the 3D Neuron Model using PMCMC.
  • Figure F.2: Trace plots for the estimated parameters of the 3D Neuron Model using MLPMCMC.
  • Figure F.3: Running mean of the increments for the estimated parameters of the 3D Neuron Model using MLPMCMC.

Theorems & Definitions (12)

  • Theorem 4.1: MSE Bound for Multilevel PMCMC Estimator
  • proof
  • Lemma S.1
  • proof
  • Lemma S.2
  • proof
  • Lemma S.3
  • proof
  • Lemma S.4
  • proof
  • ...and 2 more