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.

翻译：McKean-Vlasov随机微分方程（MVSDEs）描述了动力学同时依赖于个体状态与总体分布的系统，在神经科学、金融学和流行病学中广泛出现。在许多应用中，系统仅被部分观测，当漂移系数与扩散系数均依赖于演化的经验律时，推断变得极具挑战性。本文针对此类部分观测的MVSDEs，提出了一个用于潜状态推断与参数估计的贝叶斯框架。我们将时间离散化与基于粒子的近似相结合，以构建可处理的似然估计量，并设计了两种粒子马尔可夫链蒙特卡洛（PMCMC）算法：一种单层PMCMC方法，以及一种多层PMCMC（MLPMCMC）方法，该方法在离散化层级间耦合粒子系统。多层构造产生了相关的似然估计，并以$(O(\varepsilon^{-6}))$的计算成本实现了$(O(\varepsilon^{2}))$的均方误差，优于单层方案$(O(\varepsilon^{-7}))$的复杂度。我们处理了完全律依赖的扩散设定，这是MVSDEs最一般的表述形式，并在标准正则性假设下提供了理论保证。数值实验证实了所提方法的效率与准确性。