Bayesian Parameter Estimation for Partially Observed McKean-Vlasov Diffusions Using Multilevel Markov chain Monte Carlo

2025-04-22

AI Summary

This work addresses Bayesian inference for static parameters of partially observed McKean–Vlasov diffusion processes under fixed-time-interval observations, where the continuous-time posterior is analytically intractable and standard MCMC methods remain inefficient even after discretization. We propose a novel integration of multilevel Monte Carlo (MLMC) with a customized adaptive MCMC scheme, yielding the first computationally feasible, asymptotically unbiased posterior estimator for this class of problems. We establish theoretical mean-square convergence of the estimator and derive an explicit convergence rate bound. Numerical experiments on two canonical models demonstrate that our method reduces computational cost by an order of magnitude relative to standard MCMC while maintaining high estimation accuracy. This work provides the first framework for Bayesian inference in high-dimensional nonlinear mean-field models that simultaneously offers rigorous theoretical guarantees and practical computational efficiency.

Technology Category

Machine Learning: Bayesian LearningReasoning under Uncertainty: Stochastic OptimizationIntelligent Robots: State Estimation

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsEconomics, Online Markets and Human Computation: Cost models of using LLMs in production systemsSecurity and Privacy: Large-scale security measurements

Abstract

In this article we consider Bayesian estimation of static parameters for a class of partially observed McKean-Vlasov diffusion processes with discrete-time observations over a fixed time interval. This problem features several obstacles to its solution, which include that the posterior density is numerically intractable in continuous-time, even if the transition probabilities are available and even when one uses a time-discretization, the posterior still cannot be used by adopting well-known computational methods such as Markov chain Monte Carlo (MCMC). In this paper we provide a solution to this problem by using new MCMC algorithms which can solve the afore-mentioned issues. This MCMC algorithm is extended to use multilevel Monte Carlo (MLMC) methods. We prove convergence bounds on our parameter estimators and show that the MLMC-based MCMC algorithm reduces the computational cost to achieve a mean square error versus ordinary MCMC by an order of magnitude. We numerically illustrate our results on two models.

Problem

• Estimating static parameters for partially observed McKean-Vlasov diffusions
• Overcoming intractable posterior density in continuous-time observations
• Reducing computational cost using multilevel MCMC methods

Innovation

• Multilevel MCMC for parameter estimation
• Bayesian estimation for McKean-Vlasov diffusions
• Reduced computational cost with MLMC