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

Ajay Jasra, Amin Wu

TL;DR

The paper tackles Bayesian parameter estimation for partially observed McKean-Vlasov diffusions with discrete observations, where the posterior is intractable. It develops particle MCMC methods and extends them with multilevel MLMC to reduce computational cost, providing convergence bounds and demonstrating a cost reduction from $\mathcal{O}(\epsilon^{-7})$ to $\mathcal{O}(\epsilon^{-6})$ for attaining $\mathcal{O}(\epsilon^2)$ mean-squared error. The approach is validated on Kuramoto-type models, showing good mixing and substantial efficiency gains as the discretization level increases. This work enables scalable Bayesian inference for complex mean-field diffusion models and highlights the benefits of MLMC in PMCMC contexts.

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.

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

TL;DR

The paper tackles Bayesian parameter estimation for partially observed McKean-Vlasov diffusions with discrete observations, where the posterior is intractable. It develops particle MCMC methods and extends them with multilevel MLMC to reduce computational cost, providing convergence bounds and demonstrating a cost reduction from to for attaining mean-squared error. The approach is validated on Kuramoto-type models, showing good mixing and substantial efficiency gains as the discretization level increases. This work enables scalable Bayesian inference for complex mean-field diffusion models and highlights the benefits of MLMC in PMCMC contexts.

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.

Paper Structure

This paper contains 21 sections, 6 theorems, 63 equations, 2 figures, 2 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Model
  4. Time Discretization
  5. Methodology
  6. Introduction
  7. Approximating the Laws
  8. Markov Chain Monte Carlo Approach
  9. Modified Posterior
  10. MCMC Method
  11. Bi-Level MCMC
  12. Set-Up
  13. Algorithm
  14. Final Algorithm
  15. Theory
  16. ...and 6 more sections

Key Result

Theorem 4.1

Assume (Aass:1-ass:3). Then for any $\varphi\in\mathcal{C}_b^2(\Theta\times\mathbb{R}^{dT})\cap\mathcal{B}_b(\Theta\times\mathbb{R}^{dT})$ there exists a $C<+\infty$ such that for any $(l_{\star},L,N_{l_{\star}},I_{l_{\star}},\dots,N_L,I_L)\in\mathbb{N}^{2(L-l_{\star})+4}$ with $l_{\star}<L$ where $\mathsf{A}_{l_{\star},L}=\{(l,q)\in\{l_{\star}+1,\dots,L\}:l\neq q\}$.

Figures (2)

  • Figure 1: The convergence plot of the PMCMC (left) and MLPMCMC (right) for the Kuramoto model.
  • Figure 2: The convergence plot of the PMCMC (left) and MLPMCMC (right) for the Modified Kuramoto model.

Theorems & Definitions (12)

  • Theorem 4.1
  • proof
  • Lemma A.1
  • proof
  • Lemma A.2
  • proof
  • Lemma A.3
  • proof
  • Lemma A.4
  • proof
  • ...and 2 more