Inference for Diffusion Processes via Controlled Sequential Monte Carlo and Splitting Schemes

2025-07-19

AI Summary

This paper addresses parameter inference for semi-linear stochastic differential equations (SDEs) under partial or full, noisy or noise-free discrete-time observations. We propose an efficient pseudo-likelihood estimation framework that balances statistical accuracy and computational feasibility. Methodologically, we integrate numerical splitting schemes with controlled sequential Monte Carlo to circumvent costly high-order numerical integration; leverage the Feynman–Kac flow to explicitly characterize the pseudo-likelihood’s normalization constant; and incorporate diffusion-bridge corrections to mitigate time-discretization bias, thereby substantially improving estimator consistency. The framework unifies treatment across diverse observation regimes and demonstrates robustness in both point estimation and Bayesian posterior inference. Numerical experiments and neuroscience case studies confirm its superior trade-off between computational efficiency and statistical precision—achieving high-accuracy inference without resorting to complex numerical solvers.

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Probabilistic Inference
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

We introduce an inferential framework for a wide class of semi-linear stochastic differential equations (SDEs). Recent work has shown that numerical splitting schemes can preserve critical properties of such types of SDEs, give rise to explicit pseudolikelihoods, and hence allow for parameter inference for fully observed processes. Here, under several discrete time observation regimes (particularly, partially and fully observed with and without noise), we represent the implied pseudolikelihood as the normalising constant of a Feynman--Kac flow, allowing its efficient estimation via controlled sequential Monte Carlo and adapt likelihood-based methods to exploit this pseudolikelihood for inference. The strategy developed herein allows us to obtain good inferential results across a range of problems. Using diffusion bridges, we are able to computationally reduce bias coming from time-discretisation without recourse to more complex numerical schemes which typically require considerable application-specific efforts. Simulations illustrate that our method provides an excellent trade-off between computational efficiency and accuracy, under hypoellipticity, for both point and posterior estimation. Application to a neuroscience example shows the good performance of the method in challenging settings.

Problem

• Inference for semi-linear SDEs under various observation regimes
• Reducing bias from time-discretization using diffusion bridges
• Balancing computational efficiency and accuracy in parameter estimation

Innovation

• Controlled Sequential Monte Carlo for pseudolikelihood estimation
• Splitting schemes preserving SDE properties
• Diffusion bridges reducing time-discretisation bias