Exact Gibbs sampling for stochastic differential equations with gradient drift and constant diffusion

2026-01-31

AI Summary

This work addresses the challenge of simulating sample paths for stochastic differential equations (SDEs) with gradient drift and unit diffusion coefficients under noisy observations, where existing methods often suffer from discretization bias or high sampling complexity. The authors propose an exact Gibbs sampling framework that enables unbiased path simulation without temporal discretization and naturally integrates Gaussian process tools to facilitate parameter inference. This approach achieves, for the first time, discretization-free MCMC sampling for a broad class of SDE models, handling both univariate and multivariate cases within a unified framework—without requiring rejection sampling or debiasing techniques. Empirical evaluations on synthetic and real-world data demonstrate clear advantages over particle MCMC methods, offering superior accuracy and computational efficiency.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Stochastic differential equations (SDEs) are an important class of time-series models, used to describe stochastic systems evolving in continuous time. Simulating paths from these processes, particularly after conditioning on noisy observations of the latent path, remains a challenge. Existing methods often introduce bias through time-discretization, require involved rejection sampling or debiasing schemes or are restricted to a narrow family of diffusions. In this work, we propose an exact Markov chain Monte Carlo (MCMC) sampling algorithm that is applicable to a broad subset of all SDEs with unit diffusion coefficient; after suitable transformation, this includes an even larger class of multivariate SDEs and most 1-d SDEs. We develop a Gibbs sampling framework that allows exact MCMC for such diffusions, without any discretization error. We demonstrate how our MCMC methodology requires only fairly straightforward simulation steps. Our framework can be extended to include parameter simulation, and allows tools from the Gaussian process literature to be easily applied. We evaluate our method on synthetic and real datasets, demonstrating superior performance to particle MCMC approaches.

Problem

• stochastic differential equations
• exact sampling
• Gibbs sampling
• diffusion processes
• conditioned path simulation

Innovation

• Exact Gibbs sampling
• Stochastic differential equations
• Discretization-free inference
• Markov chain Monte Carlo
• Gaussian process