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

TL;DR

This paper introduces an exact Gibbs sampling algorithm for a broad class of stochastic differential equations, enabling bias-free simulation of paths conditioned on noisy observations, outperforming existing particle MCMC methods.

Contribution

The authors develop a novel Gibbs sampling framework that achieves exact MCMC sampling for SDEs with unit diffusion, avoiding discretization bias and simplifying simulation steps.

Findings

Outperforms particle MCMC in synthetic and real data.

Provides bias-free, exact sampling for a broad class of SDEs.

Easily extendable to parameter inference and Gaussian process tools.

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.