An Exact Auxiliary Variable Gibbs Sampler for a Class of Diffusions

TL;DR

This paper introduces an exact Gibbs sampling algorithm for simulating diffusion processes without discretization error, applicable to prior, posterior, and parameter inference, demonstrating superior performance on synthetic and real data.

Contribution

It recasts existing rejection sampling for diffusions as a latent variable model and develops an auxiliary variable Gibbs sampler that avoids time-discretization errors.

Findings

Achieves exact sampling of diffusion paths without discretization.

Demonstrates superior performance over competing methods.

Applicable to prior, posterior, and parameter inference in SDEs.

Abstract

Stochastic differential equations (SDEs) or diffusions are continuous-valued continuous-time stochastic processes widely used in the applied and mathematical sciences. Simulating paths from these processes is usually an intractable problem, and typically involves time-discretization approximations. We propose an exact Markov chain Monte Carlo sampling algorithm that involves no such time-discretization error. Our sampler is applicable to the problem of prior simulation from an SDE, posterior simulation conditioned on noisy observations, as well as parameter inference given noisy observations. Our work recasts an existing rejection sampling algorithm for a class of diffusions as a latent variable model, and then derives an auxiliary variable Gibbs sampling algorithm that targets the associated joint distribution. At a high level, the resulting algorithm involves two steps: simulating a random grid of times from an inhomogeneous Poisson process, and updating the SDE trajectory conditioned on this grid. Our work allows the vast literature of Monte Carlo sampling algorithms from the Gaussian process literature to be brought to bear to applications involving diffusions. We study our method on synthetic and real datasets, where we demonstrate superior performance over competing methods.