An Exact Auxiliary Variable Gibbs Sampler for a Class of Diffusions. Issue 2 (9th October 2020)

• Record Type: Journal Article
• Authors: Wang, Qi; Rao, Vinayak; Teh, Yee Whye
• Is Part Of: Journal of computational and graphical statistics. Volume 30:Issue 2(2021)
• Page Start: 297
• Page End: 311
• Publication Date: 2020-10-09
• Subjects: Brownian motion -- Markov chain Monte Carlo -- Poisson process -- Stochastic differential equations; Mathematical statistics -- Data processing -- Periodicals; Mathematical statistics -- Graphic methods -- Periodicals; 519.50285
• DOI: 10.1080/10618600.2020.1816177
• Languages: English
• ISSNs: 1061-8600

Abstract

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. Supplementary materials for this article are available online.