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An Exact Auxiliary Variable Gibbs Sampler for a Class of Diffusions
Journal of Computational and Graphical Statistics ( IF 2.4 ) Pub Date : 2020-10-09 , DOI: 10.1080/10618600.2020.1816177
Qi Wang 1 , Vinayak Rao 1 , Yee Whye Teh 2
Affiliation  

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 an intractable problem, and usually involves time-discretization approximations. We propose an asymptotically exact Markov chain Monte Carlo sampling algorithm that involves no such time-discretization error. Our sampler is applicable both to the problem of prior simulation from an SDE, as well as posterior simulation conditioned on noisy observations. Our work recasts an existing rejection sampling algorithm for 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.

中文翻译:

一类扩散的精确辅助变量吉布斯采样器

随机微分方程 (SDE) 或扩散是在应用科学和数学科学中广泛使用的连续值连续时间随机过程。模拟来自这些过程的路径是一个棘手的问题,通常涉及时间离散化近似。我们提出了一种渐近精确的马尔可夫链蒙特卡罗采样算法,它不涉及这种时间离散化误差。我们的采样器既适用于来自 SDE 的先验模拟问题,也适用于以噪声观测为条件的后验模拟。我们的工作将现有的用于扩散的拒绝采样算法改写为潜在变量模型,然后推导出针对相关联合分布的辅助变量 Gibbs 采样算法。在高层次上,由此产生的算法包括两个步骤:模拟来自非均匀泊松过程的随机时间网格,并更新以该网格为条件的 SDE 轨迹。我们的工作允许将来自高斯过程文献的大量蒙特卡罗采样算法文献用于涉及扩散的应用。我们在合成数据集和真实数据集上研究我们的方法,在这些数据集上我们展示了优于竞争方法的性能。
更新日期:2020-10-09
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