MEASURING SAMPLE QUALITY WITH DIFFUSIONS

• 成果类型: Article
• 署名作者: Gorham, Jackson; Duncan, Andrew B.; Vollmer, Sebastian J.; Mackey, Lester
• 署名单位: Stanford University; Imperial College London; University of Warwick; Microsoft
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/19-AAP1467
• 发表日期: 2019
• 页码: 2884-2928
• 关键词: multivariate normal approximation; steins method; variance reduction; exchangeable pairs; CONVERGENCE; langevin; equation; regression; METRICS; rates

摘要

Stein's method for measuring convergence to a continuous target distribution relies on an operator characterizing the target and Stein factor bounds on the solutions of an associated differential equation. While such operators and bounds are readily available for a diversity of univariate targets, few multivariate targets have been analyzed. We introduce a new class of characterizing operators based on Ito diffusions and develop explicit multivariate Stein factor bounds for any target with a fast-coupling Ito diffusion. As example applications, we develop computable and convergence-determining diffusion Stein discrepancies for log-concave, heavy-tailed and multimodal targets and use these quality measures to select the hyperparameters of biased Markov chain Monte Carlo (MCMC) samplers, compare random and deterministic quadrature rules and quantify bias-variance tradeoffs in approximate MCMC. Our results establish a near-linear relationship between diffusion Stein discrepancies and Wasserstein distances, improving upon past work even for strongly log-concave targets. The exposed relationship between Stein factors and Markov process coupling may be of independent interest.