WASSERSTEIN-BASED METHODS FOR CONVERGENCE COMPLEXITY ANALYSIS OF MCMC WITH APPLICATIONS

• 成果类型: Article
• 署名作者: Qin, Qian; Hobert, James P.
• 署名单位: University of Minnesota System; University of Minnesota Twin Cities; State University System of Florida; University of Florida
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1673
• 发表日期: 2022
• 页码: 124-166
• 关键词: monte-carlo algorithms; markov-chain; quantitative bounds; gibbs samplers; rates; minorization; regression

摘要

Over the last 25 years, techniques based on drift and minorization (d&m) have been mainstays in the convergence analysis of MCMC algorithms. However, results presented herein suggest that d&m may be less useful in the emerging area of convergence complexity analysis, which is the study of how the convergence behavior of Monte Carlo Markov chains scales with sample size, n, and/or number of covariates, p. The problem appears to be that minorization can become a serious liability as dimension increases. Alternative methods of constructing convergence rate bounds (with respect to total variation distance) that do not require minorization are investigated. Based on Wasserstein distances and random mappings, these methods can produce bounds that are substantially more robust to increasing dimension than those based on d&m. The Wasserstein-based bounds are used to develop strong convergence complexity results for MCMC algorithms used in Bayesian probit regression and random effects models in the challenging asymptotic regime where n and p are both large.