ON THE LIMITATIONS OF SINGLE-STEP DRIFT AND MINORIZATION IN MARKOV CHAIN CONVERGENCE ANALYSIS

• 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: 10.1214/20-AAP1628
• 发表日期: 2021
• 页码: 1633-1659
• 关键词: subgeometric rates; bounds; wasserstein; algorithm

摘要

Over the last three decades, there has been a considerable effort within the applied probability community to develop techniques for bounding the convergence rates of general state space Markov chains. Most of these results assume the existence of drift and minorization (d&m) conditions. It has often been observed that convergence rate bounds based on single-step d&m tend to be overly conservative, especially in high-dimensional situations. This article builds a framework for studying this phenomenon. It is shown that any convergence rate bound based on a set of d&m conditions cannot do better than a certain unknown optimal bound. Strategies are designed to put bounds on the optimal bound itself, and this allows one to quantify the extent to which a d&m-based convergence rate bound can be sharp. The new theory is applied to several examples, including a Gaussian autoregressive process (whose true convergence rate is known), and a Metropolis adjusted Langevin algorithm. The results strongly suggest that convergence rate bounds based on singlestep d&m conditions are quite inadequate in high-dimensional settings.