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Activity Number: 41 - Topics on Bayesian Inference
Type: Contributed
Date/Time: Monday, August 3, 2020 : 10:00 AM to 2:00 PM
Sponsor: IMS
Abstract #312668
Title: Limitations of Single-Step Drift and Minorization in Markov Chain Convergence Analysis
Author(s): Qian Qin* and James P. Hobert
Companies: University of Minnesota and University of Florida
Keywords: coupling; quantitative bound; geometric ergodicity; high-dimensional; optimal bound; convergence rate
Abstract:

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. We build a frame-work 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. The results strongly suggest that convergence rate bounds based on single-step d&m conditions are quite inadequate in high-dimensional settings.


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