On the limitations of single-step drift and minorization in Markov chain convergence analysis

Qian Qin, James P. Hobert

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

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. 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 single-step d&m conditions are quite inadequate in high-dimensional settings.

Original languageEnglish (US)
Pages (from-to)1633-1659
Number of pages27
JournalAnnals of Applied Probability
Volume31
Issue number4
DOIs
StatePublished - Aug 2021

Bibliographical note

Funding Information:
The second author was supported by NSF Grant DMS-15-11945.

Publisher Copyright:
© Institute of Mathematical Statistics, 2021

Keywords

  • Convergence rate
  • Coupling
  • Geometric ergodicity
  • High-dimensional inference
  • Optimal bound
  • Quantitative bound
  • Renewal theory

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