April 2023 Complexity results for MCMC derived from quantitative bounds
Jun Yang, Jeffrey S. Rosenthal
Author Affiliations +
Ann. Appl. Probab. 33(2): 1459-1500 (April 2023). DOI: 10.1214/22-AAP1846


This paper considers how to obtain MCMC quantitative convergence bounds which can be translated into tight complexity bounds in high-dimensional settings. We propose a modified drift-and-minorization approach, which establishes generalized drift conditions defined in subsets of the state space. The subsets are called the “large sets”, and are chosen to rule out some “bad” states which have poor drift property when the dimension of the state space gets large. Using the “large sets” together with a “fitted family of drift functions”, a quantitative bound can be obtained which can be translated into a tight complexity bound. As a demonstration, we analyze several Gibbs samplers and obtain complexity upper bounds for the mixing time. In particular, for one example of Gibbs sampler which is related to the James–Stein estimator, we show that the number of iterations required for the Gibbs sampler to converge is constant under certain conditions on the observed data and the initial state. It is our hope that this modified drift-and-minorization approach can be employed in many other specific examples to obtain complexity bounds for high-dimensional Markov chains.

Funding Statement

This research is supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada.


The authors thank Jim Hobert and Gareth Roberts for helpful discussions, and two referees for their valuable comments which have significantly improved the quality of the paper. J.Y. also thanks Quan Zhou and Aaron Smith for helpful comments on the proof.


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Jun Yang. Jeffrey S. Rosenthal. "Complexity results for MCMC derived from quantitative bounds." Ann. Appl. Probab. 33 (2) 1459 - 1500, April 2023. https://doi.org/10.1214/22-AAP1846


Received: 1 February 2020; Revised: 1 November 2021; Published: April 2023
First available in Project Euclid: 21 March 2023

zbMATH: 1515.60262
MathSciNet: MR4564431
Digital Object Identifier: 10.1214/22-AAP1846

Primary: 60J20 , 60J22
Secondary: 65C05

Keywords: convergence complexity , drift and minorization , high dimensions , Markov chain Monte Carlo , mixing time

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 2 • April 2023
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