Drift, Minorization, and Hitting Times

TL;DR

This paper proposes replacing the traditional minorization condition in the drift-and-minorization method for Markov chain convergence analysis with a hitting condition based on a single chain, simplifying the process.

Contribution

It introduces a new hitting condition as an alternative to the minorization condition, making convergence bounds easier to derive and apply.

Findings

Hitting condition can replace minorization in convergence bounds

Simplifies the analysis of Markov chain convergence

Provides comparable bounds using the new approach

Abstract

The "drift-and-minorization" method, introduced and popularized in (Rosenthal, 1995; Meyn and Tweedie, 1994; Meyn and Tweedie, 2012), remains the most popular approach for bounding the convergence rates of Markov chains used in statistical computation. This approach requires estimates of two quantities: the rate at which a single copy of the Markov chain "drifts" towards a fixed "small set", and a "minorization condition" which gives the worst-case time for two Markov chains started within the small set to couple with moderately large probability. In this paper, we build on (Oliveira, 2012; Peres and Sousi, 2015) and our work (Anderson, Duanmu, Smith, 2019a; Anderson, Duanmu, Smith, 2019b) to replace the "minorization condition" with an alternative "hitting condition" that is stated in terms of only one Markov chain, and illustrate how this can be used to obtain similar bounds that can be easier to use.