Open Access
August 2019 Convergence complexity analysis of Albert and Chib’s algorithm for Bayesian probit regression
Qian Qin, James P. Hobert
Ann. Statist. 47(4): 2320-2347 (August 2019). DOI: 10.1214/18-AOS1749


The use of MCMC algorithms in high dimensional Bayesian problems has become routine. This has spurred so-called convergence complexity analysis, the goal of which is to ascertain how the convergence rate of a Monte Carlo Markov chain scales with sample size, $n$, and/or number of covariates, $p$. This article provides a thorough convergence complexity analysis of Albert and Chib’s [J. Amer. Statist. Assoc. 88 (1993) 669–679] data augmentation algorithm for the Bayesian probit regression model. The main tools used in this analysis are drift and minorization conditions. The usual pitfalls associated with this type of analysis are avoided by utilizing centered drift functions, which are minimized in high posterior probability regions, and by using a new technique to suppress high-dimensionality in the construction of minorization conditions. The main result is that the geometric convergence rate of the underlying Markov chain is bounded below 1 both as $n\rightarrow\infty$ (with $p$ fixed), and as $p\rightarrow\infty$ (with $n$ fixed). Furthermore, the first computable bounds on the total variation distance to stationarity are byproducts of the asymptotic analysis.


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Qian Qin. James P. Hobert. "Convergence complexity analysis of Albert and Chib’s algorithm for Bayesian probit regression." Ann. Statist. 47 (4) 2320 - 2347, August 2019.


Received: 1 December 2017; Revised: 1 April 2018; Published: August 2019
First available in Project Euclid: 21 May 2019

zbMATH: 07082288
MathSciNet: MR3953453
Digital Object Identifier: 10.1214/18-AOS1749

Primary: 60J05
Secondary: 65C05

Keywords: drift condition , geometric ergodicity , High dimensional inference , large $p$-small $n$ , Markov chain Monte Carlo , minorization condition

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.47 • No. 4 • August 2019
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