The Annals of Statistics

Convergence complexity analysis of Albert and Chib’s algorithm for Bayesian probit regression

Qian Qin and James P. Hobert

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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.

Article information

Ann. Statist., Volume 47, Number 4 (2019), 2320-2347.

Received: December 2017
Revised: April 2018
First available in Project Euclid: 21 May 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 65C05: Monte Carlo methods

Drift condition geometric ergodicity high dimensional inference large $p$-small $n$ Markov chain Monte Carlo minorization condition


Qin, Qian; Hobert, James P. Convergence complexity analysis of Albert and Chib’s algorithm for Bayesian probit regression. Ann. Statist. 47 (2019), no. 4, 2320--2347. doi:10.1214/18-AOS1749.

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Supplemental materials

  • Supplementary material for “Convergence complexity analysis of Albert and Chib’s algorithm for Bayesian probit regression”. Section 6 provides some basic results on Hermitian matrices and truncated normal distributions. Section 7 gives some technical results, and the proofs for Corollary 5, Proposition 13, Proposition 16, Proposition 20 and Proposition 23.