Convergence complexity analysis of albert and chib's algorithm for Bayesian probit regression

Qian Qin, James P. Hobert

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

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 → 8 (with p fixed), and as p → 8 (with n fixed). Furthermore, the first computable bounds on the total variation distance to stationarity are byproducts of the asymptotic analysis.

Original languageEnglish (US)
Pages (from-to)2320-2347
Number of pages28
JournalAnnals of Statistics
Volume47
Issue number4
DOIs
StatePublished - Jan 1 2019
Externally publishedYes

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Keywords

  • Drift condition
  • Geometric ergodicity
  • High dimensional inference
  • Large p-small n
  • Markov chain Monte Carlo
  • Minorization condition

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