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.
Bibliographical noteFunding Information:
Received December 2017; revised April 2018. 1Supported by NSF Grant DMS-15-11945. Primary 60J05; secondary 65C05. Key words and phrases. Drift condition, geometric ergodicity, high dimensional inference, large p-small n, Markov chain Monte Carlo, minorization condition.
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- Drift condition
- Geometric ergodicity
- High dimensional inference
- Large p-small n
- Markov chain Monte Carlo
- Minorization condition