  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 (1993)
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.
