  Over the last 25 years, techniques based on drift and minorization (d&m) have
been mainstays in the convergence analysis of MCMC algorithms. However, results
presented herein suggest that d&m may be less useful in the emerging area of
convergence complexity analysis, which is the study of how the convergence
behavior of Monte Carlo Markov chains scale with sample size, $n$, and/or
number of covariates, $p$. The problem appears to be that minorization can
become a serious liability as dimension increases. Alternative methods for
constructing convergence rate bounds (with respect to total variation distance)
that do not require minorization are investigated. Based on Wasserstein
distances and random mappings, these methods can produce bounds that are
substantially more robust to increasing dimension than those based on d&m. The
Wasserstein-based bounds are used to develop strong convergence complexity
results for MCMC algorithms used in Bayesian probit regression and random
effects models in the challenging asymptotic regime where $n$ and $p$ are both
large.
