  To avoid poor empirical performance in Metropolis-Hastings and other
accept-reject-based algorithms practitioners often tune them by trial and
error. Lower bounds on the convergence rate are developed in both total
variation and Wasserstein distances in order to identify how the simulations
will fail so these settings can be avoided, providing guidance on tuning.
Particular attention is paid to using the lower bounds to study the convergence
complexity of accept-reject-based Markov chains and to constrain the rate of
convergence for geometrically ergodic Markov chains. The theory is applied in
several settings. For example, if the target density concentrates with a
parameter n (e.g. posterior concentration, Laplace approximations), it is
demonstrated that the convergence rate of a Metropolis-Hastings chain can be
arbitrarily slow if the tuning parameters do not depend carefully on n. This is
demonstrated with Bayesian logistic regression with Zellner's g-prior when the
dimension and sample increase together and flat prior Bayesian logistic
regression as n tends to infinity.
