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

Lower bounds on the rate of convergence for accept-reject-based Markov chains in Wasserstein and total variation distances