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.

翻译：为避免Metropolis-Hastings及其他接受-拒绝类算法在实证中的不良表现，实践者常通过试错法进行参数调节。本文在全变差距离与Wasserstein距离下建立了收敛速率的下界，以识别模拟失败的条件，从而避免此类参数设置，为算法调优提供指导。研究重点在于利用下界分析接受-拒绝类马尔可夫链的收敛复杂度，并对几何遍历马尔可夫链的收敛速率进行约束。该理论在多种场景中得到应用。例如，当目标密度随参数n集中时（如后验集中、拉普拉斯近似），研究表明：若调节参数未根据n进行精细设置，Metropolis-Hastings链的收敛速率可能任意缓慢。此结论通过以下案例得到验证：维度与样本量同步增长时采用Zellner g先验的贝叶斯逻辑回归，以及n趋于无穷时采用平坦先验的贝叶斯逻辑回归。