Convergence rate analyses of random walk Metropolis-Hastings Markov chains on general state spaces have largely focused on establishing sufficient conditions for geometric ergodicity or on analysis of mixing times. Geometric ergodicity is a key sufficient condition for the Markov chain Central Limit Theorem and allows rigorous approaches to assessing Monte Carlo error. The sufficient conditions for geometric ergodicity of the random walk Metropolis-Hastings Markov chain are refined and extended, which allows the analysis of previously inaccessible settings such as Bayesian Poisson regression. The key technical innovation is the development of explicit drift and minorization conditions for random walk Metropolis-Hastings, which allows explicit upper and lower bounds on the geometric rate of convergence. Further, lower bounds on the geometric rate of convergence are also developed using spectral theory. The existing sufficient conditions for geometric ergodicity, to date, have not provided explicit constraints on the rate of geometric rate of convergence because the method used only implies the existence of drift and minorization conditions. The theoretical results are applied to random walk Metropolis-Hastings algorithms for a class of exponential families and generalized linear models that address Bayesian Regression problems.

翻译：随机游走Metropolis-Hastings马尔可夫链在一般状态空间上的收敛速率分析主要集中于建立几何遍历性的充分条件或分析混合时间。几何遍历性是马尔可夫链中心极限定理的关键充分条件，并为严格评估蒙特卡洛误差提供了方法。本文改进了随机游走Metropolis-Hastings马尔可夫链几何遍历性的充分条件并加以扩展，从而能够分析此前难以处理的场景（如贝叶斯泊松回归）。关键技术突破在于为随机游走Metropolis-Hastings算法开发了显式的漂移条件与极小化条件，从而给出了几何收敛速率的显式上界与下界。进一步地，利用谱理论推导了几何收敛速率的下界。此前现有的几何遍历性充分条件未能提供几何收敛速率的显式约束，因其采用的方法仅能证明漂移条件与极小化条件的存在性。本文将理论结果应用于一类指数族分布及广义线性模型（解决贝叶斯回归问题）的随机游走Metropolis-Hastings算法。