Markov chain Monte Carlo (MCMC) methods are one of the most common classes of algorithms to sample from a target probability distribution $\pi$. A rising trend in recent years consists in analyzing the convergence of MCMC algorithms using tools from the theory of large deviations. One such result is a large deviation principle for algorithms of Metropolis-Hastings (MH) type (Milinanni & Nyquist, 2024), which are a broad and popular sub-class of MCMC methods. A central object in large deviation theory is the rate function, through which we can characterize the speed of convergence of MCMC algorithms. In this paper we consider the large deviation rate function from (Milinanni & Nyquist, 2024), of which we prove an alternative representation. We also determine upper and lower bounds for the rate function, based on which we design schemes to tune algorithms of MH type.

翻译：马尔可夫链蒙特卡罗（MCMC）方法是从目标概率分布 $\pi$ 中采样的最常用算法类别之一。近年来的一个新兴趋势是利用大偏差理论中的工具分析MCMC算法的收敛性。其中一项成果是针对Metropolis-Hastings（MH）型算法的大偏差原理（Milinanni & Nyquist, 2024），这是MCMC方法中一个广泛且流行的子类。大偏差理论的核心对象是率函数，通过该函数我们可以表征MCMC算法的收敛速度。本文研究了（Milinanni & Nyquist, 2024）提出的大偏差率函数，并证明了其替代表示形式。我们还确定了该率函数的上下界，并基于这些界限设计了MH型算法的调优方案。