With an aim to analyse the performance of Markov chain Monte Carlo (MCMC) methods, in our recent work we derive a large deviation principle (LDP) for the empirical measures of Metropolis-Hastings (MH) chains on a continuous state space. One of the (sufficient) assumptions for the LDP involves the existence of a particular type of Lyapunov function, and it was left as an open question whether or not such a function exists for specific choices of MH samplers. In this paper we analyse the properties of such Lyapunov functions and investigate their existence for some of the most popular choices of MCMC samplers built on MH dynamics: Independent Metropolis Hastings, Random Walk Metropolis, and the Metropolis-adjusted Langevin algorithm. We establish under what conditions such a Lyapunov function exists, and from this obtain LDPs for some instances of the MCMC algorithms under consideration. To the best of our knowledge, these are the first large deviation results for empirical measures associated with Metropolis-Hastings chains for specific choices of proposal and target distributions.

翻译：为分析马尔可夫链蒙特卡洛（MCMC）方法的性能，我们在近期工作中推导了连续状态空间上Metropolis-Hastings（MH）链经验测度的大偏差原理（LDP）。该LDP的（充分）假设之一涉及特定类型Lyapunov函数的存在性，而此类函数对特定MH采样方法是否存在此前仍是一个开放问题。本文分析了此类Lyapunov函数的性质，并考察了基于MH动力学构建的几种最常用MCMC采样方法（独立Metropolis-Hastings、随机游走Metropolis及Metropolis调整Langevin算法）中该函数的存在性。我们明确了此类Lyapunov函数存在的条件，并据此推导了所研究的MCMC算法部分实例的LDP。据我们所知，这是首次针对特定提议分布与目标分布组合下的Metropolis-Hastings链经验测度获得的大偏差结果。