To sample from a given target distribution, Markov chain Monte Carlo (MCMC) sampling relies on constructing an ergodic Markov chain with the target distribution as its invariant measure. For any MCMC method, an important question is how to evaluate its efficiency. One approach is to consider the associated empirical measure and how fast it converges to the stationary distribution of the underlying Markov process. Recently, this question has been considered from the perspective of large deviation theory, for different types of MCMC methods, including, e.g., non-reversible Metropolis-Hastings on a finite state space, non-reversible Langevin samplers, the zig-zag sampler, and parallell tempering. This approach, based on large deviations, has proven successful in analysing existing methods and designing new, efficient ones. However, for the Metropolis-Hastings algorithm on more general state spaces, the workhorse of MCMC sampling, the same techniques have not been available for analysing performance, as the underlying Markov chain dynamics violate the conditions used to prove existing large deviation results for empirical measures of a Markov chain. This also extends to methods built on the same idea as Metropolis-Hastings, such as the Metropolis-Adjusted Langevin Method or ABC-MCMC. In this paper, we take the first steps towards such a large-deviations based analysis of Metropolis-Hastings-like methods, by proving a large deviation principle for the the empirical measures of Metropolis-Hastings chains. In addition, we characterize the rate function and its properties in terms of the acceptance- and rejection-part of the Metropolis-Hastings dynamics.

翻译：为了从给定的目标分布中采样，马尔可夫链蒙特卡洛（MCMC）方法依赖于构建一个以目标分布为不变测度的遍历马尔可夫链。对于任何MCMC方法，一个关键问题是如何评估其效率。一种方法是考虑相关的经验测度及其收敛到基础马尔可夫过程平稳分布的速度。近年来，这一问题已从大偏差理论的角度得到了研究，涉及不同类型的MCMC方法，例如有限状态空间上的非可逆Metropolis-Hastings、非可逆Langevin采样器、zig-zag采样器以及并行回火。这种基于大偏差的方法已被证明在分析现有方法和设计新型高效方法方面卓有成效。然而，对于更一般状态空间上的Metropolis-Hastings算法（MCMC采样的核心工具），由于基础马尔可夫链的动力学违反了现有用于证明马尔可夫链经验测度大偏差结果的条件，同类的技术尚未能用于分析其性能。这也延伸至基于与Metropolis-Hastings相同思想构建的方法，例如Metropolis调整的Langevin方法或ABC-MCMC。在本文中，我们通过证明Metropolis-Hastings链经验测度的大偏差原理，迈出了对Metropolis-Hastings类方法进行基于大偏差分析的第一步。此外，我们还根据Metropolis-Hastings动力学中的接受部分和拒绝部分，刻画了速率函数及其性质。