Markov chain Monte Carlo (MCMC) algorithms have played a significant role in statistics, physics, machine learning and others, and they are the only known general and efficient approach for some high-dimensional problems. The random walk Metropolis (RWM) algorithm as the most classical MCMC algorithm, has had a great influence on the development and practice of science and engineering. The behavior of the RWM algorithm in high-dimensional problems is typically investigated through a weak convergence result of diffusion processes. In this paper, we utilize the Mosco convergence of Dirichlet forms in analyzing the RWM algorithm on large graphs, whose target distribution is the Gibbs measure that includes any probability measure satisfying a Markov property. The abstract and powerful theory of Dirichlet forms allows us to work directly and naturally on the infinite-dimensional space, and our notion of Mosco convergence allows Dirichlet forms associated with the RWM chains to lie on changing Hilbert spaces. Through the optimal scaling problem, we demonstrate the impressive strengths of the Dirichlet form approach over the standard diffusion approach.

翻译：马尔可夫链蒙特卡洛（MCMC）算法在统计学、物理学、机器学习等领域发挥了重要作用，是已知解决某些高维问题的唯一通用且高效的方法。随机游走Metropolis（RWM）算法作为最经典的MCMC算法，对科学和工程的发展与实践产生了深远影响。高维问题中RWM算法的行为通常通过扩散过程的弱收敛性结果进行研究。本文利用狄利克雷形式的Mosco收敛性，分析大图上的RWM算法，其目标分布为满足马尔可夫性质的任意概率测度构成的吉布斯测度。狄利克雷形式抽象而强大的理论使我们能够直接在无穷维空间上自然开展工作，而本文提出的Mosco收敛性概念允许与RWM链相关的狄利克雷形式存在于变化的希尔伯特空间上。通过最优尺度问题，我们展示了狄利克雷形式方法相较于标准扩散方法的显著优势。