Convergence analysis of Markov chain Monte Carlo methods in high-dimensional statistical applications is increasingly recognized. In this paper, we develop general mixing time bounds for Metropolis-Hastings algorithms on discrete spaces by building upon and refining some recent theoretical advancements in Bayesian model selection problems. We establish sufficient conditions for a class of informed Metropolis-Hastings algorithms to attain relaxation times that are independent of the problem dimension. These conditions are grounded in high-dimensional statistical theory and allow for possibly multimodal posterior distributions. We obtain our results through two independent techniques: the multicommodity flow method and single-element drift condition analysis; we find that the latter yields a tighter mixing time bound. Our results and proof techniques are readily applicable to a broad spectrum of statistical problems with discrete parameter spaces.

翻译：马尔可夫链蒙特卡洛方法在高维统计应用中的收敛性分析日益受到关注。本文通过借鉴并改进贝叶斯模型选择问题中的最新理论进展，建立了离散空间上Metropolis-Hastings算法的通用混合时间界。我们为一类知情Metropolis-Hastings算法建立了充分条件，使其弛豫时间与问题维度无关。这些条件基于高维统计理论，并允许后验分布可能呈现多模态特性。我们通过两种独立技术获得结果：多商品流方法和单元素漂移条件分析；发现后者能给出更紧的混合时间界。我们的结果与证明技术可广泛应用于具有离散参数空间的各类统计问题。