The widespread use of Markov Chain Monte Carlo (MCMC) methods for high-dimensional applications has motivated research into the scalability of these algorithms with respect to the dimension of the problem. Despite this, numerous problems concerning output analysis in high-dimensional settings have remained unaddressed. We present novel quantitative Gaussian approximation results for a broad range of MCMC algorithms. Notably, we analyse the dependency of the obtained approximation errors on the dimension of both the target distribution and the feature space. We demonstrate how these Gaussian approximations can be applied in output analysis. This includes determining the simulation effort required to guarantee Markov chain central limit theorems and consistent variance estimation in high-dimensional settings. We give quantitative convergence bounds for termination criteria and show that the termination time of a wide class of MCMC algorithms scales polynomially in dimension while ensuring a desired level of precision. Our results offer guidance to practitioners for obtaining appropriate standard errors and deciding the minimum simulation effort of MCMC algorithms in both multivariate and high-dimensional settings.

翻译：随着马尔可夫链蒙特卡洛（MCMC）方法在高维应用中的广泛使用，关于这些算法相对于问题维度的可扩展性研究已成为重要课题。尽管如此，高维场景下的输出分析仍存在诸多未解决的问题。本文针对广泛的MCMC算法提出了新颖的定量高斯近似结果，特别分析了所得近似误差对目标分布维度与特征空间维度的依赖关系。我们论证了这些高斯近似如何应用于输出分析，包括确定保证马尔可夫链中心极限定理所需模拟量、高维场景下的方差一致性估计等问题。通过建立终止准则的定量收敛界，我们证明在确保预定精度水平的前提下，一大类MCMC算法的终止时间随维度呈多项式规模增长。本研究为实践者提供了在多变量与高维场景下获取合适标准误差、确定MCMC算法最小模拟量的理论指导。