Abstract

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 approximations for a broad range of both continuous and discrete time 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 central limit theorems and variance estimation in the high-dimensional setting. 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. Co-authors: Jun Yang (University of Copenhagen) and Zhou Zhou(University of Toronto)