Convergence rate and concentration inequalities for Gibbs sampling in high dimension

TL;DR

This paper analyzes the convergence rate and concentration inequalities of Gibbs sampling in high-dimensional settings, providing explicit bounds and Gaussian concentration results under Dobrushin's condition.

Contribution

It offers new explicit estimates of convergence rates and Gaussian concentration inequalities for Gibbs sampling in high-dimensional spaces under Dobrushin's condition.

Findings

Explicit exponential convergence rate estimates

Gaussian concentration inequalities for empirical means

Applicability under Dobrushin's uniqueness condition

Abstract

The objective of this paper is to study the Gibbs sampling for computing the mean of observable in very high dimension - a powerful Markov chain Monte Carlo method. Under the Dobrushin's uniqueness condition, we establish some explicit and sharp estimate of the exponential convergence rate and prove some Gaussian concentration inequalities for the empirical mean.