Convergence of Griddy Gibbs Sampling and other perturbed Markov chains

TL;DR

This paper establishes the theoretical convergence and stability of the Griddy Gibbs sampling method, providing guarantees for its invariant measure and its approximation to the true distribution.

Contribution

It proves the existence and uniqueness of the invariant measure for Griddy Gibbs and offers $L^p$ estimates on its distance to the true distribution, also analyzing perturbations of Markov chains.

Findings

Proves the invariant measure for Griddy Gibbs is unique under natural conditions.

Provides $L^p$ bounds on the distance between the invariant measure and the target distribution.

Analyzes the stability of invariant measures under small perturbations of transition probabilities.

Abstract

The Griddy Gibbs sampling was proposed by Ritter and Tanner (1992) as a computationally efficient approximation of the well-known Gibbs sampling method. The algorithm is simple and effective and has been used successfully to address problems in various fields of applied science. However, the approximate nature of the algorithm has prevented it from being widely used: the Markov chains generated by the Griddy Gibbs sampling method are not reversible in general, so the existence and uniqueness of its invariant measure is not guaranteed. Even when such an invariant measure uniquely exists, there was no estimate of the distance between it and the probability distribution of interest, hence no means to ensure the validity of the algorithm as a means to sample from the true distribution. In this paper, we show, subject to some fairly natural conditions, that the Griddy Gibbs method has a unique, invariant measure. Moreover, we provide $L^p$ estimates on the distance between this invariant measure and the corresponding measure obtained from Gibbs sampling. These results provide a theoretical foundation for the use of the Griddy Gibbs sampling method. We also address a more general result about the sensitivity of invariant measures under small perturbations on the transition probability. That is, if we replace the transition probability $P$ of any Monte Carlo Markov chain by another transition probability $Q$ where $Q$ is close to $P$, we can still estimate the distance between the two invariant measures. The distinguishing feature between our approach and previous work on convergence of perturbed Markov chain is that by considering the invariant measures as fixed points of linear operators on function spaces, we don't need to impose any further conditions on the rate of convergence of the Markov chain.