  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.
