If we wish to estimate efficiently the expectation of an arbitrary function on the basis of the output of a Gibbs sampler, which is better: deterministic or random sweep? In each case we calculate the asymptotic variance of the empirical estimator, the average of the function over the output, and determine the minimal asymptotic variance for estimators that use no information about the underlying distribution. The empirical estimator has noticeably smaller variance for deterministic sweep. The variance bound for random sweep is in general smaller than for deterministic sweep, but the two are equal if the target distribution is continuous. If the components of the target distribution are not strongly dependent, the empirical estimator is close to efficient under deterministic sweep, and its asymptotic variance approximately doubles under random sweep.
"Information bounds for Gibbs samplers." Ann. Statist. 26 (6) 2128 - 2156, December 1998. https://doi.org/10.1214/aos/1024691464