  We study control variate methods for Markov chain Monte Carlo (MCMC) in the
setting of deterministic sweep sampling using $K\geq 2$ transition kernels. New
variance reduction results are provided for MCMC averages based on sweeps over
general transition kernels, leading to a particularly simple control variate
estimator in the setting of deterministic sweep Gibbs sampling. Theoretical
comparisons of our proposed control variate estimators with existing literature
are made, and a simulation study is performed to examine the amount of variance
reduction in some example cases. We also relate control variate approaches to
approaches based on conditioning (or Rao-Blackwellization), and show that the
latter can be viewed as an approximation of the former. Our theoretical results
hold for Markov chains under standard geometric drift assumptions.
