Control variates and Rao-Blackwellization for deterministic sweep Markov chains

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.