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A general methodology is presented for the construction and effective use ofcontrol variates for reversible MCMC samplers. The values of the coefficientsof the optimal linear combination of the control variates are computed, andadaptive, consistent MCMC estimators are derived for these optimalcoefficients. All methodological and asymptotic arguments are rigorouslyjustified. Numerous MCMC simulation examples from Bayesian inferenceapplications demonstrate that the resulting variance reduction can be quitedramatic.

Notes on Using Control Variates for Estimation with Reversible MCMC Samplers