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The recently proposed L-lag coupling for unbiased Markov chain Monte Carlo(MCMC) calls for a joint celebration by MCMC practitioners and theoreticians.For practitioners, it circumvents the thorny issue of deciding the burn-inperiod or when to terminate an MCMC sampling process, and opens the door forsafe parallel implementation. For theoreticians, it provides a powerful tool toestablish elegant and easily estimable bounds on the exact error of an MCMCapproximation at any finite number of iterates. A serendipitous observationabout the bias-correcting term leads us to introduce naturally availablecontrol variates into the L-lag coupling estimators. In turn, this extensionenhances the coupled gains of L-lag coupling, because it results in moreefficient unbiased estimators, as well as a better bound on the total variationerror of MCMC iterations, albeit the gains diminish as L increases.Specifically, the new upper bound is theoretically guaranteed to never exceedthe one given previously. We also argue that L-lag coupling represents acoupling for the future, breaking from the coupling-from-the-past type ofperfect sampling, by reducing the generally unachievable requirement of beingperfect to one of being unbiased, a worthwhile trade-off for ease ofimplementation in most practical situations. The theoretical analysis issupported by numerical experiments that show tighter bounds and a gain inefficiency when control variates are introduced.

Double Happiness: Enhancing the Coupled Gains of L-lag Coupling via Control Variates