  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-in
period or when to terminate an MCMC sampling process, and opens the door for
safe parallel implementation. For theoreticians, it provides a powerful tool to
establish elegant and easily estimable bounds on the exact error of an MCMC
approximation at any finite number of iterates. A serendipitous observation
about the bias-correcting term leads us to introduce naturally available
control variates into the L-lag coupling estimators. In turn, this extension
enhances the coupled gains of L-lag coupling, because it results in more
efficient unbiased estimators, as well as a better bound on the total variation
error of MCMC iterations, albeit the gains diminish as L increases.
Specifically, the new upper bound is theoretically guaranteed to never exceed
the one given previously. We also argue that L-lag coupling represents a
coupling for the future, breaking from the coupling-from-the-past type of
perfect sampling, by reducing the generally unachievable requirement of being
perfect to one of being unbiased, a worthwhile trade-off for ease of
implementation in most practical situations. The theoretical analysis is
supported by numerical experiments that show tighter bounds and a gain in
efficiency when control variates are introduced.
