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

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