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

Radu V. Craiu, X. Meng
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引用次数: 5

Abstract

The recently proposed L-lag coupling for unbiased MCMC \citep{biswas2019estimating, jacob2020unbiased} 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 iteration, 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 MCMC approximation at any finite number of iteration. A serendipitous observation about the bias correcting term led 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 with the numerical value of L. Specifically, the new bound is theoretically guaranteed to never exceed the one given previously. We also argue that L-lag coupling represents a long sought after coupling for the future, breaking a logjam of the coupling-from-the-past type of perfect sampling, by reducing the generally un-achievable requirement of being \textit{perfect} to being \textit{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.
双喜:通过控制变量增强l -滞后耦合的耦合增益。
最近提出的无偏MCMC \citep{biswas2019estimating, jacob2020unbiased}的l滞后耦合需要MCMC实践者和理论家共同庆祝。对于实践者来说,它规避了决定老化期或何时终止MCMC迭代的棘手问题,并为安全的并行实现打开了大门。对于理论家来说,它提供了一个强大的工具,可以在任意有限次迭代下建立精确误差的优雅且易于估计的边界。对偏差校正项的偶然观察使我们将自然可用的控制变量引入到l滞后耦合估计器中。反过来,这种扩展增强了L-lag耦合的耦合增益,因为它产生了更有效的无偏估计器以及MCMC迭代的总变异误差的更好的界,尽管增益随着l的数值而减小。具体来说,理论上保证新的界永远不会超过先前给出的界。我们还认为,l滞后耦合代表了未来长期追求的耦合,通过将通常无法实现的要求从\textit{完美}降低到\textit{不偏不倚},打破了过去完美采样类型耦合的僵局,这是在大多数实际情况下易于实现的值得权衡的代价。数值实验结果支持了理论分析,表明在引入控制变量时边界更紧,效率更高。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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