The theory of two projections is utilized to study two-component Gibbs samplers. Through this theory, previously intractable problems regarding the asymptotic variances of two-component Gibbs samplers are reduced to elementary matrix algebra exercises. It is found that in terms of asymptotic variance, the two-component random-scan Gibbs sampler is never much worse, and could be considerably better than its deterministic-scan counterpart, provided that the selection probability is appropriately chosen. This is especially the case when there is a large discrepancy in computation cost between the two components. The result contrasts with the known fact that the deterministic-scan version has a faster convergence rate, which can also be derived from the method herein. On the other hand, a modified version of the deterministic-scan sampler that accounts for computation cost can outperform the random-scan version.

翻译：利用双投影理论研究双分量吉布斯采样器。通过该理论，先前关于双分量吉布斯采样器渐近方差的棘手问题简化为初等矩阵代数运算。研究发现，在渐近方差方面，若适当选择选择概率，双分量随机扫描吉布斯采样器的性能绝不会显著劣于确定性扫描采样器，甚至可能大幅优于后者。当两个分量的计算代价差异较大时，这一优势尤为明显。该结果与确定性扫描版本具有更快收敛速度的已知结论形成对比（该收敛速度也可通过本文方法推导得出）。另一方面，考虑计算代价的改进型确定性扫描采样器可能超越随机扫描版本。