Gibbs sampling repeatedly samples from the conditional distribution of one variable, x_i, given other variables, either choosing i randomly, or updating sequentially using some systematic or random order. When x_i is discrete, a Gibbs sampling update may choose a new value that is the same as the old value. A theorem of Peskun indicates that, when i is chosen randomly, a reversible method that reduces the probability of such self transitions, while increasing the probabilities of transitioning to each of the other values, will decrease the asymptotic variance of estimates. This has inspired two modified Gibbs sampling methods, originally due to Frigessi, et al and to Liu, though these do not always reduce self transitions to the minimum possible. Methods that do reduce the probability of self transitions to the minimum, but do not satisfy the conditions of Peskun's theorem, have also been devised, by Suwa and Todo. I review past methods, and introduce a broader class of reversible methods, based on what I call "antithetic modification", which also reduce asymptotic variance compared to Gibbs sampling, even when not satisfying the conditions of Peskun's theorem. A modification of one method in this class reduces self transitions to the minimum possible, while still always reducing asymptotic variance compared to Gibbs sampling. I introduce another new class of non-reversible methods based on slice sampling that can also minimize self transition probabilities. I provide explicit, efficient implementations of all these methods, and compare their performance in simulations of a 2D Potts model, a Bayesian mixture model, and a belief network with unobserved variables. The non-reversibility produced by sequential updating can be beneficial, but no consistent benefit is seen from the individual updates being done by a non-reversible method.

翻译：吉布斯采样反复从一个变量x_i的条件分布中采样（给定其他变量），其中i可随机选择，或按某种系统顺序或随机顺序进行顺序更新。当x_i是离散变量时，吉布斯采样更新可能选择与旧值相同的新值。Peskun的一个定理表明，在随机选择i的情况下，若采用可逆方法降低此类自转移的概率，同时增加向其他每个值转移的概率，则会减小估计的渐近方差。这促使了两种改进的吉布斯采样方法（最初由Frigessi等人以及Liu提出），尽管它们并非总能将自转移降至最低可能。由Suwa和Todo设计的方法虽能将自转移概率降至最低，但不符合Peskun定理的条件。本文回顾了已有方法，并引入一类更广泛的基于我称之为"对抗修正"的可逆方法，即使不满足Peskun定理的条件，也能比吉布斯采样进一步降低渐近方差。对该类中某一方法的修正可最大程度降低自转移，同时始终保持比吉布斯采样更小的渐近方差。我还引入了另一类基于切片采样的非可逆方法，同样能最小化自转移概率。我为所有这些方法提供了显式高效的实现，并在二维Potts模型、贝叶斯混合模型及含隐变量的信念网络模拟中比较了它们的性能。顺序更新产生的非可逆性可能具有优势，但未见通过非可逆方法执行单个更新能带来一致性能提升的证据。