A Peskun ordering between two samplers, implying a dominance of one over the other, is known among the Markov chain Monte Carlo community for being a remarkably strong result, but it is also known for being one that is notably difficult to establish. Indeed, one has to prove that the probability to reach a state $\mathbf{y}$ from a state $\mathbf{x}$, using a sampler, is greater than or equal to the probability using the other sampler, and this must hold for all pairs $(\mathbf{x}, \mathbf{y})$ such that $\mathbf{x} \neq \mathbf{y}$. We provide in this paper a weaker version that does not require an inequality between the probabilities for all these states: essentially, the dominance holds asymptotically, as a varying parameter grows without bound, as long as the states for which the probabilities are greater than or equal to belong to a mass-concentrating set. The weak ordering turns out to be useful to compare lifted samplers for partially-ordered discrete state-spaces with their Metropolis--Hastings counterparts. An analysis in great generality yields a qualitative conclusion: they asymptotically perform better in certain situations (and we are able to identify them), but not necessarily in others (and the reasons why are made clear). A thorough study in a specific context of graphical-model simulation is also conducted.

翻译：在两个采样器之间的Peskun序意味着一个采样器对另一个的支配性，这在马尔可夫链蒙特卡洛社区中因其极强效的结论而闻名，但也因其难以建立而著称。实际上，必须证明从状态$\mathbf{x}$出发，使用一个采样器到达状态$\mathbf{y}$的概率大于或等于使用另一个采样器的概率，且这一性质对所有满足$\mathbf{x} \neq \mathbf{y}$的状态对$(\mathbf{x}, \mathbf{y})$成立。本文提出一个较弱的版本，无需对所有状态对建立概率不等式：本质上，当某个变化参数无界增长时，只要概率满足大于或等于关系的状态属于一个质量集中集合，该支配性便渐近成立。此弱序被证明在将部分有序离散状态空间上的提升采样器与其Metropolis-Hastings对应方法进行比较时具有实用价值。一个高度一般性的分析得出了定性结论：在某些情况下（我们能够识别这些情况），这些采样器渐近表现更优，但在其他情况下则未必（其原因已清晰阐明）。此外，本文还在图模型模拟的具体背景下进行了深入研究。