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. It is however also known for being a result 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 quantitative study in a specific context of graphical-model simulation is also conducted.

翻译：在马尔可夫链蒙特卡洛领域中，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，这一结果以其极强的结论性而闻名。然而，其难以证明的特性同样广为人知：研究者需要证明从状态$\mathbf{x}$到达状态$\mathbf{y}$的概率不小于另一采样器的对应概率，且此不等式须对所有满足$\mathbf{x} \neq \mathbf{y}$的配对$(\mathbf{x}, \mathbf{y})$成立。本文提出该性质的弱化版本，无需保持对所有状态的概率不等式：本质上，随着某个变参数无界增长，只要满足概率不等式的状态属于质量集中集合，支配性即可渐近成立。该弱序对于将偏序离散状态空间上的提升采样器与对应的Metropolis-Hastings采样器进行比较具有实用价值。广泛的一般性分析得出定性结论：提升采样器在特定情形下（可被明确识别）具有渐近优势，但在其他情形下未必成立（其原因也将被阐明）。此外，本文在图形模型模拟的具体背景下开展了定量研究。