一种渐近Peskun序及其在提升采样器中的应用
An asymptotic Peskun ordering and its application to lifted samplers
AI总结：
本文提出一种弱化的渐近Peskun序，无需所有状态对的转移概率满足不等式，仅需质量集中集内的状态满足且随参数趋于无穷时渐近成立，可用于比较部分有序离散空间的提升采样器与Metropolis–Hastings采样器，得出定性结论并开展了图模型仿真的定量研究。
摘要：
在马尔可夫链蒙特卡洛领域，两个采样器之间的Peskun序（意味着一个采样器优于另一个）是一项极为重要的强结论，但它也因难以确立而闻名。实际上，研究者必须证明使用某一采样器从状态$\boldsymbol{x}$到达状态$\boldsymbol{y}$的概率，大于或等于使用另一采样器的对应概率，且该结论必须对所有满足$\boldsymbol{x} \neq \boldsymbol{y}$的状态对$(\boldsymbol{x}, \boldsymbol{y})$均成立。本文提出了一种更弱的Peskun序版本，它不要求所有状态的转移概率都满足不等式：本质上，当一个可变参数趋于无穷大时，只要概率大于或等于的状态属于一个质量集中集，这种优势就会渐近成立。这种弱序可用于比较部分有序离散状态空间的提升采样器（lifted samplers）与其对应的Metropolis–Hastings采样器。一般性分析得出了定性结论：在某些可被识别的场景下，提升采样器的渐近性能更优，但在其他场景中未必如此，且本文阐明了其中的原因。此外，本文还在图模型仿真的特定场景下开展了定量研究。
英文摘要：
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.
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