论文标题
渐近Peskun订购及其应用于提起的采样器
An asymptotic Peskun ordering and its application to lifted samplers
论文作者
论文摘要
在马尔可夫链蒙特卡洛社区中，在两个采样器之间进行了佩斯肯的订购，这意味着一个占优势，这是一个非常强烈的结果。但是，它也以很难建立的结果而闻名。实际上，必须证明，使用Sampler的状态$ \ Mathbf {X} $从状态$ \ MATHBF {y} $达到状态的可能性大于或等于使用其他采样器的概率，并且必须适用于所有Pairs $（\ Mathbf {x}}，\ mathbf {y Mathbf {y y neq） \ mathbf {y} $。我们在本文中提供了一个较弱的版本，该版本不需要所有这些状态的概率之间的不平等：基本上，只要概率大于或等于属于质量浓度集的概率的状态，主导地位就渐近地存在，因为变化的参数会增长而无约束力。事实证明，弱排序对于将部分订购的离散状态空间与其大都市（Hastings-Hastings对应物）进行比较的采样器非常有用。一般性的分析得出了定性的结论：它们在某些情况下渐近表现更好（并且我们能够识别它们），但不一定在其他情况下（以及为什么明确的原因）。还在图形模拟的特定情况下进行了定量研究。
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.
