关于对称划分格与概率交互作用

中文摘要 AI 辅助

划分格的Möbius函数可用于定义概率测度的交互作用。当下集在变量的置换下不变时，我们称之为对称划分格。我们通过整数划分来刻画这些格，并得到其顶部元素处Möbius系数的递推关系。通过构造划分乘积线性独立的概率测度，我们证明了在坐标空间足够大的情况下，交互作用的置换不变性等价于其底层下集的置换不变性。我们还刻画了迫使交互作用对所有概率测度都消失的下集上的划分，并构造了具有零交互作用的不可分解分布。我们定义了对称划分格的阶，并将其与消失的边缘分布联系起来。我们研究了广义Lancaster，定义了Streitberg和大小受限划分格，并获得了它们的显式Möbius系数。最后，我们用特征函数来表达交互作用方程。对于高斯分布，我们证明了一个交互作用恰好在下集中的一个非平凡划分下分布因子化时消失。我们还给出了由Bernstein函数生成的径向特征函数的非消失准则。

英文摘要

The Möbius function of a partition lattice can be used to define interactions of probability measures. We call a lower set a symmetric partition lattice when it is invariant under permutations of the variables. We characterize these lattices through integer partitions and obtain a recursion for their Möbius coefficients at the top element. By constructing probability measures whose partition products are linearly independent, we show that permutation invariance of an interaction is equivalent to that of its underlying lower set, provided the coordinate spaces are sufficiently large. We also characterize the partitions on a lower set that force an interaction to vanish for every probability measure, and construct indecomposable distributions with zero interaction. We define the order of a symmetric partition lattice and relate it to vanishing marginals. We study generalized Lancaster and define the Streitberg and size-limited partition lattices, obtaining explicit Möbius coefficients for them. Finally, we express interaction equations in terms of characteristic functions. For Gaussian distributions, we show that an interaction vanishes exactly when the distribution factorizes according to a nontrivial partition in the underlying lower set. We also give a nonvanishing criterion for radial characteristic functions generated by Bernstein functions.

发表机构

• Universidade Estadual de Mato Grosso do Sul（南马托格罗索州立大学）

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