Lattice conditional independence models and Hibi ideals
Lattice conditional independence models and Hibi ideals
复制标题
格条件独立模型和 Hibi 理想
DOI：
10.1112/tlm3.12041
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发表时间：
2021
影响因子：
0.8
通讯作者：
H. Wynn
中科院分区：
文献类型：
--
作者：
P. Caines;F. Mohammadi;E. Sáenz;H. Wynn
点阵条件独立模型[Andersson和Perlman，多元正态分布中条件独立的点阵模型，Ann.]Statist. 21(1993)， 1318-1358]是一类首先为高斯情况开发的模型，其中分布格对所有条件独立语句进行分类。主要结果是，这些模型可以等价地通过传递有向无环图（TDAG）来描述，其中，与因果模型的正常情况一样，条件独立性是根据图中祖先的条件来描述的。我们证明了代数中的平行研究流，Hibi理想理论，不仅直接映射到格条件无关模型，而且提供了一个从线性高斯情况推广理论的工具。给定一个分配格(i)每个条件独立语句与格上定义的Hibi关系相关联，（ii）有向图由格中的链给出，这些链对应于条件独立链，（iii）链中乘积项的消去理想给出Hibi理想，（iv） TDAG可以从Hibi理想的Alexander对偶构造的特殊二部图中恢复。简要地证明了统计对数线性模型、时间序列和香农信息流的自然应用。
Lattice conditional independence models [Andersson and Perlman, Lattice models for conditional independence in a multivariate normal distribution, Ann. Statist. 21 (1993), 1318–1358] are a class of models developed first for the Gaussian case in which a distributive lattice classifies all the conditional independence statements. The main result is that these models can equivalently be described via a transitive directed acyclic graph (TDAG) in which, as is normal for causal models, the conditional independence is in terms of conditioning on ancestors in the graph. We demonstrate that a parallel stream of research in algebra, the theory of Hibi ideals, not only maps directly to the lattice conditional independence models but gives a vehicle to generalise the theory from the linear Gaussian case. Given a distributive lattice (i) each conditional independence statement is associated with a Hibi relation defined on the lattice, (ii) the directed graph is given by chains in the lattice which correspond to chains of conditional independence, (iii) the elimination ideal of product terms in the chains gives the Hibi ideal and (iv) the TDAG can be recovered from a special bipartite graph constructed via the Alexander dual of the Hibi ideal. It is briefly demonstrated that there are natural applications to statistical log‐linear models, time series and Shannon information flow.