用低阶条件性确定完全条件独立性

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摘要翻译：
与随机向量相关联的集中图是一个无向图，其中每个顶点对应于向量中的一个随机变量。在任意一对顶点（或变量）之间没有边，就等于在给定所有其他变量的情况下，这两个变量之间完全条件独立。在多元高斯情况下，没有边对应于精度矩阵中的零系数，该系数是协方差矩阵的逆。众所周知，这个浓度图代表了相关随机向量分布中的一些条件独立性。这些条件独立性对应于该图中边的“分离”或缺失。在本文中，我们假定在概率分布中除了图所表示的独立性之外，没有其他独立性存在。这个性质被称为概率分布相对于相关浓度图的完全马尔可夫性。本文证明了这种特殊的浓度图，即与完全马尔可夫分布相关的浓度图，只需限定有限的变量数即可确定。我们证明这个数目等于浓度图中最小分离器的最大尺寸。
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英文标题：
《Determining full conditional independence by low-order conditioning》
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作者：
Dhafer Malouche
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最新提交年份：
2010
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分类信息：

一级分类：Mathematics        数学
二级分类：Statistics Theory        统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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一级分类：Statistics        统计学
二级分类：Machine Learning        机器学习
分类描述：Covers machine learning papers (supervised, unsupervised, semi-supervised learning, graphical models, reinforcement learning, bandits, high dimensional inference, etc.) with a statistical or theoretical grounding
覆盖机器学习论文（监督，无监督，半监督学习，图形模型，强化学习，强盗，高维推理等）与统计或理论基础
--
一级分类：Statistics        统计学
二级分类：Statistics Theory        统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
--

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英文摘要：
  A concentration graph associated with a random vector is an undirected graph where each vertex corresponds to one random variable in the vector. The absence of an edge between any pair of vertices (or variables) is equivalent to full conditional independence between these two variables given all the other variables. In the multivariate Gaussian case, the absence of an edge corresponds to a zero coefficient in the precision matrix, which is the inverse of the covariance matrix. It is well known that this concentration graph represents some of the conditional independencies in the distribution of the associated random vector. These conditional independencies correspond to the "separations" or absence of edges in that graph. In this paper we assume that there are no other independencies present in the probability distribution than those represented by the graph. This property is called the perfect Markovianity of the probability distribution with respect to the associated concentration graph. We prove in this paper that this particular concentration graph, the one associated with a perfect Markov distribution, can be determined by only conditioning on a limited number of variables. We demonstrate that this number is equal to the maximum size of the minimal separators in the concentration graph.
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PDF链接：
https://arxiv.org/pdf/705.1613

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关键词：独立性 distribution Multivariate Conditioning independence edge 浓度 conditional absence 矩阵