The fundamental concepts underlying Markov networks are the conditional independence and the set of rules called Markov properties that translate conditional independence constraints into graphs. We introduce the concept of mutual conditional independence in an independent set of a Markov network, and we prove its equivalence to the Markov properties under certain regularity conditions. This extends the notion of similarity between separation in graph and conditional independence in probability to similarity between the mutual separation in graph and the mutual conditional independence in probability. Model selection in graphical models remains a challenging task due to the large search space. We show that mutual conditional independence property can be exploited to reduce the search space. We present a new forward model selection algorithm for graphical log-linear models using mutual conditional independence. We illustrate our algorithm with a real data set example. We show that for sparse models the size of the search space can be reduced from O(n3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {O} (n^{3})$\end{document} to O(n2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {O}(n^{2})$\end{document} using our proposed forward selection method rather than the classical forward selection method. We also envision that this property can be leveraged for model selection and inference in different types of graphical models.

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相互条件独立及其在马尔可夫网络模型选择中的应用

马尔可夫网络的基本概念是条件独立性和称为马尔可夫属性的规则集，这些规则将条件独立性约束转换为图形。我们在一个独立的马尔可夫网络集合中引入了相互条件独立的概念，并证明了它在一定正则性条件下与马尔可夫性质的等价性。这将图分离和概率条件独立之间的相似性概念扩展到图的相互分离和概率上的相互条件独立之间的相似性。由于搜索空间很大，图形模型中的模型选择仍然是一项具有挑战性的任务。我们表明可以利用相互条件独立属性来减少搜索空间。我们提出了一种新的前向模型选择算法，用于使用相互条件独立的图形对数线性模型。我们用一个真实的数据集例子来说明我们的算法。我们表明，对于稀疏模型，搜索空间的大小可以从 O(n3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage 减少{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {O} (n^{3})$\end{document} 到 O( n2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin }{-69pt} \begin{document}$\mathcal {O}(n^{2})$\end{document} 使用我们提出的前向选择方法而不是经典的前向选择方法。我们还设想可以利用此属性在不同类型的图形模型中进行模型选择和推理。