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Gaussian Faithful Markov Trees

Abstract : We study two types of graphical models in this paper: concentration an covariance graphical models. Graphical models use graphs to encode or capture the multivariate dependencies that are present in a given multivariate distribution. A concentration graph associated with a multivariate probability distribution of a given random vector is an undirected graph where each vertex represents each of the different components of the random vector and where the absence of an edge between any pair of variables implies conditional independence between these two variables given the remaining ones. Similarly, a covariance graph reflects marginal independences in the sense that the absence of an edge between any pair of variables implies marginal independence between these two variables. These two graphical models do not encode only pairwise relationship between variables, but they allow us to read many other conditional independence statements present in the probability distribution through separation criteria in this graph. In general, the graph may omit some of these conditional independence statements and when the graph encode exactly all of these conditional independences we say that the probability distribution is faithful to its corresponding graphical model. We present here two mathematical results concerning Markov trees, graphical models corresponding to trees: Gaussian Markov trees are necessarily faithful to their concentration and covariance graph. More formally this means that Gaussian distributions that have trees as concentration graphs are necessarily faithful (see [1]). Similarly an equivalent result can be proved for covariance graphs (see [2]). However the methods of proofs used for these two results are completely different.
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Submitted on : Thursday, June 24, 2010 - 8:54:52 AM
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Dhafer Malouche, Bala Rajaratnam. Gaussian Faithful Markov Trees. 42èmes Journées de Statistique, 2010, Marseille, France, France. ⟨inria-00494743⟩

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