Gaussian Covariance Faithful Markov Trees
Graphical models are useful for characterizing conditional and marginal independence structures in high-dimensional distributions. An important class of graphical models is covariance graph models, where the nodes of a graph represent different components of a random vector, and the absence of an ed...
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| Format: | Article |
| Language: | English |
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Wiley
2011-01-01
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| Series: | Journal of Probability and Statistics |
| Online Access: | http://dx.doi.org/10.1155/2011/152942 |
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| _version_ | 1832551760558817280 |
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| author | Dhafer Malouche Bala Rajaratnam |
| author_facet | Dhafer Malouche Bala Rajaratnam |
| author_sort | Dhafer Malouche |
| collection | DOAJ |
| description | Graphical models are useful for characterizing conditional and marginal independence structures in high-dimensional distributions. An important class of graphical models is covariance graph models, where the nodes of a graph represent different components of a random vector, and the absence of an edge between any pair of variables implies marginal independence. Covariance graph models also represent more complex conditional independence relationships between subsets of variables. When the covariance graph captures or reflects all the conditional independence statements present in the probability distribution, the latter is said to be faithful to its covariance graph—though in general this is not guaranteed. Faithfulness however is crucial, for instance, in model selection procedures that proceed by testing conditional independences. Hence, an analysis of the faithfulness assumption is important in understanding the ability of the graph, a discrete object, to fully capture the salient features of the probability distribution it aims to describe. In this paper, we demonstrate that multivariate Gaussian distributions that have trees as covariance graphs are necessarily faithful. |
| format | Article |
| id | doaj-art-c7a854e34b254590912d0889262e5244 |
| institution | Kabale University |
| issn | 1687-952X 1687-9538 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Probability and Statistics |
| spelling | doaj-art-c7a854e34b254590912d0889262e52442025-02-03T06:00:43ZengWileyJournal of Probability and Statistics1687-952X1687-95382011-01-01201110.1155/2011/152942152942Gaussian Covariance Faithful Markov TreesDhafer Malouche0Bala Rajaratnam1Unité de Recherche Signaux et Systémes (U2S), Ecole Supérieure de la Statistique et de l'Analyse de l'Information (ESSAI), Ecole Nationale d'Ingénieurs de Tunis (ENIT), 6 Rue des Métiers, Charguia II 2035, Tunis Carthage, Ariana, Tunis 1002, TunisiaDepartment of Statistics, Department of Environmental Earth System Science, Woods Institute for the Environment, Stanford University, Standford, CA 94305, USAGraphical models are useful for characterizing conditional and marginal independence structures in high-dimensional distributions. An important class of graphical models is covariance graph models, where the nodes of a graph represent different components of a random vector, and the absence of an edge between any pair of variables implies marginal independence. Covariance graph models also represent more complex conditional independence relationships between subsets of variables. When the covariance graph captures or reflects all the conditional independence statements present in the probability distribution, the latter is said to be faithful to its covariance graph—though in general this is not guaranteed. Faithfulness however is crucial, for instance, in model selection procedures that proceed by testing conditional independences. Hence, an analysis of the faithfulness assumption is important in understanding the ability of the graph, a discrete object, to fully capture the salient features of the probability distribution it aims to describe. In this paper, we demonstrate that multivariate Gaussian distributions that have trees as covariance graphs are necessarily faithful.http://dx.doi.org/10.1155/2011/152942 |
| spellingShingle | Dhafer Malouche Bala Rajaratnam Gaussian Covariance Faithful Markov Trees Journal of Probability and Statistics |
| title | Gaussian Covariance Faithful Markov Trees |
| title_full | Gaussian Covariance Faithful Markov Trees |
| title_fullStr | Gaussian Covariance Faithful Markov Trees |
| title_full_unstemmed | Gaussian Covariance Faithful Markov Trees |
| title_short | Gaussian Covariance Faithful Markov Trees |
| title_sort | gaussian covariance faithful markov trees |
| url | http://dx.doi.org/10.1155/2011/152942 |
| work_keys_str_mv | AT dhafermalouche gaussiancovariancefaithfulmarkovtrees AT balarajaratnam gaussiancovariancefaithfulmarkovtrees |