`Journal of Probability and StatisticsVolume 2011 (2011), Article ID 152942, 10 pageshttp://dx.doi.org/10.1155/2011/152942`
Research Article

## Gaussian Covariance Faithful Markov Trees

1Unité 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, Tunisia
2Department of Statistics, Department of Environmental Earth System Science, Woods Institute for the Environment, Stanford University, Standford, CA 94305, USA

Received 30 May 2011; Accepted 9 August 2011

Academic Editor: Junbin B. Gao

Copyright © 2011 Dhafer Malouche and Bala Rajaratnam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.