Abstract
We describe various sets of conditional independence relationships, sufficient for qualitatively comparing non-vanishing squared partial correlations of a Gaussian random vector. These sufficient conditions are satisfied by several graphical Markov models. Rules for comparing degree of association among the vertices of such Gaussian graphical models are also developed. We apply these rules to compare conditional dependencies on Gaussian trees. In particular for trees, we show that such dependence can be completely characterised by the length of the paths joining the dependent vertices to each other and to the vertices conditioned on. We also apply our results to postulate rules for model selection for polytree models. Our rules apply to mutual information of Gaussian random vectors as well.
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Notes
The author would like to thank the referee for drawing his attention to this equality which improved the original proof immensely.
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Acknowledgments
The author would like to thank Michael Perlman, Thomas Richardson, Mathias Drton, Antar Bandyopadhyay, the referees and the associate editor for their useful comments and suggestions during the preparation of this article.
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This research was partially supported by Grant R-155-000-081-112 from National University of Singapore.
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Chaudhuri, S. Qualitative inequalities for squared partial correlations of a Gaussian random vector. Ann Inst Stat Math 66, 345–367 (2014). https://doi.org/10.1007/s10463-013-0417-x
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DOI: https://doi.org/10.1007/s10463-013-0417-x