Qualitative inequalities for squared partial correlations of a Gaussian random vector
This work addresses the need for understanding and comparing association degrees in Gaussian models, particularly for graphical Markov models, but it is incremental as it builds on existing theory without introducing new paradigms.
The paper tackles the problem of comparing conditional dependencies in Gaussian graphical models by deriving sufficient conditions for qualitatively comparing squared partial correlations, and applies these rules to characterize dependencies in Gaussian trees based on path lengths and to propose model selection rules for polytrees.
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 characterized 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.