  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.
