  The Gaussian model equips strong properties that facilitate studying and
interpreting graphical models. Specifically it reduces conditional independence
and the study of positive association to determining partial correlations and
their signs. When Gaussianity does not hold partial correlation graphs are a
useful relaxation of graphical models, but it is not clear what information
they contain (besides the obvious lack of linear association). We study
elliptical and transelliptical distributions as middle-ground between the
Gaussian and other families that are more flexible but either do not embed
strong properties or do not lead to simple interpretation. We characterize the
meaning of zero partial correlations in the elliptical family and
transelliptical copula models and show that it retains much of the dependence
structure from the Gaussian case. Regarding positive dependence, we prove
impossibility results to learn (trans)elliptical graphical models, including
that an elliptical distribution that is multivariate totally positive of order
two for all dimensions must be essentially Gaussian. We then show how to
interpret positive partial correlations as a relaxation, and obtain important
properties related to faithfulness and Simpson's paradox. We illustrate the
transelliptical model potential to study tail dependence in S&P500 data, and of
positivity to improve regularized inference.
