Conditional Dependence and Partial Correlation

In the multivariate normal case, conditional independence is the same as zero partial correlation.  (See below.) That makes a lot of things a lot simpler.  In particular, determining ordering in a DAG is just a matter of assessing partial correlations. Of course in many applications normality may not hold, but still...

Aust. N.Z. J. Stat. 46(4), 2004, 657–664
PARTIAL CORRELATION AND CONDITIONAL CORRELATION AS MEASURES OF CONDITIONAL INDEPENDENCE
Kunihiro Baba1∗, Ritei Shibata1 and Masaaki Sibuya2
Keio University and Takachiho University
Summary
This paper investigates the roles of partial correlation and conditional correlation as mea-sures of the conditional independence of two random variables. It first establishes a suffi-cientconditionforthecoincidenceofthepartialcorrelationwiththeconditionalcorrelation. The condition is satisfied not only for multivariate normal but also for elliptical, multi-variate hypergeometric, multivariate negative hypergeometric, multinomial and Dirichlet distributions. Such families of distributions are characterized by a semigroup property as a parametric family of distributions. A necessary and sufficient condition for the coinci-dence of the partial covariance with the conditional covariance is also derived. However, a known family of multivariate distributions which satisfies this condition cannot be found, except for the multivariate normal. The paper also shows that conditional independence has no close ties with zero partial correlation except in the case of the multivariate normal distribution; it has rather close ties to the zero conditional correlation. It shows that the equivalence between zero conditional covariance and conditional independence for normal variables is retained by any monotone transformation of each variable. The results suggest that care must be taken when using such correlations as measures of conditional indepen-dence unless the joint distribution is known to be normal. Otherwise a new concept of conditional independence may need to be introduced in place of conditional independence through zero conditional correlation or other statistics.
Keywords: elliptical distribution; exchangeability; graphical modelling; monotone transformation.