  In this paper, the defining properties of a valid measure of the dependence
between two random variables are reviewed and complemented with two original
ones, shown to be more fundamental than other usual postulates. While other
popular choices are proved to violate some of these requirements, a class of
dependence measures satisfying all of them is identified. One particular
measure, that we call the Hellinger correlation, appears as a natural choice
within that class due to both its theoretical and intuitive appeal. A simple
and efficient nonparametric estimator for that quantity is proposed. Synthetic
and real-data examples finally illustrate the descriptive ability of the
measure, which can also be used as test statistic for exact independence
testing.
