  We provide a unifying framework linking two classes of statistics used in
two-sample and independence testing: on the one hand, the energy distances and
distance covariances from the statistics literature; on the other, distances
between embeddings of distributions to reproducing kernel Hilbert spaces
(RKHS), as established in machine learning. The equivalence holds when energy
distances are computed with semimetrics of negative type, in which case a
kernel may be defined such that the RKHS distance between distributions
corresponds exactly to the energy distance. We determine the class of
probability distributions for which kernels induced by semimetrics are
characteristic (that is, for which embeddings of the distributions to an RKHS
are injective). Finally, we investigate the performance of this family of
kernels in two-sample and independence tests: we show in particular that the
energy distance most commonly employed in statistics is just one member of a
parametric family of kernels, and that other choices from this family can yield
more powerful tests.
