  The measure of Jensen-Fisher divergence between probability distributions is
introduced and its theoretical grounds set up. This quantity, in contrast to
the remaining Jensen divergences, is very sensitive to the fluctuations of the
probability distributions because it is controlled by the (local) Fisher
information, which is a gradient functional of the distribution. So, it is
appropriate and informative when studying the similarity of distributions,
mainly for those having oscillatory character. The new Jensen-Fisher divergence
shares with the Jensen-Shannon divergence the following properties:
non-negativity, additivity when applied to an arbitrary number of probability
densities, symmetry under exchange of these densities, vanishing if and only if
all the densities are equal, and definiteness even when these densities present
non-common zeros. Moreover, the Jensen-Fisher divergence is shown to be
expressed in terms of the relative Fisher information as the Jensen-Shannon
divergence does in terms of the Kullback-Leibler or relative Shannon entropy.
Finally the Jensen-Shannon and Jensen-Fisher divergences are compared for the
following three large, non-trivial and qualitatively different families of
probability distributions: the sinusoidal, generalized gamma-like and
Rakhmanov-Hermite distributions.
