  The present paper shows that warped Riemannian metrics, a class of Riemannian
metrics which play a prominent role in Riemannian geometry, are also of
fundamental importance in information geometry. Precisely, the paper features a
new theorem, which states that the Rao-Fisher information metric of any
location-scale model, defined on a Riemannian manifold, is a warped Riemannian
metric, whenever this model is invariant under the action of some Lie group.
This theorem is a valuable tool in finding the expression of the Rao-Fisher
information metric of location-scale models defined on high-dimensional
Riemannian manifolds. Indeed, a warped Riemannian metric is fully determined by
only two functions of a single variable, irrespective of the dimension of the
underlying Riemannian manifold. Starting from this theorem, several original
contributions are made. The expression of the Rao-Fisher information metric of
the Riemannian Gaussian model is provided, for the first time in the
literature. A generalised definition of the Mahalanobis distance is introduced,
which is applicable to any location-scale model defined on a Riemannian
manifold. The solution of the geodesic equation is obtained, for any Rao-Fisher
information metric defined in terms of warped Riemannian metrics. Finally,
using a mixture of analytical and numerical computations, it is shown that the
parameter space of the von Mises-Fisher model of $n$-dimensional directional
data, when equipped with its Rao-Fisher information metric, becomes a Hadamard
manifold, a simply-connected complete Riemannian manifold of negative sectional
curvature, for $n = 2,\ldots,8$. Hopefully, in upcoming work, this will be
proved for any value of $n$.
