Warped Riemannian metrics for locationscale models
Abstract
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 RaoFisher information metric of any locationscale 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 RaoFisher information metric of locationscale models defined on highdimensional 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 RaoFisher 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 locationscale model defined on a Riemannian manifold. The solution of the geodesic equation is obtained, for any RaoFisher 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 MisesFisher model of $n$dimensional directional data, when equipped with its RaoFisher information metric, becomes a Hadamard manifold, a simplyconnected complete Riemannian manifold of negative sectional curvature, for $n = 2,\ldots,8$. Hopefully, in upcoming work, this will be proved for any value of $n$.
 Publication:

arXiv eprints
 Pub Date:
 July 2017
 arXiv:
 arXiv:1707.07163
 Bibcode:
 2017arXiv170707163S
 Keywords:

 Mathematics  Statistics Theory
 EPrint:
 first version, before submission