Warped Riemannian metrics for location-scale models

TL;DR

This paper demonstrates that Rao-Fisher information metrics of location-scale models on Riemannian manifolds are warped Riemannian metrics, providing new formulas, a generalized Mahalanobis distance, and insights into the geometry of statistical models.

Contribution

It proves that Rao-Fisher metrics are warped Riemannian metrics for invariant models, derives new formulas, and analyzes the geometry of the von Mises-Fisher model.

Findings

Rao-Fisher information metric of Riemannian Gaussian model derived for the first time

Introduces a generalized Mahalanobis distance for Riemannian models

Shows the von Mises-Fisher model's parameter space is a Hadamard manifold for certain dimensions

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 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 $n$.