Uniqueness of Fisher–Rao geodesics

Determine whether Fisher–Rao geodesics are unique in general for Fisher–Rao manifolds arising from parametric statistical models, and resolve the open question concerning potential non-uniqueness of these geodesics.

Background

While geodesics on manifolds with non-positive sectional curvature are known to be unique, the general situation for Fisher–Rao manifolds is unclear. The paper notes that non-uniqueness of Fisher geodesics is an open question; resolving it would clarify fundamental geometric properties of Fisher–Rao manifolds and inform algorithms that rely on geodesic uniqueness.