Fisher-Rao geometry of Dirichlet distributions

Alice Le Brigant (1) , Stephen C. Preston (2) , Stéphane Puechmorel (3)

1 SAMM - Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne)
2 Brooklyn College [CUNY]
3 ENAC - Ecole Nationale de l'Aviation Civile

In this paper, we study the geometry induced by the Fisher-Rao metric on the parameter space of Dirichlet distributions. We show that this space is geodesically complete and has everywhere negative sectional curvature. An important consequence of this negative curvature for applications is that the Fréchet mean of a set of Dirichlet distributions is uniquely defined in this geometry.

Domaines: Géométrie différentielle [math.DG]; Statistiques [math.ST]

Alice Le Brigant, Stephen C. Preston, Stéphane Puechmorel. Fisher-Rao geometry of Dirichlet distributions. Differential Geometry and its Applications, 2021, 74, pp.101702. ⟨10.1016/j.difgeo.2020.101702⟩. ⟨hal-02568473v2⟩