Geometric mean of probability measures and geodesics of Fisher information metric

Abstract

The space of all probability measures having positive density function on a connected compact smooth manifold $M$, denoted by $\mathcal{P}(M)$, carries the Fisher information metric $G$. We define the geometric mean of probability measures by the aid of which we investigate information geometry of $\mathcal{P}(M)$, equipped with $G$. We show that a geodesic segment joining arbitrary probability measures ${\mu }_1$ and ${\mu }_2$ is expressed by using the normalized geometric mean of its endpoints. As an application, we show that any two points of $\mathcal{P}(M)$ can be joined by a unique geodesic. Moreover, we prove that the function $\ell$ defined by $\ell ({\mu }_1,{\mu }_2):=2\arccos {\int }_M\sqrt{p_1{p}_2}d\lambda$, ${\mu }_i=p_i\lambda$, $i=1,2$ gives the Riemannian distance function on $\mathcal{P}(M)$. It is shown that geodesics are all minimal.

33 pages, 1 figure

References in corpus (3)

• New results on mixture and exponential models by Orlicz spaces
• Generalized Gibbs Ensembles in Discrete Quantum Gravity
• Information geometry of the space of probability measures and barycenter maps

Cited by in corpus (2)

• Information geometry of the space of probability measures and barycenter maps
• Generating the support with extreme value losses