Geometric mean of probability measures and geodesics of Fisher information metric

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 $\mathrm{\ell }$ 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.