Title: 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 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 distance
function on $\mathcal{P}(M)$. It is shown that geodesics are all minimal.