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 P(M)$\mathcal {P}(M)$, carries the Fisher information metric G. We define the geometric mean of probability measures by the aid of which we investigate information geometry of P(M)$\mathcal {P}(M)$, equipped with G. We show that a geodesic segment joining arbitrary probability measures μ1 and μ2 is expressed by using the normalized geometric mean of its endpoints. As an application, we show that any two points of P(M)$\mathcal {P}(M)$ can be joined by a unique geodesic. Moreover, we prove that the function ℓ defined by ℓ(μ1,μ2):=2arccos∫Mp1p2dλ$\ell \!\big (\mu _1, \mu _2\big ):=2\arccos \int \nolimits _M \sqrt {p_1p_2}\,d\lambda$, μi=piλ$\mu _i=p_i \lambda$, i=1,2$i=1,2$, gives the Riemannian distance function on P(M)$\mathcal {P}(M)$. It is shown that geodesics are all minimal.