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 (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 (𝑀), equipped with 𝐺. We show that a geodesic segment joining arbitrary probability measures 𝜇 1 and 𝜇 2 is expressed by using the normalized geometric mean of its endpoints. As an application, we show that any two points of (𝑀) can be joined by a unique geodesic. Moreover, we prove that the function 𝓁 defined by 𝓁 ( 𝜇 1 , 𝜇 2 ) ∶= 2 arccos ∫ 𝑀 √ 𝑝 1 𝑝 2 𝑑𝜆, 𝜇 𝑖 = 𝑝 𝑖 𝜆, 𝑖 = 1, 2, gives the Riemannian distance function on (𝑀). It is shown that geodesics are all minimal.