A geometry on the space of probabilities I. The finite dimensional case

Abstract

In this note we provide a natural way of defining exponential coordinates on the class of probabilities on the set $\Omega =[1,n]$ or on $\mathbb{P}=\{p=(p_1,\dots ,p_n)\in {\mathbb{R}}^n|p_i>0;{\Sigma }_{i=1}^n{p}_i=1\}$. For that we have to regard $\mathbb{P}$ as a projective space and the exponential coordinates will be related to geodesic flows in ${\mathbb{C}}^n$.