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 $\mathbb{C}^n$.