Geometry of the probability simplex and its connection to the maximum entropy method

The use of geometrical methods in statistics has a long and rich history highlighting many different aspects. These methods are usually based on a Riemannian structure defined on the space of parameters that characterize a family of probabilities. In this paper, we consider the finite dimensional case but the basic ideas can be extended similarly to the infinite-dimensional case. Our aim is to understand exponential families of probabilities on a finite set from an intrinsic geometrical point of view and not through the parameters that characterize some given family of probabilities.

For that purpose, we consider a Riemannian geometry defined on the set of positive vectors in a finite-dimensional space. In this space, the probabilities on a finite set comprise a submanifold in which exponential families correspond to geodesic surfaces. We shall also obtain a geometric/dynamic interpretation of Jaynes’ method of maximum entropy.