Similarities between Information geometry and Euclidean geometry

Abstract

Dually flat spaces play a key role in the differential geometrical approach in statistics (information geometry) and many divergences have been studied as an amount which measures the discrepancy between two probability distributions. In a dually flat space, a canonical divergence is naturally introduced and it satisfies a relational expression called triangular relation. This can be regarded as a generalization of the law of cosines in Euclidean space. We introduce a new divergence corresponding to the squared distance in a dually flat space. For this divergence, we show that more relational equations and theorems similar to Euclidean space hold and study the relationship with other divergences.