Information Geometry and Hamiltonian Systems on Lie Groups

The present paper deals with a class of left-invariant semi-definite metrics, called Fisher-Rao semi-definite metrics, on Lie groups appearing in transformation models. It is assumed that a family of invariant probability density functions on the sample manifold is given and that these probability density functions are invariant under a smooth Lie group action. As have been studied by Barndorff-Nielsen and his coauthors, as well as Amari and his collaborators, the Fisher-Rao semi-definite metric is naturally induced as a left-invariant semi-definite metric on the Lie group, which is regarded as the parameter space of the family of probability density functions. For a specific choice of family of probability density functions on compact semi-simple Lie group, the equation for the geodesic flow is derived through the Euler-Poincare reduction. Certain perspectives are mentioned about the geodesic equation on the basis of its similarity with the Brockett double bracket equation and with the Euler-Arnol’d equation for a generalized free rigid body dynamics.