Geodesic flows of the Fisher-Rao metrics for the statistical transformation models

The statistical transformation models appear in the statistical inference for the mani-

folds of samples on which a Lie group acts smoothly. It is natural to consider a family of

probability density functions on the sample manifold with the parameter in the Lie group.

Being (relatively) invariant, this family gives rise to the Fisher-Rao (semi-definite) metric,

as well as the Amari-Chentsov cubic tensor, on the Lie group, both of which are funda-

mental objects in the information geometry. This talk gives an overview on the general

framework of the statistical transformation models and then deals with the geodesic flows

of the Fisher-Rao metrics for specific examples from the viewpoint of geometric mechanics.

Some relation with sub-Riemannian structures will also be mentioned.