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.
