🤖 AI Summary
This study investigates the invariance properties of distributional divergence measures within group-symmetric statistical models. By endowing both the sample and parameter spaces with a group action and assuming that density functions transform according to a multiplier representation, the authors integrate tools from group representation theory, transformation models, $f$-divergence analysis, and Fisher–Rao information geometry. They establish that all $f$-divergences and the Fisher–Rao distance are invariant under the induced group action. The key contribution lies in showing that such invariant divergences reduce to functions depending solely on the maximal invariants of the parameter pair. This framework is successfully extended to multivariate location-scale families, where the invariant geometric structure of the parameter space is characterized via double coset decompositions.
📝 Abstract
Many statistical models have natural symmetries described by a group action. We study how such symmetries affect the comparison of two distributions. We work with a transformation model in which a group acts on both the sample space and the parameter space, and the densities transform with a multiplier. Under this assumption, we show that every $f$-divergence is invariant under the group action. As a consequence, an invariant divergence depends only on a maximal invariant of the pair of parameters. When the action on the parameter space is transitive, this maximal invariant is given by a double coset. We apply this result to multidimensional location-scale families, and we show that the same reduction applies to the Fisher--Rao distance.