INFORMATION GEOMETRY 01/01/2025

Mathematics

Information geometry is a branch of Mathematics that applies the techniques of differential geometry to the field of probability theory. The family of probability distributions which constitutes a statistical model has a rich geometric structure as a manifold with Riemannian metric and dual connections. Using this geometric interpretation one can obtain new insight into the framework of statistical inference and can develop new techniques for inference. In the geometric aspects of statistical manifolds, our effort is to understand (i) geometry of immersions and statistical manifolds (ii) geometry of submersions and statistical manifolds, and (iii) tangent bundles and harmonic maps of statistical manifolds. In the case of immersions into statistical manifolds, a necessary and sufficient condition for the inherited statistical manifold structures to be dual to each other is obtained. Then, statistical immersion is defined and proved a necessary condition for a statistical manifold to be a statistical hypersurface. Its converse is also proved. A necessary and sufficient condition for the inherited statistical manifold structures to be dual to each other is proved for a centro-affine immersion of codimension two into a dually flat statistical manifold. Then proved that the inherited statistical manifold structure is conformally-projectively flat in this case. We introduced the concept of a conformal submersion with horizontal distribution for Riemannian manifolds, which is a generalization of the affine submersion with horizontal distribution. A necessary condition for the existence of such a map is proved. Then compares the geodesics for a conformal submersion with horizontal distribution. A necessary and sufficient condition for the horizontal lift of a geodesic to be geodesic is obtained. In the case of conformal submersion with horizontal distribution, proved a necessary and sufficient condition for (M, ∇, gm) to become a statistical manifold. We obtained a necessary and sufficient condition for the tangent bundle TM to be a statistical manifold with the complete lift connection and the Sasaki lift metric. Then, proved a necessary and sufficient condition for the harmonicity of the identity map for conformally-projectively equivalent statistical manifolds. The conformal statistical submersion is defined which is a generalization of the statistical submersion and proved that harmonicity and conformality cannot coexist. Then, given a necessary condition for the harmonicity of the tangent map with respect to the complete lift structure on the tangent bundles. Also, proved a necessary and sufficient condition for the tangential map to be a statistical submersion.