Statistical structures on manifolds and their immersions

A. A. Rylov

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference «Classical and Modern Geometry» dedicated to the 100th anniversary of the birth of Professor Levon Sergeyevich Atanasyan (July 15, 1921—July 5, 1998). Moscow, November 1–4, 2021. Part 1, Tome 220 (2023), pp. 113-124

Résumé

An important example of structures of information geometry is a statistical structure. This is a Riemannian metric $g$ on a smooth manifold $M$ with a completely symmetric tensor field $K$ of type $(2,1)$. On a manifold endowed with the statistical structure $(g,K)$, a one-parameter family of $\alpha$-connections $\nabla^{\alpha}=D+\alpha\cdot K$ is defined invariantly, where $D$ is the Levi-Civita connection of the metric $g$ and $\alpha$ is a parameter. In this paper, we characterize conjugate symmetric statistical structures and their particular case—structures of constant $\alpha$-curvature. As an example, a description of a structure with $\alpha$-connection of constant curvature on a two-dimensional statistical Pareto model is given. We prove that the two-dimensional logistic model has a $2$-connection of constant negative curvature and the two-dimensional Weibull—Gnedenko model has a $1$-connection of constant positive curvature. Both these models possess conjugate symmetric statistical structures. For the case of a manifold $\widehat{M}$ with a torsion-free linear connection $\widehat{\nabla}$ immersed in a Riemannian manifold with statistical structure $(g,K)$, a criterion is obtained that a statistical structure with an appropriate $\widehat{\alpha}$-connection $\widehat{\nabla}$ is induced on the preimage.

Détail

Keywords: Riemannian metric, statistical structure, conjugate symmetric statistical manifold, statistical model, second fundamental form, relatively affine mapping.
Mots-clés : $\alpha$-connection