Statistical structures on manifolds and their immersions

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.