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 α-connections ∇^α = D + α • K is defined invariantly, where D is the Levi-Civita connection of the metric g and α is a parameter. In this paper, we characterize conjugate symmetric statistical structures and their particular case—structures of constant α-curvature. As an example, a description of a structure with α-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 $\hat{M}$ with a torsion-free linear connection $\hat{\nabla }$ immersed in a Riemannian manifold with statistical structure (g,K), a criterion is obtained that a statistical structure with an appropriate а-connection $\hat{\nabla }$ is induced on the preimage.