Canonical foliations of statistical manifolds with statistical models

Constructing a realisation of a statistical manifold as a statistical model, i.e a manifold with dual connections with respect to a Fisher metric, is an important question in information geometry. While a positive answer to this problem was given by the work on Hong Van Lé [1], writing explicitly the probability family giving rise to the Fisher metric is generally a difficult task. In this work, starting with the sheaf of solutions J_∇ J ∇ of the Hessian equation on a gauge structure (M,∇ ) ( M , ∇ ) , a canonical representation of the group associated to the Lie algebra formed by its sections is introduced. On the foliation it defines, a characterization of compact hyperbolic leaves is then obtained. Furthermore, these leaves can be provided with an explicit statistical model structure, that is a probability density defining a Fisher metric.