Transversely Hessian foliations and information geometry

A family of probability distributions parametrized by an open domain Λ in Rn defines the Fisher information matrix on this domain which is positive semi-definite. In information geometry the standard assumption has been that the Fisher information matrix is positive definite defining in this way a Riemannian metric on Λ. If we replace the "positive definite" assumption by "0-deformable" condition a foliation with a transvesely Hessian structure appears naturally. We develop the study of transversely Hessian foliations in view of applications in information geometry.