Transversely Hessian foliations and information geometry

Abstract

A family of probability distributions parametrized by an open domain Λ in R n 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 tensor is positive definite defining in this way a Riemannian metric on Λ. If we replace the "positive definite" assumption by the existence of a suitable torsion-free connection, a foliation with a transversely Hessian structure appears naturally. In the paper we develop the study of transversely Hessian foliations in view of applications in information geometry.2011 Mathematics Subject Classification. 53C12

Amari Functors and Dynamics in Gauge Structures

“…When we deal with the vector tangent bundles the differential operator D ∇∇ * plays many outstanding roles in the global analysis of the base manifold [6]. In general though every vector bundle admits positive metric structures this same claim is far from being true for symplectic structure and for positive signature metric structures.…”

“…We deduce from Corollary 2 that every special statistical manifold supports a canonical (quasi) 4-web, viz 4 foliations in (quasi) general position. Before pursing we remark that among formalisms introduce in [6], many (of them) walk in the category of vector bundles. We go to perform this remark.…”

“…The functor just mentioned is called the general Amari functor of the vector bundle E. According to [6], the general Amari functor yield two restrictions :…”

“…We go to perform a formalism which is developed in [5]. See also [6] for the case of tangent bundles of a manifolds. The concerns are metric structures in vector bundles.…”

“…Inspired by the appendix to [4] and by [6] and by we define a map from pairs of gauge structures in the space of differential operators DO(Eo * ⊗ E, T * M ⊗ E * ⊗E). To every pair of gauge structures [(E, ∇), (E, ∇ ⋆ )] we introduce the first order differential operator…”