Transversely Hessian foliations and information geometry

TL;DR

This paper explores the geometric structures of probability distribution families in information geometry, focusing on transversely Hessian foliations that generalize the standard Riemannian metric assumption.

Contribution

It introduces the concept of transversely Hessian foliations in information geometry, extending the classical framework beyond positive definite Fisher information matrices.

Findings

Develops the theory of transversely Hessian foliations in the context of information geometry.

Connects foliation structures with the geometry of probability distribution families.

Provides a framework for analyzing statistical models with indefinite Fisher information matrices.

Abstract

A family of probability distributions parametrized by an open domain $\Lambda$ 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 $\Lambda$. 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.