Doubly autoparallel structure on the probability simplex

TL;DR

This paper explores the geometric properties and classification of doubly autoparallel submanifolds within the probability simplex, revealing their algebraic structure and common features in information geometry.

Contribution

It provides an algebraic characterization and classification of doubly autoparallel submanifolds, advancing understanding in information geometric structures.

Findings

Identification of common properties of doubly autoparallel submanifolds

Algebraic characterization of these submanifolds

Complete classification within the probability simplex

Abstract

On the probability simplex, we can consider the standard information geometric structure with the e- and m-affine connections mutually dual with respect to the Fisher metric. The geometry naturally defines submanifolds simultaneously autoparallel for the both affine connections, which we call {\em doubly autoparallel submanifolds}. In this note we discuss their several interesting common properties. Further, we algebraically characterize doubly autoparallel submanifolds on the probability simplex and give their classification.