Asymmetric Topologies on Statistical Manifolds

TL;DR

This paper explores the use of asymmetric information distances to define novel asymmetric topologies on statistical manifolds, focusing on the properties generated by the Kullback-Leibler divergence.

Contribution

It introduces a new framework for asymmetric topologies on statistical manifolds using information distances, particularly analyzing the topology induced by KL divergence.

Findings

Quasimetric topology generated by KL divergence is characterized.

Topological properties of the asymmetric norms are investigated.

The framework enhances understanding of asymmetry in statistical manifolds.

Abstract

Asymmetric information distances are used to define asymmetric norms and quasimetrics on the statistical manifold and its dual space of random variables. Quasimetric topology, generated by the Kullback-Leibler (KL) divergence, is considered as the main example, and some of its topological properties are investigated.