New Metric and Connections in Statistical Manifolds

TL;DR

This paper introduces a new metric and a family of $oldsymbol{ heta}$-connections in statistical manifolds based on $oldsymbol{ extit{ ext{phi}}}$-divergence, generalizing classical concepts like Fisher information and Amari's $oldsymbol{ extit{ ext{alpha}}}$-connections.

Contribution

It proposes a novel metric and $oldsymbol{ extit{ ext{alpha}}}$-connections derived from $oldsymbol{ extit{ ext{phi}}}$-divergence, extending existing geometric structures in statistical manifolds.

Findings

Defined a new metric based on $ extit{ ext{phi}}$-divergence.

Generalized Fisher information metric and Amari's $ extit{ ext{alpha}}$-connections.

Analyzed parallel transport for $ extit{ ext{alpha}}=1$.

Abstract

We define a metric and a family of $\alpha$-connections in statistical manifolds, based on $\varphi$-divergence, which emerges in the framework of $\varphi$-families of probability distributions. This metric and $\alpha$-connections generalize the Fisher information metric and Amari's $\alpha$-connections. We also investigate the parallel transport associated with the $\alpha$-connection for $\alpha=1$.