Hooke's law in statistical manifolds and divergences

Abstract

The concept of the canonical divergence is defined for dually flat statistical manifolds in terms of the Legendre transform between dual affine coordinates. In this article, we introduce a new two point function defined for any triple $(g, \nabla, \nabla^{*})$ of a Riemannian metric $g$ and two affine connections $\nabla$ and $\nabla^{*}$. We show that this interprets the canonical divergence without refering to the existence of special coordinates (dual affine coordinates) but in terms of only classical mechanics concerning $\nabla$- and $\nabla^{*}$-geodesics. We also discuss the properties of the two point function and show that this shares some important properties with the canonical divergence defined on dually flat statistical manifolds.