Explicit Information Geometric Calculations of the Canonical Divergence of a Curve

Information geometry concerns the study of a dual structure ( g , ∇ , ∇ * ) upon a smooth manifold M . Such a geometry is totally encoded within a potential function usually referred to as a divergence or contrast function of ( g , ∇ , ∇ * ) . Even though infinitely many divergences induce on M the same dual structure, when the manifold is dually flat, a canonical divergence is well defined and was originally introduced by Amari and Nagaoka. In this pedagogical paper, we present explicit non-trivial differential geometry-based proofs concerning the canonical divergence for a special type of dually flat manifold represented by an arbitrary 1 -dimensional path γ . Highlighting the geometric structure of such a particular canonical divergence, our study could suggest a way to select a general canonical divergence by using the information from a general dual structure in a minimal way.