In this talk, I explore the mathematical relationships between duality in information geometry, convex analysis, and divergence functions. First, from the fundamental inequality of a convex function, a family of divergence measures can be constructed, which specializes to the familiar Bregman divergence, Jenson difference, beta-divergence, and alpha-divergence, etc.

Second, the mixture parameter turns out to correspond to the alpha <-> -alpha duality in information geometry (which I call "referential duality", since it is related to the choice of a reference point for computing divergence).

Third, convex conjugate operation induces another kind of duality in information geometry, namely, that of biorthogonal coordinates and their transformation (which I call "representational duality", since it is related to the expression of geometric quantities, such as metric, affine connection, curvature, etc of the underlying manifold). Under this analysis, what is traditionally called "+1/-1 duality" and "e/m duality" in information geometry reflect two very different meanings of duality that are nevertheless intimately interwined for dually flat spaces.