Information Geometry with (Para-)Kähler Structures

Abstract

We investigate conditions under which a statistical manifold $\mathfrak {M}$ (with a Riemannian metric g and a pair of torsion-free conjugate connections $\nabla , \nabla ^*$) can be enhanced to a (para-)Kähler structure. Assuming there exists an almost (para-)complex structure L compatible with g on a statistical manifold $\mathfrak {M}$ (of even dimension), then we show $(\mathfrak {M}, g, L, \nabla )$ is (para-)Kähler if $\nabla$ and L are Codazzi coupled. Other equivalent characterizations involve a symplectic form $\omega \equiv g(L \cdot , \cdot )$. In terms of the compatible triple $(g, \omega , L)$, we show that (i) each object in the triple induces a conjugate transformation on $\nabla$ and becomes an element of an (Abelian) Klein group; (ii) the compatibility of any two objects in the triple with $\nabla$ leads to the compatible quadruple $(g, \omega , L , \nabla )$ in which any pair of objects are mutually compatible. This is what we call Codazzi-(para-)Kähler manifold [8] which admits the family of torsion-free $\alpha$-connections (convex mixture of $\nabla , \nabla ^*$) compatible with $(g, \omega , L)$. Finally, we discuss the properties of divergence functions on $\mathfrak {M}\times \mathfrak {M}$ that lead to Kähler (when $L=J, J^2=-id$) and para-Kähler (when $L=K, K^2= id$) structures.