Dually flat geometries of the deformed exponential family

An exponential family is dually flat with respect to Amari's ±1 connection. A deformed exponential family which is a generalization of the exponential family has two dually flat structures called the U-geometry and the χ-geometry. In the case of an exponential family invariant α-geometry gives the dually flat structure. But for a deformed exponential family, one need to consider generalized geometric structures other than the invariant α-geometry. The (F,G)-geometry on a statistical manifold is such a generalized geometry defined using a general embedding function F and a positive smooth function G. In this paper, we present the role of the (F,G)-geometry in the study of a deformed exponential family. We show that the dually flat U-geometry is the (F,G)-geometry for suitable choices of F and G. Further we show that the dully flat χ-geometry is the conformal flattening of the (F,G)-geometry for suitable F and G.