Hessian Structures And Non-Invariant (F, G)-Geometry On A Deformed Exponential Family

A deformed exponential family has two kinds of dual Hessian structures, the U-geometry and the chi-geometry. In this paper, we discuss the relation between the non-invariant (F, G)-geometry and the two Hessian structures on a deformed exponential family. A generalized likelihood function called F-likelihood function is defined and proved that the Maximum F-likelihood estimator is a Maximum a posteriori estimator.