Normalization Problems for Deformed Exponential Families

In this study, after reviewing fundamental properties of ordinary exponential families, we consider geometry of deformed exponential families. We redefine a deformed logarithm function and a deformed exponential function. In fact, by adjusting the initial condition of deformed logarithm, we find that the α -representations in information geometry and the rectified linear unit (ReLU) belong the class of q-exponentials. Under the new definition of deformed exponential function, we discuss dually flat structures for deformed exponential families. We also consider normalization problems for those families using the ReLU.