Rényi Relative Entropy from Homogeneous Kullback-Leibler Divergence Lagrangian

We study the homogeneous extension of the Kullback-Leibler divergence associated to a covariant variational problem on the statistical bundle. We assume a finite sample space. We show how such a divergence can be interpreted as a Finsler metric on an extended statistical bundle, where the time and the time score are understood as extra random functions defining the model—-. We find a relation between the homogeneous generalisation of the Kullback-Leibler divergence and the Rényi relative entropy, the Rényi parameter being related to the time-reparametrization lapse of the Lagrangian model. We investigate such intriguing relation with an eye to applications in physics and quantum information theory.