Information geometry is a differential geometric framework to study spaces of probability distributions. Central tools of this framework are the Fisher-Rao metric, a Riemannian metric induced by the Fisher information on parametric statistical models, and the family of dual alpha-connections. Both the Fisher-Rao metric and the alpha-connections have non-parametric counterparts, which we discuss in this talk. We introduce the $L^p$-Fisher-Rao metrics, a family of Finsler metrics that generalize the Fisher-Rao metric, and show that their geodesics coincide with those of the alpha-connections on the space of smooth densities (for some relation between p and alpha). This result no longer holds on the space of probability densities. This gives a new variational interpretation of alpha-geodesics as being energy minimizing curves.