On a prior based on the Wasserstein information matrix

We introduce a prior for the parameters of univariate continuous distributions, based on the Wasserstein information matrix, which is invariant under reparameterisations. We briefly discuss the links between the proposed prior with information geometry. We present several examples where we can either obtain this prior in closed-form, or propose a numerically tractable approximation for cases where the prior is not available in closed-form. Since this prior is improper in some cases, we present sufficient conditions for the propriety of the posterior distribution for general classes of models.