Fisher–rao and Wasserstein–otto Geometries of Parametric Probability Families: A Comparative Atlas

We compare the Fisher–Rao and Wasserstein–Otto met- rics by pulling them back onto the same parametric families of proba- bility densities. After recalling the measure-theoretic and differential- geometric framework, we derive the two ambient constructions and make their relation explicit on the real line. The comparative atlas covers location–dispersion models, multivariate Gaussians, linear and nonlin- ear Gibbs families, and push-forward families. For each model we specify its parameter and tangent spaces, compute the density differential and both induced metrics, and interpret the resulting geometry through den- sities, metric indicatrices, and geodesics. Exact paths are distinguished from numerical approximations and from paths constrained to a sub- family.