On Closed-Form expressions for the Fisher-Rao Distance
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The Fisher-Rao distance is the geodesic distance between probability distributions in a statistical manifold equipped with the Fisher metric, which is the natural choice of Riemannian metric on such manifolds. Finding closed-form expressions for the Fisher-Rao distance is a non-trivial task, and those are available only for a few families of probability distributions. In this survey, we collect explicit examples of known Fisher-Rao distances for both discrete (binomial, Poisson, geometric, negative binomial, categorical, multinomial, negative multinomial) and continuous distributions (exponential, Gaussian, log-Gaussian, Pareto). We expand this list by deducing those expressions for Rayleigh, Erlang, Laplace, Cauchy and power function distributions.
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