  We examine properties of the Concrete (or Gumbel-softmax) distribution on the
simplex. Using the natural vector space structure of the simplex, the Concrete
distribution can be regarded as a transformation of the uniform distribution
through a reflection and a location-scale transformation. The Fisher
information is computed and the corresponding information metric is hyperbolic
space. We explicitly give an explicit transformation of the parameters of the
distribution to Poincar\'e half-space coordinates, which correspond to an
orthogonal parameterization, and the Fisher-Rao geodesic distance is computed.
