Three-by-three correlation matrices: its exact shape and a family of distributions

We give a novel and simple convex construction of three-by-three correlation matrices. This construction reveals the exact shape of the volume for these matrices: it is a tetrahedron point-wise transformed through the sine function. Hence the space of three-by-three correlation matrices is isomorphic to the standard three-simplex, and the matrices can be sampled by placing distributions on the three-simplex. This gives densities on the matrices that are flexible and easily interpreted; these will be useful in Bayesian analysis of correlation matrices. Examples using Dirichlet distributions are provided. We show the uniqueness of the construction, and we also prove that there is no parallel construction for higher order correlation matrices.