Enriched conjugate and reference priors for the Wishart family on symmetric cones

A general Wishart family on a symmetric cone is a natural exponential family (NEF) having a homogeneous quadratic variance function. Using results in the abstract theory of Euclidean Jordan algebras, the structure of conditional reducibility is shown to hold for such a family, and we identify the associated parameterization phi and analyze its properties. The enriched standard conjugate family forphi and the mean parameter u are defined and discussed. This family is considerably more flexible than the standard conjugate one. The reference priors for phi and mu are obtained and shown to belong to the enriched standard conjugate family; in particular, this allows us to verify that reference posteriors are always proper. The above results extend those available for NEFs having a simple quadratic variance function. Specifications of the theory to the cone of real symmetric and positive-definite matrices are discussed in detail and allow us to perform Bayesian inference on the covariance matrix Sigma of a multivariate normal model under the enriched standard conjugate family. In particular, commonly employed Bayes estimates, such as the posterior expectation of Sigma and Sigma^{-1} are provided in closed form.