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 φ and analyze its properties. The enriched standard conjugate family for φ and the mean parameter μ. are defined and discussed. This family is considerably more flexible than the standard conjugate one. The reference priors for φ and μ 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 Σ of a multivariate normal model under the enriched standard conjugate family. In particular, commonly employed Bayes estimates, such as the posterior expectation of Σ and Σ-1, are provided in closed form.