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

Abstract

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 for $\phi$ and the mean parameter $\mu$ 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.