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Enriched conjugate and reference priors for the Wishart family on symmetric cones. (English) Zbl 1046.62054
Summary: 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 \(\varphi\) and analyze its properties.
The enriched standard conjugate family for \(\varphi\) and the mean parameter \(\mu\) are defined and discussed. This family is considerably more flexible than the standard conjugate one. The reference priors for \(\varphi\) 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.

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62F15 Bayesian inference
62E15 Exact distribution theory in statistics
60E05 Probability distributions: general theory
17C99 Jordan algebras (algebras, triples and pairs)
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