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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.

##### MSC:
 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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