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October 2003 Enriched conjugate and reference priors for the Wishart family on symmetric cones
Guido Consonni, Piero Veronese
Ann. Statist. 31(5): 1491-1516 (October 2003). DOI: 10.1214/aos/1065705116

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.

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Guido Consonni. Piero Veronese. "Enriched conjugate and reference priors for the Wishart family on symmetric cones." Ann. Statist. 31 (5) 1491 - 1516, October 2003. https://doi.org/10.1214/aos/1065705116

Information

Published: October 2003
First available in Project Euclid: 9 October 2003

zbMATH: 1046.62054
MathSciNet: MR2012823
Digital Object Identifier: 10.1214/aos/1065705116

Subjects:
Primary: 62E15, 62F15
Secondary: 60E05

Rights: Copyright © 2003 Institute of Mathematical Statistics

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Vol.31 • No. 5 • October 2003
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