Enriched conjugate and reference priors for the Wishart family on symmetric cones
Enriched conjugate and reference priors for the Wishart family on symmetric cones
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对称锥体上 Wishart 系列的丰富共轭和参考先验
DOI：
10.1214/aos/1065705116
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发表时间：
2003
影响因子：
4.5
通讯作者：
Piero Veronese
中科院分区：
文献类型：
--
作者：
G. Consonni;Piero Veronese
对称锥上的一般Wishart族是具有齐次二次方差函数的自然指数族（NEF）。利用欧几里德约当代数抽象理论的结果，证明了这类代数的条件可约性结构是成立的，并识别了相关的参数化Φ，分析了其性质。定义并讨论了Φ的富标准共轭族和平均参数μ。这个族比标准共轭族灵活得多。得到了Φ和μ的参考先验，表明它们属于富集标准共轭族；特别地，这允许我们验证参考后置总是正确的。上述结果扩展了具有简单二次方差函数的nef的可用结果。详细讨论了该理论对实对称正定矩阵锥的说明，并允许我们对富标准共轭族下多元正态模型的协方差矩阵Σ进行贝叶斯推理。特别是，常用的贝叶斯估计，如Σ和Σ -1的后验期望，以封闭形式提供。
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.
影响因子：
2.4
作者：
José M Bernardo and Adrian F M Smith-José-M-Bernardo-and-Adrian-F-M-Smith-2177748935
通讯作者：
José M Bernardo and Adrian F M Smith-José-M-Bernardo-and-Adrian-F-M-Smith-2177748935