多元统一偏斜椭圆分布的共轭性质
Conjugacy properties of multivariate unified skew-elliptical distributions
AI总结：
本文证明多元统一偏斜椭圆（SUE）分布在更广泛的多元回归与删失、二分化数据模型中具有共轭性，将贝叶斯共轭推断扩展至椭圆及偏斜椭圆族。
摘要：
近期研究表明，广泛的多元统一偏斜正态（SUN）分布类具有重要的共轭性质。当这些分布在 probit、tobit 和多项 probit 模型中用作系数向量的先验时，所得后验仍属于 SUN 族。尽管这一结果推动了贝叶斯推断与计算的重要进展，但其在多元高斯模型的完全观测、离散化或删失实现所对应似然之外的适用性仍未得到探索。本文填补了这一空白，证明更广泛的多元统一偏斜椭圆（SUE）分布族将 SUN 扩展到椭圆密度的更一般扰动，可为更广泛的模型类保证共轭性，而不限于依赖完全观测、离散化或删失高斯分布的模型。该结果利用 SUE 在线性组合、条件化和边缘化下的封闭性，证明该分布族与多元回归模型所诱导的似然共轭，这些模型处理来自偏斜椭圆分布的完全观测、删失或二分化实现。这一进展将支持共轭贝叶斯推断的模型集合扩大到由椭圆族和偏斜椭圆族产生的一般形式，包括多元 Student's t 和偏斜 t 等。
英文摘要：
The broad class of multivariate unified skew-normal (SUN) distributions has been recently shown to possess important conjugacy properties. When used as priors for the coefficients vector in probit, tobit, and multinomial probit models, these distributions yield posteriors that still belong to the SUN family. Although this result has led to important advancements in Bayesian inference and computation, its applicability beyond likelihoods associated with fully-observed, discretized, or censored realizations from multivariate Gaussian models remains yet unexplored. This article covers such a gap by proving that the wider family of multivariate unified skew-elliptical (SUE) distributions, which extends SUNs to more general perturbations of elliptical densities, guarantees conjugacy for broader classes of models, beyond those relying on fully-observed, discretized or censored Gaussians. Such a result leverages the closure under linear combinations, conditioning and marginalization of SUE to prove that this family is conjugate to the likelihood induced by multivariate regression models for fully-observed, censored or dichotomized realizations from skew-elliptical distributions. This advancement enlarges the set of models that enable conjugate Bayesian inference to general formulations arising from elliptical and skew-elliptical families, including the multivariate Student's t and skew-t, among others.
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