The broad class of multivariate unified skew-normal (SUN) distributions has been recently shown to possess fundamental conjugacy properties. When used as priors for the vector of parameters in general probit, tobit, and multinomial probit models, these distributions yield posteriors that still belong to the SUN family. Although such a core 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 an important 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 such a family is conjugate to the likelihood induced by general multivariate regression models for fully-observed, censored or dichotomized realizations from skew-elliptical distributions. This advancement substantially 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.

翻译：近期研究表明，多元统一偏正态分布族具有基本共轭性质。当该分布族作为广义probit、tobit及多项probit模型中参数向量的先验分布时，其后验仍属于同一分布族。尽管这一核心结论已推动贝叶斯推断与计算的重大进展，但其在多元高斯模型完全观测、离散化或截断实现似然函数之外的适用性仍有待探索。本文通过证明多元统一偏椭圆分布族（可视为统一偏正态分布对椭圆密度更一般摄动的推广）在更广泛模型类别（不限于基于完全观测、离散化或截断高斯分布的模型）中具有共轭性，填补了这一重要空白。该结论利用统一偏椭圆分布在线性组合、条件分布和边缘化下的封闭性质，证明该分布族与偏椭圆分布完全观测、截断或二分实现对应的广义多元回归模型诱导的似然函数具有共轭性。这一突破性进展将适用于共轭贝叶斯推断的模型范畴显著扩展至椭圆及偏椭圆族的一般形式，其中涵盖多元t分布与偏t分布等。