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 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 this 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 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：多元统一偏斜正态（SUN）分布这一广泛类别最近被证明具有重要的共轭性质。当将其用作一般概率单位模型、托比特模型及多项概率单位模型中参数向量的先验分布时，所得后验分布仍属于SUN族。尽管这一核心结论已推动贝叶斯推断与计算领域取得重要进展，但其在超越多元高斯模型对应的完全观测、离散化或删失似然函数方面的适用性仍有待探索。本文通过证明更广泛的多元统一偏斜椭圆（SUE）分布族——该族将SUN推广至椭圆密度更一般的扰动形式——能够保证在更广泛的模型类别中保持共轭性，从而填补了这一重要空白。该结论利用SUE分布在线性组合、条件化与边缘化运算下的封闭性，证明了该分布族对由一般多元回归模型诱导的似然函数具有共轭性，这些模型处理来自偏斜椭圆分布的完全观测、删失或二值化实现。这一进展将适用于共轭贝叶斯推断的模型范围扩展至椭圆族与偏斜椭圆族衍生的普遍形式，包括多元t分布与偏斜t分布等。