Conjugacy properties of multivariate unified skew-elliptical distributions

Abstract

The family of multivariate unified skew-normal (SUN) distributions has been recently shown to possess fundamental conjugacy properties. When used as priors for the vector of coefficients 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 regression models for fully-observed, censored or dichotomized realizations from skew-elliptical distributions. This key 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.