  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.
