Bayesian Conjugacy in Probit, Tobit, Multinomial Probit and Extensions: A Review and New Results

A broad class of models that routinely appear in several fields can be expressed as partially or fully discretized Gaussian linear regressions. Besides including classical Gaussian response settings, this class also encompasses probit, multinomial probit and tobit regression, among others, thereby yielding one of the most widely-implemented families of models in routine applications. The relevance of such representations has stimulated decades of research in the Bayesian field, mostly motivated by the fact that, unlike for Gaussian linear regression, the posterior distribution induced by such models does not seem to belong to a known class, under the commonly assumed Gaussian priors for the coefficients. This has motivated several solutions for posterior inference relying either on sampling-based strategies or on deterministic approximations that, however, still experience computational and accuracy issues, especially in high dimensions. The scope of this article is to review, unify and extend recent advances in Bayesian inference and computation for this core class of models. To address such a goal, we prove that the likelihoods induced by these formulations share a common analytical structure implying conjugacy with a broad class of distributions, namely the unified skew-normal (SUN), that generalize Gaussians to include skewness. This result unifies and extends recent conjugacy properties for specific models within the class analyzed, and opens new avenues for improved posterior inference, under a broader class of formulations and priors, via novel closed-form expressions, iid samplers from the exact SUN posteriors, and more accurate and scalable approximations from variational Bayes and expectation-propagation. Such advantages are illustrated in simulations and are expected to facilitate the routine-use of these core Bayesian models, while providing novel frameworks for studying theoretical properties and developing future extensions. Supplementary materials for this article are available online.