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