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Multinomial probit models are routinely-implemented representations forlearning how the class probabilities of categorical response data change with pobserved predictors. Although several frequentist methods have been developedfor estimation, inference and classification within such a class of models,Bayesian inference is still lagging behind. This is due to the apparent absenceof a tractable class of conjugate priors, that may facilitate posteriorinference on the multinomial probit coefficients. Such an issue has motivatedincreasing efforts toward the development of effective Markov chain Monte Carlomethods, but state-of-the-art solutions still face severe computationalbottlenecks, especially in high dimensions. In this article, we show that theentire class of unified skew-normal (SUN) distributions is conjugate to severalmultinomial probit models. Leveraging this result and the SUN properties, weimprove upon state-of-the-art solutions for posterior inference andclassification both in terms of closed-form results for several functionals ofinterest, and also by developing novel computational methods relying either onindependent and identically distributed samples from the exact posterior or onscalable and accurate variational approximations based on blockedpartially-factorized representations. As illustrated in simulations and in agastrointestinal lesions application, the magnitude of the improvementsrelative to current methods is particularly evident, in practice, when thefocus is on high-dimensional studies.

A Class of Conjugate Priors for Multinomial Probit Models which Includes the Multivariate Normal One