Multinomial probit (mnp) models are fundamental and widely-applied regression models for categorical data. Fasano and Durante (2022) proved that the class of unified skew-normal distributions is conjugate to several mnp sampling models. This allows to develop Monte Carlo samplers and accurate variational methods to perform Bayesian inference. In this paper, we adapt the abovementioned results for a popular special case: the discrete-choice mnp model under zero mean and independent Gaussian priors. This allows to obtain simplified expressions for the parameters of the posterior distribution and an alternative derivation for the variational algorithm that gives a novel understanding of the fundamental results in Fasano and Durante (2022) as well as computational advantages in our special settings.

Multinomial probit models are widely-implemented representations which allow both classification and inference by learning changes in vectors of class probabilities with a set of p observed predictors. Although various frequentist methods have been developed for estimation, inference and classification within such a class of models, Bayesian inference is still lagging behind. This is due to the apparent absence of a tractable class of conjugate priors, that may facilitate posterior inference on the multinomial probit coefficients. Such an issue has motivated increasing efforts toward the development of effective Markov chain Monte Carlo methods, but state-of-the-art solutions still face severe computational bottlenecks, especially in large p settings. In this article, we prove that the entire class of unified skew-normal (SUN) distributions is conjugate to a wide variety of multinomial probit models, and we exploit the SUN properties to improve upon state-of-art-solutions for posterior inference and classification both in terms of closed-form results for key functionals of interest, and also by developing novel computational methods relying either on independent and identically distributed samples from the exact posterior or on scalable and accurate variational approximations based on blocked partially-factorized representations. As illustrated in a gastrointestinal lesions application, the magnitude of the improvements relative to current methods is particularly evident, in practice, when the focus is on large p applications.

Multinomial probit (MNP) model for discrete choice modeling is often used in economics, market research, political sciences and transportation. It models the choices made by agents given their demographic characteristics and/or the features of the K 2 available choice alternatives. Examples include the study of consumer's purchasing behavior (e.g., McCulloch et al. (2000); Imai and van Dyk (2005)); voting behavior in multi-party elections (e.g., Quinn et al. (1999)); and choice of different modes of transportation (e.g., Bolduc (1999)). Details of the MNP model in which choices depend on predictors in a linear fashion is studied in McFadden et al.(1973); McFadden(1989); Keane(1992); McCulloch and Rossi (1994); Nobile (1998); McCulloch et al. (2000); Imai and van Dyk (2005); Train (2009); Burgette and Nordheim (2012) among others. Among widely used multinomial choice modeling procedures are the multinomial logit model (e.g., McFadden et al. (1973); Train (2009)) and multinomial probit model (e.g., McFadden (1989); McCulloch and Rossi (1994); Imai and van Dyk (2005)). The former relies on an assumption that a choice outcome is independent of removal (or introduction) of an irrelevant choice alternative while the latter including MPBART does not make this restrictive assumption.