  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.
