  Fitted probabilities from widely used Bayesian multinomial probit models can
depend strongly on the choice of a base category, which is used to uniquely
identify the parameters of the model. This paper proposes a novel
identification strategy, and associated prior distribution for the model
parameters, that renders the prior symmetric with respect to relabeling the
outcome categories. The new prior permits an efficient Gibbs algorithm that
samples rank-deficient covariance matrices without resorting to
Metropolis-Hastings updates.
