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.
Lane F. Burgette. David Puelz. P. Richard Hahn. "A Symmetric Prior for Multinomial Probit Models." Bayesian Anal. 16 (3) 991 - 1008, September 2021. https://doi.org/10.1214/20-BA1233