Symmetric Bayesian Multinomial Probit Models

Standard Bayesian multinomial probit (MNP) models that are fit using different base categories can give different predictions. Therefore, we propose the symmetric MNP model, which does not make reference to a base category. To achieve this, we employ novel sum-to-zero identifying restrictions on the latent utilities and regression coefficients that define the model. This results in a model whose prior and likelihood are symmetric with respect to relabeling the outcome categories. As part of this model, we define a prior on the space of symmetric, positive-semidefinite matrices that allows for an efficient marginal data augmentation Gibbs sampling algorithm. We demonstrate our methods on two consumerchoice datasets where different base categories give different posterior inferences. The symmetric MNP sensibly gives predictions that are between those of the differing standard MNP specifications, while improving mixing in the Gibbs sampler. We also propose a symmetric MNP that assumes independent but heteroscedastic errors, which may be a useful compromise between a model that assumes independence of irrelevant alternatives, and one that allows arbitrary substitution patterns.