Globally and symmetrically identified Bayesian multinomial probit model

Bayesian multinomial probit models have been widely used to analyze discrete choice data. Existing methods have some shortcomings in parameter identification or sensitivity of posterior inference to labeling of choice objects. The main task of this study is to simultaneously deal with these problems. First we propose a globally and symmetrically identified multinomial probit model with covariance matrix positive semidefinite. However, it is difficult to design an efficient Bayesian algorithm to fit the model directly because it is infeasible to sample a positive semidefinite matrix from a regular distribution. Then we develop a projected model for the above proposed model by projection technique. This projected model is equivalent to the former one, but equips with a positive definite covariance matrix. Finally, based on the latter model, we develop an efficient Bayesian algorithm to fit it by using modern Markov chain Monte Carlo techniques. Through simulation studies and an analysis of clothes detergent purchases data, we demonstrated that our approach not only solved the identifiability problem, but also showed robustness and satisfactory estimation accuracy, while the computation costs were comparable.