Parameter-expanded data augmentation for analyzing multinomial probit models

The multinomial probit model has been a prominent tool to analyze nominal categorical data, but the computational complexity of maximum likelihood functions presents challenges in the usage of this model. Furthermore, the model identification is extremely tenuous and usually necessitates the covariance matrix of the latent multivariate normal variables to be a restricted covariance matrix, which brings a rigorous task for both likelihood-based estimation and Markov chain Monte Carlo (MCMC) sampling. We tackle this issue by constructing a non-identifiable model and developing parameter-expanded data augmentation. Our proposed methods circumvent sampling a restricted covariance matrix commonly implemented by a painstaking Metropolis-Hastings (MH) algorithm and enable to sample a covariance matrix without restriction through a Gibbs sampler. Therefore, our proposed methods advance the convergence and mixing of the MCMC components considerably. We investigate our proposed methods along with the method based on the identifiable model through simulation studies and further illustrate their performance by an application to consumer choice on liquid laundry detergents data.