A novel hierarchical constrained covariance model in multinomial probit regression for single- and multi-response datasets

Bayesian inference for multinomial probit (MNP) regression requires specifying appropriate restriction to identify the dispersion parameters of the underlying multivariate normal distribution. The standard practice is to use an Inverse-Wishart prior distribution for the covariance matrix with suitable constraint. In this article, we develop a new kind of prior distribution for the MNP covariance where the dispersion parameters are blocked into standard deviations and correlations with an identifiability constraint imposed on the first block. The linearity of the constraint implies that the prior joint distribution for this block can be expressed in closed form allowing us to easily update them using Markov chain Monte Carlo algorithms. The correlation parameters in the other block are modeled with a joint uniform prior which makes it easier to explore support for simpler substructures. The proposed method is subsequently generalized to work with datasets having multivariate nominal outcomes. Multiple simulation studies and a real-world public perception data analysis are presented to illustrate the performance of the proposed method in terms of parameter estimation and predictive accuracy.