Hierarchical Bayesian variable selection in the probit model with mixture of nominal and ordinal responses

Multi-class classification problems have been studied for pure nominal and pure ordinal responses. However, there are some cases where the multi-class responses are a mixture of nominal and ordinal. To address this problem we build a hierarchical multinomial probit model with a mixture of both types of responses using latent variables. The nominal responses are each associated to distinct latent variables whereas the ordinal responses have a single latent variable. Our approach first treats the ordinal responses as a single nominal category and then separates the ordinal responses within this category. We introduce sparsity into the model using Bayesian variable selection (BVS) within the regression in order to improve variable selection classification accuracy. Two indicator vectors (indicating presence of the covariate) are used, one for nominal and one for ordinal responses. We develop efficient posterior sampling. Using simulated data, we compare the classification accuracy of our method to existing ones.