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A multinomial probit model is proposed to examine a categorical response variable, with the main objective being the identification of influential variables in the model. To this end, a Bayesian selection technique is employed, featuring two hierarchical indicators where the first indicator denotes a variable's relevance to the categorical response, and the subsequent indicator relates to the variable's importance at a specific categorical level, aiding in assessing its impact at that level. The selection process relies on posterior indicator samples generated through an MCMC algorithm. The efficacy of our Bayesian selection strategy is demonstrated through both simulation and an application to a real-world example.

Indicator, MCMC Algorithm, Multi-task Learning