A recent development in penalized probit modelling using a hierarchical Bayesian approach has led to a sparse binomial (two-class) probit classifier that can be trained via an EM algorithm. A key advantage of the formulation is that no tuning of hyperparameters relating to the penalty is needed thus simplifying the model selection process. The resulting model demonstrates excellent classification performance and a high degree of sparsity when used as a kernel machine. It is, however, restricted to the binary classification problem and can only be used in the multinomial situation via a one-against-all or one-against-many strategy. To overcome this, we apply the idea to the multinomial probit model. This leads to a direct multi-classification approach and is shown to give a sparse solution with accuracy and sparsity comparable with the current state-of-the-art. Comparative numerical benchmark examples are used to demonstrate the method
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