Sparse Bayesian Learning (SBL) is a widely used framework which helps us to deal with two basic problems of machine learning, to avoid overfitting of the model and to incorporate prior knowledge into it. In this work, multiple linear regression models under the SBL framework are used for the problem of multiclass classification when multiple subjects are available. As a case study, we apply our method to the detection of Steady State Visual Evoked Potentials (SSVEP), a problem that arises frequently into the Brain Computer Interface (BCI) paradigm. The multiclass classification problem is decomposed into multiple regression problems. By solving these regression problems, a discriminant vector is learned for further processing. In addition the adoption of the kernel trick and the special treatment of produced similarity matrix provides us with the ability to use a Leave-One-Subject-Out training procedure resulting in a classification system suitable for subject independent classification. Extensive comparisons are carried out between the proposed algorithm, the SVM classifier and the CCA based methodology. The experimental results demonstrate that the proposed algorithm outperforms the competing approaches, in terms of classification accuracy and Information Transfer Rate (ITR), when the number of utilized EEG channels is small.