When dealing with normally distributed classes, it is well known that the optimal discriminant function for two-classes is linear when the covariance matrices are equal. In this paper, we determine conditions for the optimal linear classifier when the covariance matrices are non-equal. In all the cases discussed here, the classifier is given by a pair of straight lines which is a particular case of the general equation of second degree. One of these cases is when we have two overlapping classes with equal means, which is a general case of the Minsky's Paradox for the Perceptron. Our results, which to our knowledge are the pioneering results for pairwise linear classifiers, yield a general linear classifier for this particular case, which can be obtained directly from the parameters of the distribution. Numerous other analytic results for two and d-dimensional normal vectors have been derived. Finally, we have also provided some empirical results in all the cases, and demonstrated that these linear classifiers achieve very good performance.

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Series Lecture Notes in Computer Science
Rueda, L. (Luis), & Oommen, J. (2000). The foundational theory of optimal Bayesian pairwise linear classifiers. In Lecture Notes in Computer Science.