The Fisher criterion defines the optimal one-dimensional projection direction w for linear discriminant analysis as the maximiser of the ratio J(w) = (w^T S_B w) / (w^T S_W w), where S_B is the between-class scatter formed from the class mean differences and S_W is the pooled within-class scatter of the projected data. Maximising separation of the projected class means alone is ill-posed because w can be scaled without bound, so either a norm constraint or the within-class variance in the denominator is required; solving the constrained problem yields w proportional to (m2 - m1), while the full ratio yields w proportional to S_W^{-1}(m2 - m1). This direction coincides, up to scale, with the discriminant direction obtained from Gaussian class-conditional densities with shared covariance, establishing that LDA has a well-defined justification in terms of sample means and sample scatter alone, without assuming Gaussianity.
The Fisher criterion defines the optimal one-dimensional projection direction w for linear discriminant analysis as the maximiser of the ratio J(w) = (w^T S_B w) / (w^T S_W w), where S_B is the betwe…