Prior-dependent Gaussian discriminant boundary ($L^{T}x-\frac{1}{2}{L}^T({\mu }_1+{\mu }_2)+\log ({\pi }_1/{\pi }_2)=0$)

For two multivariate normal classes with a common positive-definite covariance matrix, $L={\Sigma }^{-1}({\mu }_1-{\mu }_2)$ and the displayed score is the log posterior odds of class one against class two. With equal error costs, positive scores favor class one. Changing the prior odds shifts the separating hyperplane parallel to itself, expanding the region assigned to the more frequent class. On the boundary either decision has the same conditional risk.