 Solution

Let $f_i$ be the multivariate normal density of $N_p({\mu }_i,{\Sigma }_i)$. Under zero-one loss, the Bayes classifier chooses the class with the larger posterior probability. ThusEquivalently, it chooses class one when the log posterior oddsis nonnegative. The decision boundary is $D_{{\pi }_0}(x)=0$.

If ${\Sigma }_0={\Sigma }_1=\Sigma$, the $x^T{\Sigma }^{-1}x$ terms cancel, leavingan affine function. This is the linear boundary of linear discriminant analysis. If ${\Sigma }_0\ne {\Sigma }_1$, the quadratic part iswhich is nonzero, so the boundary is a possibly degenerate quadratic hypersurface, as in quadratic discriminant analysis. This is the Gaussian Bayes classifier.