Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 43 / 5 / i / Solution

Under equal costs for either wrong assignment, the Bayes classifier chooses the class of greatest posterior probability. With positive prior probabilities, it assigns class one when ${\pi }_1{f}_1(x)>{\pi }_2{f}_2(x)$; ties can be resolved arbitrarily. For the two multivariate normal densities with common positive-definite covariance matrix $V$, their normalizing factors cancel. The log posterior odds areTherefore the Gaussian Bayes classifier has the required linear form, withA larger prior probability for class one decreases $b$, expanding its decision region. The prior-dependent Gaussian discriminant boundary is the hyperplane $a^{T}x=b$.