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

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 $a^{T}x=b$.

For equal prior probabilities, $b=\frac{1}{2}{a}^T({\mu }_1+{\mu }_2)$. The projected score is normal under either class because it is a linear image of a multivariate normal vector, and its variance isMoreover $a^T{\mu }_1-a^T{\mu }_2={\delta }^2$, so the midpoint threshold lies ${\delta }^2/2$ from either projected mean. HenceStandardizing the two normal distributions, with $\Phi$ the standard normal distribution function, givesThus the equal-covariance Gaussian classification error is $p(\delta )=\Phi (-\delta /2)$. The Mahalanobis distance $\delta$ between the class means is their separation in units of within-class noise. It yields $p\to 1/2$ as $\delta \downarrow 0$ and $p\to 0$ as $\delta \to \infty$. The stipulated $\delta >0$ means the two class means are distinct.