Derivation of LDA log likelihood ratio one-to-one classifier

In order to be able to design the likelihood ratio classifier, we make the following assumptions: • The conditional probability density functions p(x|ci) are normal with mean μi and covariance Ci • All classes have different mean, but the same covariance, i.e. Ci = Cw. This covariance is called the within class variance. The variance within a class is e.g. caused by expression and illumination variations in face recognition and assumed to have the same impact on the appearance for every face. • There is an infinite number of classes ci, the means of which are normally distributed with mean μb and covariance Cb where the subscript b denotes between class. We can now rewrite equation 4 by substituting the summation by an integral (there is an infinite number of classes) and writing pw(x|μ) for the conditional probability on a sample x given it is of a class with mean μ. The subscript w means within class.