Pattern Recognition, 4th Edition

2.4.2. The Bayesian Classifier for Normally Distributed Classes

Our goal in this section is to study the optimal Bayesian classifier when the involved pdfs, p(x|ωⁱ), i = 1, 2,…, M (likelihood functions of ωⁱ with respect to x), describing the data distribution in each one of the classes, are multivariate normal distributions, that is, N(μⁱ, Σⁱ), i = 1, 2,…, M. Because of the exponential form of the involved densities, it is preferable to work with the following discriminant functions, which involve the (monotonic) logarithmic function ln(·):(2.33)or(2.34)where cⁱ is a constant equal to –(l/2) ln 2π – (1/2) ln|Σⁱ|. Expanding, we obtain(2.35)