Pattern Recognition, 4th Edition

2.2. Bayes Decision Theory

We will initially focus on the two-class case. Let ω¹, ω² be the two classes in which our patterns belong. In the sequel, we assume that the a priori probabilities P(ω¹), P(ω²) are known. This is a very reasonable assumption, because even if they are not known, they can easily be estimated from the available training feature vectors. Indeed, if N is the total number of available training patterns, and N¹, N² of them belong to ω¹ and ω², respectively, then P(ω¹) ≈ N¹/N and P(ω²) ≈ N²/N.

The other statistical quantities assumed to be known are the class-conditional probability density functions p(x|ωⁱ), i = 1, 2, describing the distribution of the feature vectors in each of the classes. If these are not known, they ...