Discrimination and Classification, Round 2

In this chapter we continue the theory of classification developed in Chapter 5 on a somewhat more general level. We start out with some basic consideration of optimality. In the notation introduced in Section 5.4, Y will denote a p-variate random vector measured in k groups (or populations). Let X denote a discrete random variable that indicates group membership, i.e., takes values 1, … , k. The probabilities (1)$$ {\pi _j} = \Pr \left[ {X = j} \right]\quad j = 1, \ldots ,k, $$ will be referred to as prior probabilities, as usual. Suppose that the distribution of Y in the jth group is given by a pdf f _j (y), which may be regarded as the conditional pdf of Y, given X = j. Assume for simplicity that Y is continuous with sample space ℝp in each group. Then the joint pdf of X and Y, as seen from Sec tion 2.8, is (2)$$ {f_{XY}}\left( {j,y} \right) = \left\{ \begin{gathered} {\pi _j}{f_j}\left( y \right)\;for{\kern 1pt} j = 1, \ldots ,k,y \in {\mathbb{R}^p} \hfill \\ 0\quad otherwise. \hfill \\ \end{gathered} \right. $$