Page 164 - Introduction to Statistical Pattern Recognition

146 Introduction to Statistical Pattern Recognition

(4.65)
=-P,q, +p2r12 ’
On the other hand, the first term of (4.64), E(h2(X)}, is the second order
moment of h (X), and is expressed by

Therefore, r2 is a function of ql, q2, o:, and 0;. Since ar2lao? =Pi, the
optimum V according to (4.27) is

v = [PIE, + P2C2]-I(M2 - MI) . (4.67)

(2) y(X) = qz(X) - q,(X): This is the Bayes discriminant function
(recalling q I (X) 6 q2(X) or h (X) = q2(X)-q I (X) 3 0). Therefore, if we can
match the designed discriminant function with the Bayes one, the result must
be desirable. For this y(X),

which is identical to (4.65). Also, note that E(h2(X)} is not affected by y(X),
and is equal to (4.66).
Thus, for both cases, the mean-square error expressions become the same
except for the constant term E{f(X)), and the resulting optimum classifiers
are the same. Also, note that the mean-square errors for these y(X)’s are a spe-
cial case of a general criterion function f(ql,q2,0:,o:). This is not surpris-
ing, since the mean-square error consists of the first and second order moments
of h(X). However, it is possible to make the mean-square error a different
type of criterion than f(ql,qz,oy,o;) by selecting y(X) in a different way.
This is the subject of the next section.