Page 255 - Introduction to Statistical Pattern Recognition

5 Parameter Estimation 237

where

- (X-M I )%;I (X-M 1) - (Y-M )%;I (Y-M , )] (5.159)

Likewise, when Y comes from 02,
n 1
hy(X) = h(X) - ygz(X,Y) for YEW^ , (5.160)
where g2 is the same as (5.159) except that M2 and X2 are used instead of M,
and El.
When this modified classifier is used on an independent set of test sam-
ples, the resulting error is, using (5.59),

(5.161)

where + and i = 1 are used for YEW, and - and i = 2 are for YEW?. The
approximation in the last line involves replacing e/oh(X) by elo"(x). Unlike the
case of the R error, (5.161) keeps F(X) in its integrand. This makes the
integral in (5.161) particularly easy to handle. If the quadratic classifier is the
Bayes classifier, the integration with respect to w results in