Page 134 - Introduction to Statistical Pattern Recognition

116 Introduction to Statistical Pattern Recognition

(3.189)

The left side of (3.188) includes all X's which belong to w1 and are classified
correctly; hence, it should be l-~~. On the other hand, the right side of (3.188)
includes all X's which belong to 02 and are misclassified as 01; hence, it
should be E~. By the same argument, the left and right sides of (3.189)
become and l-~~, respectively. Therefore, (3.188) and (3.189) are rewritten
as

(3.190)

(3.191)

or

1 -E1
>A, (3.192)
E2
El
IB. (3.193)
1 - E2

Thus, for any given and ~2, A and B are obtained by (3.192) and (3.193).
When the minus-log likelihood ratio is used, A and B should be converted to

1 -E1
Q =- InA 2-In - (3.194)
,
E2
b=-1nB <-In -. El (3.195)
1 - €2

When the increments h (Xi) are small, the likelihood ratio will exceed the thres-
hold values A and B by only a small amount at the stage where 0; is chosen.