Page 129 - Introduction to Statistical Pattern Recognition
P. 129
3 Hypothesis Testing 111
(3.175)
where h (X) = - In p I (X)/p2(X) is the likelihood ratio for an individual obser-
vation vector. The s of (3.175) is compared with a threshold such as InPIIP2
for the Bayes classifier, and the group of the samples (XI, . . . ,Xm) is
classified to wI or 02, depending on s c 0 or s > 0 (assuming 1nP I/P2 = 0).
The expected values and variances of s for w1 and w2 are
m
E(slw;} = ~.E(h(Xj)lwj) =m qi , (3.176)
j=l
since the h (X,)’s are also independent and identically distributed with mean q;
and variance 0;.
When the Bayes classifier is used for h (X), it can be proved that q I I 0
and q2 2 0 as follows:
(3.178)
