Page 75 - Introduction to Statistical Pattern Recognition

3 Hypothesis Testing 57

(3.18)

The Bayes decision rule becomes a linear function of xk’s.

The Bayes Decision Rule for Minimum Cost

Often in practice, minimizing the probability of error is not the best cri-
terion to design a decision rule because the misclassifications of ol - and 02-
samples may have different consequences. For example, the misclassification
of a cancer patient to normal may have a more damaging effect than the
misclassification of a normal patient to cancer. Therefore, it is appropriate to
assign a cost to each situation as
cIj = cost of deciding X E o, when X E o, . (3.19)

Then, the conditional cost of deciding X E o, given X, r,(X), is
i’i(X) = ci 14 (XI + cj2q2(X) . (3.20)
I
The decision rule and the resulting conditional cost given X, i’ (X), are

(3.21)

and

r(X) = min[rI(X), r.z(X)I . (3.22)
The total cost of this decision is