Page 76 - Introduction to Statistical Pattern Recognition

58 Introduction to Statistical Pattern Recognition

The boundary which minimizes r of (3.23) can be found as follows.
First, rewrite (3.23) as a function of Ll alone. This is done by replacing
[ pi(X)CUr with 1 - I, pi(X)dX, since L I and L2 do not overlap and cover the
I
entire domain. Thus,

Now our problem becomes one of choosing L I such that r is minimized. Sup-
pose, for a given value of X, that the integrand of (3.24) is negative. Then we
can decrease r by assigning X to Ll. If the integrand is positive, we can
decrease r by assigning X to L2. Thus the minimum cost decision rule is to
assign to Ll those X’s and only those X’s, for which the integrand of (3.24) is
negative. This decision rule can be stated by the following inequality:

01
(c 12-C22)P2P 2W) 3 (C2I --c I I )P IP I (X) (3.25)
0-
or

(3.26)

This decision rule is called the Bayes test for- minimum cost.
Comparing (3.26) with (3.4), we notice that the Bayes test for minimum
cost is a likelihood ratio test with a different threshold from (3.4), and that the
selection of the cost functions is equivalent to changing the a priori probabili-
ties Pi. Equation (3.26) is equal to (3.4) for the special selection of the cost
functions

This is called a symmetrical cost function. For a symmetrical cost function, the
cost becomes the probability of error, and the test of (3.26) minimizes the pro-
bability of error.
Different cost functions are used when a wrong decision for one class is
more critical than one for the other class.