Page 154 - Introduction to Statistical Pattern Recognition

136 Introduction to Statistical Pattern Recognition

v, = -VT[P,M, + P,M2]. (4.37)
Equation (4.37) indicates that, if V is multiplied by a constant a, v, is also
changed by a factor of a. The decision made by VTX + v, >< 0 is equivalent
to the decision of aVTX + avo >< 0 for any positive a. This confirms that the
scale of V is irrelevant in our discussion.

Optimum Design for Normal Distributions

Theoretical approach: When the distributions of h (X) are normal, we
can find the V and vo which minimize the Bayes error in the h-space. The
Bayes error in the h-space is expressed as a function of qj and 0' as

(4.38)

For this criterion, the derivatives of E are

(4.39)

(4.40)

(4.41)

(4.42)

Therefore, from (4.26)

(4.43)

That is, v, must be selected to make the two density functions of h (X) equal at
h(X) = 0. Substituting (4.43) into (4.39) and (4.40), and using (4.28)