Page 72 - Introduction to Statistical Pattern Recognition

54 Introduction to Statistical Pattern Recognition

b* I *L2
+--1-->c2
I
Fig. 3-1 Bayes decision rule for minimum error.

boundary is shifted to the left. This argument can be extended to a general n-
dimensional case.
The computation of the Bayes error is a very complex problem except in
some special cases. This is due to the fact that E is obtained by integrating
high-dimensional density functions in complex regions as seen in (3.8). There-
fore, it is sometimes more convenient to integrate the density function of
h = h (X) of (3.5), which is one-dimensional:

(3.9)

(3.10)

where ph(h I mi) is the conditional density of h for mi. However, in general,
the density function of h is not available, and very difficult to compute.

Example 1: When the pi(X)'s are normal with expected vectors Mi and
covariance matrices C;, the decision rule of (3.5) becomes

h (X) = - In 1(X)

(3.1 1)

Equation (3.1 1) shows that the decision boundary is given by a quadratic form
in X. When CI = C2 = C, the boundary becomes a linear function of X as