Page 103 - Introduction to Statistical Pattern Recognition

3 Hypothesis Testing 85

3.3 Error Probability in Hypothesis Testing

Associated with any decision rule is a probability of error.. The proba-
bility of error is the most effective measure of a decision rule's usefulness. In
general, the calculation of error probability is very difficult, although the con-
cept is quite simple. In order to evaluate (3.8), we must perform an n-
dimensional integration in a complicated region. A more promising procedure
is to determine the density function of the likelihood ratio, and integrate it in
the one-dimensional h-space, as in (3.9) and (3.10). This is possible for nor-
mal distributions, and will be discussed in this section. However, if the distri-
butions are not normal, finding the density function of h is very difficult.
Thus, in many practical problems, we either employ experimental techniques
such as Monte Carlo simulation, or we seek bounds on the error probabilities.
We will discuss error bounds in the next section.

Linear Boundaries

When the distributions are normal with equal covariance matrices,
Zl Z2 = Z, the minus-log likelihood ratio becomes a linear function of X as
=
shown in (3.12). Since (3.12) is a linear transformation from an n-dimensional
space to one-dimension, h(X) is a normal random variable when X is a nor-
mally distributed random vector. The expected value and variance of h (X) can
be calculated as follows:

(3.96)

Since E { X I mi } = Mi, (3.96) becomes

I
q, = --(A42 - MI)Y(M2 - M,) = -q , (3.97)
2
1
q2 =+-(A42 -M,)Y(M2 -M,)=+q. (3.98)
2

Also,