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4 Parametric Classifiers 125

best classifier in all cases. No parametric classifier will exceed the perfor-
mance of the likelihood ratio test.

4.1 The Bayes Linear Classifier

For two normal distributions, the Bayes decision rule can be expressed as
a quadratic function of the observation vector X as

1 1
-(X - MdTC;'(X -MI) - -(X - M2)7C,'(X - M2)
2
2
(4.1)

When both covariance matrices are equal, that is when C, =C2 =C, (4.1)
reduces to a linear function of X as

(4.2)

Furthermore, if the covariance matrix is the identity matrix, I, then we can
view X as an observation corrupted by white noise. The components of X are
uncorrelated and have unit variance. The Bayes decision rule reduces to

There have been a number of classifiers, such as the correlation classifier
and the matched filter, developed in the communication field for signal detec-
tion problems [l]. We will discuss here how these classifiers are related to the
Bayes classifier.

Correlation Classifier

The product MTX is called the correlation between Mi and X. When X
consists of time-sampled values taken from a continuous random process, x(f),
we can write the correlation as

M:X = &?i(fj)xq) . (4.4)
j=l
In the continuous case, the correlation becomes an integral, that is