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4.2 Bayesian Classification 103

Table 4.2. Covariances per class and pooled covariance for the cork stoppers (two
classes).

CI cz Pooled C

As already mentioned in section 2.2, using decision functions based on the
individual covariance matrices, instead of a pooled covariance matrix, will produce
quadratic decision boundaries. However, a quadratic classifier is less robust (more
sensitive to parameter deviations) than a linear one, especially in high dimensional
spaces, and needs a much larger training set for adequate design (see e.g. Fukunaga
and Hayes, 1989).

4.2.3 Reject Region
In practical applications of pattern recognition, it often happens that simply using a
decision rule such as (4-13a) or (4-18c) will produce many borderline decisions,
very sensitive to noise present in the data and to the numerical accuracy of the
classifiers. For instance, for the cork stoppers data with the decision border
depicted in Figure 4.1 1, many patterns lying near the border can change the
assigned class by only a slight adjustment. This means that, in fact, such patterns
largely share the characteristics of both classes. For such patterns, it is often more
advisable to place them in a special class for further inspection. This is certainly a
must in some applications, e.g., in the medical field, where borderline cases
between normal and abnormal states deserve further analysis. One way to do this is
to attach qualifications to the computed posterior probabilities P(a(x) for the
decided class q. We could, for instance, attach the qualitative "definite" if the
probability is bigger than 0.9, "probable" if it is between 0.9 and 0.8 and "possible"
if it is below 0.8. In this way, the cork stopper case 55 (Figure 4.22) would be
classified as a "possible" cork of class "super", and case 54 as a "probable" cork of
class "average".
Instead of attaching qualitative descriptions to the obtained classifications, a
method used in certain circumstances is to stipulate the existence of a special class,
the reject class or region.
Let us denote:

w*: the decided class;
Ui: the class with maximum posterior probability, i.e., P(uilx) = max P(w,lx)
for all classes w, # mi.

The Bayes rule can then be written simply @=ai.