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4.2 Bayesian Classification 101

Figure 4.20. Two classes with symmetric distributions and the same standard
deviation. (a) Normal; (b) Cauchy; (c) Logistic.

The optimum classifier for the three cases uses the same decision threshold
value: 1.15. However, the classification errors are different:

Normal: Pe = 1 - erfl2.312) = 12.5%
Cauchy: Pe = 22.7%
Logistic: Pe = 24.0%

As previously mentioned, statistical software products use a pooled covariance
matrix when performing discriminant analysis. The influence of this practice on the
obtained error, compared with the theoretical optimal Bayesian error, is discussed
in detail in Fukunaga (1990). Experimental results show that when the covariance
matrices exhibit limited deviations from the pooled covariance matrix, then the
designed classifier has a performance similar to the optimal performance with
equal covariances. This is reasonable, since for covariance matrices that are not
very distinct, the difference between the optimum quadratic solution and the sub-
optimum linear solution should only be noticeable for the patterns that are far away
from the prototypes, as shown in Figure 4.21.

Figure 4.21. Discrimination of two classes with optimum quadratic classifier (solid
line) and sub-optimum linear classifier (dotted line).