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4.2 Bayesian Classification 99

Using these decision functions, clearly dependent on the Mahalanobis metric,
one can design a minimum risk Bayes classifier, usually called an optimum
classifier. In general, one obtains quadratic discriminants.
Notice that formula (4-23b) uses the true value of the Mahalanobis distance,
whereas before we have used estimates of this distance.
For the situation of equal covariance for all classes (C,=Z), neglecting constant
terms, one obtains:

For a two-class problem, the discriminant d(x)=h,(x)-h2(x) is easily computed
as:

Therefore, one obtains linear decision functions (notice the similarity to
formulas (4-5b) and (4-5c)).

Figure 4.19. Error probability of a Bayesian two-class discrimination with normal
distributions and equal prevalences and covariance.