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4.2 Bayesian Classification 93

Figure 4.13. Influence of the prevalence threshold on the classification errors,
represented by the shaded areas (dark grey represents the errors for class a). (a)
Equal prevalences; (b) Unequal prevalences.

Figure 4.14 shows the classification matrix obtained with the prevalences
computed in (4-12), which are indicated in the Group row.
We see that indeed the decision threshold deviation led to a better performance
for class y than for class 4. This seems reasonable since class & now occurs
more often. Since the overall error has increased, one may wonder if this influence
of the prevalences was beneficial after all. The answer to this question is related to
the topic of class~~cation risks, which we will present now.
Let us assume that the cost of a wl ("super") cork stopper is 0.025 € and the cost
of a WL ("average") cork stopper is 0.015 €. Suppose that the wl cork stoppers are
to be used in special bottles whereas the y cork stoppers are used in normal
bottles.
A wrong classification of a super quality cork stopper will amount to a loss of
0.025-0.015=0.01 € (see Figure 4.15). A wrong classification of an average cork
stopper leads to its rejection with a loss of 0.015 €.
Denote:

SB - Action of using a cork stopper in special bottles.
NB - Action of using a cork stopper in normal bottles.
q=S (class super); w2=A (class average)

1 Total 1 73.00000 4 1 59 1
Figure 4.14. Classification results of the cork stoppers with unequal prevalences:
0.4 for class wl and 0.6 for class y.