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4.2 Bayesian Classification 95

We are obviously interested in minimizing an average risk computed for an
arbitrarily large number of cork stoppers. The Bayes rule for minimum risk
achieves this through the minimization of the individual conditional risks R(a, 1 x).
Let us assume first that wrong decisions imply the same loss, which can be
scaled to a unitary loss:

In this situation, since all posterior probabilities add up to one, we have to
minimize:

This corresponds to maximizing P(oi 1 x), i.e., the Bayes decision rule for
minimum risk corresponds to the generalized version of (4- 13a):

Decide wi if P(u, 1 x) > P(u, ( x). b'j * i . (4- 1%)

In short:
The Bayes decision rule for minimum risk, when correct decisions have zero loss
and wrong decisions have equal losses, corresponds to selecting the class with
maximum posterior probability.

The decision function for class mi is therefore:

Let us now consider the situation of different losses for wrong decisions,
assuming first, for the sake of simplicity, that c=2. Taking into account expressions
(4-17a) and (4-17b), it is readily concluded that we will decide wl if:

Therefore, the decision threshold with which the likelihood ratio is compared is
inversely weighted by the losses (compare with 4-15a). This decision rule can be
implemented as shown in Figure 4.16 (see also Figure 4.6).
Equivalently, we can use the following adjustedprevalences: