Page 258 - Applied Statistics Using SPSS, STATISTICA, MATLAB and R

6.3 Bayesian Classification 239

Let us assume, first, that wrong decisions imply the same loss, which can be
scaled to a unitary loss:

0 if i = j
λ ij = (αλ i |ω j ) =  . 6.22a
 1 if i ≠ j

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

R(α i | ) x = ∑ P(ω j | ) x =1 − P(ω i | ) x . 6.22b
j ≠i

This corresponds to maximising P(ω i | x), i.e., the Bayes decision rule for
minimum risk corresponds to the generalised version of 6.15a:

Decide ω i if P(ω i | ) x > P(ω j | x ), ∀ j ≠ i . 6.22c

Thus, the decision function for class ω i is the posterior probability,
P
g i (x ) = (ω i | ) x , and the classification rule amounts to selecting the class with
maximum posterior probability.
Let us now consider the situation of different losses for wrong decisions,
assuming, for the sake of simplicity, that c = 2. Taking into account expressions
6.20a and 6.20b, it is readily concluded that we will decide ω 1 if:

>
λ P (ω 1 | x ) λ P (ω 2 | ) x ⇒ ( p x |ω 1 )λ P (ω 1 ) > ( p x |ω 2 )λ P (ω 2 ) . 6.23
21
21
12
12

This is equivalent to formula 6.17 using the following adjusted prevalences:

λ P (ω ) λ P (ω )
P * (ω 1 ) = 21 1 ; P * (ω 2 ) = 12 2 . 6.23a
+
+
λ P (ω 1 ) λ P (ω 2 ) λ P (ω 1 ) λ P (ω 2 )
21
12
12
21

STATISTICA and SPSS allow specifying the priors as estimates of the sample
composition (as in 6.14) or by user assignment of specific values. In the latter the
user can adjust the priors in order to cope with specific classification risks.

Example 6.7
Q: Redo Example 6.6 using adjusted prevalences that take into account 6.14 and
the loss matrix 6.19. Compare the classification risks with and without prevalence
adjustment.
A: The losses are λ 12 = 0.015 and λ 21 = 0.01. Using the prevalences 6.14, one
*
*
obtains P (ω 1) = 0.308 and P (ω 2 ) = 0.692. The higher loss associated with a
*
wrong classification of a ω 2 cork stopper leads to an increase of P ( ω 2 ) compared
*
with P ( ω 1). The consequence of this adjustment is the decrease of the number of