Page 236 - Introduction to Statistical Pattern Recognition

218 Introduction to Statistical Pattern Recognition

-- (5.107)
1
n
Equation (5.107) can be derived by replacing ;(X) in (5.97) with pi(X). Qua-
tion (5.107) is proportional to Iln because Ed(Ah(X)Ah(Y)) is proportional to
11%.

Substituting (5.104)-(5.107) into (5.103), and ignoring the terms propor-
tional to l/Nin,

As we discussed in the previous section, Vard { 61 is proportional to lln2
when the Bayes classifier is used for normal distributions. Therefore, Var(;]
of (5.108) is dominated by the first two terms which are due to the finite test
set. A comparison of (5.108) and (5.49) shows that the effect of the finite
design set appears in El and E2 of (5.108) instead of EI and ~2 of (5.49). That
is, the bias due to the finite design set increases the variance proportionally.
However, since (Ei -&,)-I/%, this effect can be ignored. It should be noted
,.
that Vard( E} could be proportional to 117l if the classifier is not the Bayes.
Thus, we can draw the following conclusions from (5.102) and (5.108).
When both design and test sets are finite,
1. the bias of the classification error comes entirely from the finite design
set, and
2. the variance comes predominantly from the finite test set.