Page 79 - Introduction to Statistical Pattern Recognition
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3 Hypothesis Testing 61
--y$
-0.34’
p=4 2 I 1/2 114
1
Fig. 3-3 Neyman - Pearson boundaries.
The Minimax Test
In the Bayes test for minimum cost, we notice that the likelihood ratio is
compared with a threshold value which is a function of Pi. Therefore, in order
to design a decision rule which minimizes the cost, we need to know the
values of Pi beforehand. After the design is completed, the decision rule stays
optimum only if the Pj’s stay the same. Unfortunately in practice, the Pi’s
vary after the decision rule is fixed. The minimax test is designed to protect
the performance of the decision rule, even if the Pi’s vary unexpectedly.
First, let us express the cost of (3.24) in terms of PI. Since
PI + P2 = 1, P2 is uniquely determined by P Inserting P2 = 1-P I into
(3.24), and replacing [ p I (X)dX by 1 - I p I (X)dX,
I L?
