Page 84 - Introduction to Statistical Pattern Recognition

66 Introduction to Statistical Pattern Recognition

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Fig. 3-7 The operating characteristic of Data I-I on a special coordinate sys-
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Multihypothesis Tests

When the samples are known to come from L classes, we can generalize
the binary hypothesis testing problem.
First, if our decision is simply based on probabilities, the decision rule is
qk(X) = max qi(X) + X E wk . (3.43)
I
Or, by the Bayes theorem,

.
Pkpk(X) = max Pipi(X) + X E ok (3.44)
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Since X belongs to wj with the probability of qj(X), the decision rule of (3.43)
misclassifys X from oi (j # k) to mk with the same probability. Summing up
these, the conditional probability of error given X, due to (3.43), becomes