Page 77 - Introduction to Statistical Pattern Recognition

3 Hypothesis Testing 59

The Neyman - Pearson Test

The Neyman-Pearson test follows from a third formulation of the
hypothesis test problem. Recall that we can commit two types of errors in a
two-class decision problem. Let the probabilities of these two errors again be
and E~- The Neyman-Pearson decision rule is the one which minimizes
subject to E;! being equal to a constant, say Q. To determine this decision rule,
we must find the minimum of
(3.28)
I' = E1 + ME2 - Eo) 7
where p is a Lagrange multiplier. Inserting and of (3.8) into (3.28),

Using the same argument as in the derivation of (3.25) from (3.24), I' can be
minimized by selecting L and L as

(3.30)

or

(3.31)

Comparing (3.3 1) with (3.26), we can conclude that the Neyman-Pearson test
does not offer any new decision rule but relies on the likelihood ratio test, as
did the Bayes test. However, the preceding discussion shows that the likeli-
hood ratio test is the test which minimizes the error for one class, while main-
taining the error for the other class constant.
The threshold p is the solution, for a given Q, of the following equation:

E2 = ,, p*(X)dX = Eo (3.32)
I
Or, using the density function of h(X) of (3.10).