Page 273 - Schaum's Outlines - Probability, Random Variables And Random Processes

266 DECISION THEORY [CHAP. 8

which is called the likelihood ratio test, and 1 is called the threshold value of the test.
Note that the likelihood ratio A(x) is also often expressed as

B. MAP Test:
Let P(Hi (x), i = 0, 1, denote the probability that Hi was true given a particular value of x. The
conditional probability P(Hi ( x) is called a posteriori (or posterior) probability, that is, a probability
that is computed after an observation has been made. The probability P(Hi), i = 0, 1, is called a priori
(or prior) probability. In the maximum a posteriori (MAP) test, the decision regions R, and R, are
selected as
R, = (x: P(H, 1 X) > P(Hl I x))
R, = {x: P(Ho (x) < P(H, I x))

Thus, the MAP test is given by

which can be rewritten as

Using Bayes' rule [Eq. (1.42)], Eq. (8.1 3) reduces to

Using the likelihood ratio A(x) defined in Eq. (8.8)' the MAP test can be expressed in the following
likelihood ratio test as

where q = P(Ho)/P(Hl) is the threshold value for the MAP test. Note that when P(H,) = P(H,), the
maximum-likelihood test is also the MAP test.

C. Neyman-Pearson Test :
As we mentioned before, it is not possible to simultaneously minimize both a(= PI) and fl(= P,).
The Neyman-Pearson test provides a workable solution to this problem in that the test minimizes fl
for a given level of a. Hence, the Neyman-Pearson test is the test which maximizes the power of the
test 1 - /? for a given level of significance a. In the Neyman-Pearson test, the critical (or rejection)
region R, is selected such that 1 - fl = 1 - P(D, ( HI) = P(Dl I HI) is maximum subject to the con-
straint a = P(D, I H,) = a,. This is a classical problem in optimization: maximizing a function subject
to a constraint, which can be solved by the use of Lagrange multiplier method. We thus construct the
objective function
J = (1 - fl) - A(a - a,) (8.1 6)
where 1 2 0 is a Lagrange multiplier. Then the critical region R, is chosen to maximize J. It can be
shown that the Neyman-Pearson test can be expressed in terms of the likelihood ratio test as