Page 275 - Probability, Random Variables and Random Processes

CHAP. 8) DECISION THEORY

(Prob. 8.8)

where the threshold value q of the test is equal to the Lagrange multiplier A, which is chosen to satisfy
the contraint a = a,.

D. Bayes' Test :
Let Cij be the cost associated with (D, , Hi), which denotes the event that we accept Hi when Hi is
true. Then the average cost, which is known as the Bayes' risk, can be written as

where P(Di, Hi) denotes the probability that we accept Hi when Hj is true. By Bayes' rule (1.42), we
have

In general, we assume that

since it is reasonable to assume that the cost of making an incorrect decision is higher than the cost of
making a correct decision. The test that minimizes the average cost e is called the Bayes' test, and it
can be expressed in terms of the likelihood ratio test as (Prob. 8.10)

(8.21)

Note that when C,, - Coo = Col - Cll , the Bayes' test (8.21) and the MAP test (8.15) are identical.

E. Minimum Probability of Error Test:
If we set Coo = Cll = 0 and Col = Clo = 1 in Eq. (8.18), we have
e = P(Dl, Ho) + P(Do, HI) = P,
which is just the probability of making an incorrect decision. Thus, in this case, the Bayes' test yields
the minimum probability of error, and Eq. (8.21) becomes

We see that the minimum probability of error test is the same as the MAP test.

F. Minimax Test :
We have seen that the Bayes' test requires the a priori probabilities P(Ho) and P(Hl). Frequently,
these probabilities are not known. In such a case, the Bayes' test cannot be applied, and the following
minimax (min-max) test may be used. In the minimax test, we use the Bayes' test which corresponds
to the least favorable P(Ho) (Prob. 8.12). In the minimax test, the critical region RT is defined by