Page 45 - Introduction to Information Optics

30 1. Entropy Information and Optics

If we further assign the cost values C 00, C 01, C 10, C n , respectively, to the
following cases: (1) a = 0 is actually true, and the decision is to accept it; (2)
a = 0 is actually true, and the decision is to reject it; (3) a = 1 is actually true,
and the decision is to reject it; and (4) a = 1 is actually true, and the decision
is to accept it, then the overall average cost is

(1.102
= I I

where a 0 = 0, a l — 1, P(a i) is the a priori probability of a t, and P(B j/a i) is
the conditional probability that b falls in Bj if a t is actually true, as shown in
Fig. 1.10.
To minimize the average cost, it is desirable to select a region B 0, where
J5j = B — B 0, in such a manner that the average cost is minimum; in other
words, to place certain restrictions on the cost values C i;- so that C will be
minimum for a desirable B 0. For example, a miss or a false alarm may be
costlier than correct decisions:

(1.103)
and
(1.104)

In view of the cost function, we have

C = C 01P(a = 0) 4- C uP(a = 1) + P(a = 1)(C 10 - a = l)db
JBo
/»
- P(fl = 0)(C 01 - C 00)P(6/a = 0)db, (1.105)

B

Fig. 1.10. Hypothesis of the received events for two possible transmitted events.