Page 46 - Introduction to Information Optics

1.5. Signal Analysis

where

P(BJa i}=\-P(BJa i\ / = 0, 1.

To minimize the C, an optimum region B 0 can be selected. In view of the
cost in imposition of Eqs. (1.103) and (1.104), it is sufficient to select region B 0
such that the second integral of Eq. 1.105 is larger than the first integral, for
which we conclude

a = 0) Pi^ (
'
P(b/a = l) P(fl=OXC 01 -C 00 )
Let us write
^ P(b/a = 0)
a (1.107)
P(&/0 = 1)
which is the likelihood ratio, and

p a
/? A ( = iXCn, - ~n, n 10R,
/J (LI081
-p(^oxc -c )
00
01
which is simply a constant incorporated with the a priori probabilities and the
error costs. The decision rule is to select the hypothesis for which the signal is
actually absent, if a > /?.
If the inequality of Eq. 1.106 is reversed (a < /?) then one chooses B ± instead.
In other words, Bayes' decision rule (Eq. 1.106) ensures a minimum average cost
for the decision making.
Furthermore, if the costs of the errors are equal, C 10 = C 01, then the
decision rule reduces to Eqs. 1.101 and 1.102. If the decision making has
sufficient information on the error costs, then one uses Bayes's decision rule of
Eq. 1.106 to begin with. However, if information on the error costs is not
provided, then one uses the decision rule as given by Eqs. 1.101 and 1.102.
We also noted that the Bayesian decision process depends on the a priori
probabilities P(a — 0) and P(a = 1). However, if the a priori probabilities are
not provided but one wishes to proceed with decision making alone, then the
likelihood ratio test can be applied. That is, if