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Unformatted text preview: on rule is the one that minimizes pe . That is why some books call the MAP rule the minimum
probability of error rule.
Examples are given in the remainder of this section, illustrating the use of ML and MAP decision
rules for hypothesis testing.
Example 2.11.1 Suppose you have a coin and you know that either : H1 : the coin is biased,
showing heads on each ﬂip with probability 2/3; or H0 : the coin is fair. Suppose you ﬂip the coin
ﬁve times. Let X be the number of times heads shows. Describe the ML and MAP decision rules,
and ﬁnd pfalse alarm , pmiss , and pe for both of them, using the prior (π0 , π1 ) = (0.2, 0.8) for the MAP
rule and for deﬁning pe for both rules.
Solution: The rows of the likelihood matrix consist of the pmf of the binomial distribution with
n = 5 and p = 2/3 for H1 and p = 1/2 for H0 :
X=0
H1 15
3 H0 X=1 15
2 5 2
3 X=2 14
3 23
3 10 15
2 5 X=3 22
3 10 13
3 10 15
2 X=4 12
3 5 15
2 10 24
3 5 1
3 15
2 X=5
25
3
15
.
2 In computing the likelihood ratio, the...
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 Spring '08
 Zahrn
 The Land

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