Law Of Total Probability, Bayes' Formula And Binary Hypothesis Testing Worksheet With Answers - University Of Illinois, 2012 Page 4
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7The ML rules in favor of H
if Λ(k)
1. Therefore, to rule in favor of H
,
1
1
Λ(k)
1
k 1
p
(1
p
)
1
1
1
k 1
p
(1
p
)
0
0
k 1
p
(1
p
)
1
1
ln
ln 1
k 1
p
(1
p
)
0
0
p
1
p
1
1
ln
+ (k
1) ln
0
p
1
p
0
0
p
1
ln
( )
p
0
k
+ 1
1 p
1
ln
1 p
0
p
0
ln
p
1
k
1 +
1 p
1
ln
1 p
0
1 p
where ( ) follows because p
< p
so that ln
1
< 0.
0
1
1 p
0
1
1
(b) Find p
, p
, and p
under the ML decision rule assuming p
= e
, p
= 1
e
.
M D
F A
e
0
1
Solution: Notice that with the given values of p
and p
, the ML rule becomes ruling
0
1
in favor of H
if X
2. Therefore,
1
p
= P declare H
true H
= P X > 2 H
= 1
P X
2 H
0
1
1
1
M D
1 1
2 1
1
1
1
= 1
p
(1
p
)
+ p
(1
p
)
= 1
(1
e
) + (1
e
)e
1
1
1
1
2
= e
0.1353
1 1
2 1
p
= P declare H
true H
= P X
2 H
= p
(1
p
)
+ p
(1
p
)
F A
1
0
0
0
0
0
0
1
1
1
1
2
= e
+ e
(1
e
) = 2e
e
0.6004
To calculate the average error probability we use the fact that in this case 2π
= 3π
so
0
1
that π
= 3/5 and π
= 2/5.
0
1
3
2
6
1
1
2
2
1
2
p
= π
p
+ π
p
=
2e
e
+
e
=
e
e
0.4144
e
0
F A
1
M D
5
5
5
5
(c) Find the MAP decision rule in this binary hypothesis testing. Assume ties are broken in
favor of H
, and leave p
and p
as variables (do not use the values given in part (b)).
1
0
1
π
Solution: In this case, the MAP rules in favor of H
if Λ(k)
. Therefore, to rule
0
1
π
1
in favor of H
,
1
4
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