Uncertainty Theory
Baoding Liu @ Tsinghua University
What is the Conditional Uncertain Measure of A given B?
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http://orsc.edu.cn/liu/ut.pdf
http://orsc.edu.cn/liu/Lecture/Main.pdf
Uncertainty Theory
Baoding Liu @ Tsinghua University
New Information: What is the event B in practice?
Assume the strength of a bridge ranges between 100 and 200 tons.
If a truck weighing 140 tons crosses over the bridge, then B = [140, 200].
If the bridge collapses, then B = [100, 140).
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http://orsc.edu.cn/liu/ut.pdf
http://orsc.edu.cn/liu/Lecture/Main.pdf
Uncertainty Theory
Baoding Liu @ Tsinghua University
Prior Uncertain Measure
Assume that the uncertain measures
M{B},
M{A ∩ B},
M{Ac ∩ B}
are known for us by some prior knowledge.
We will call them the prior uncertain measures.
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http://orsc.edu.cn/liu/ut.pdf
http://orsc.edu.cn/liu/Lecture/Main.pdf
Uncertainty Theory
Baoding Liu @ Tsinghua University
Posterior Uncertain Measure
The updated uncertain measure M{A|B} is called posterior uncertain measure.
The updating process will be undertaken by the conditional uncertain measure.
http://orsc.edu.cn/liu
http://orsc.edu.cn/liu/ut.pdf
http://orsc.edu.cn/liu/Lecture/Main.pdf
Uncertainty Theory
Baoding Liu @ Tsinghua University
Conditional Uncertain Measure M{A|B}
First Step: M{A ∩ B} ⇒
M{A ∩ B}
M{B}
Second Step: M{A|B} ≤
M{A ∩ B}
M{B}
Third Step: M{A|B} = 1 − M{Ac |B} ≥ 1 −
M{Ac ∩ B}
M{B}
Conclusion: The conditional uncertain measure should be on the interval
M{Ac ∩ B} M{A ∩ B}
1−
,
.
M{B}
M{B}
http://orsc.edu.cn/liu
http://orsc.edu.cn/liu/ut.pdf
http://orsc.edu.cn/liu/Lecture/Main.pdf
Uncertainty Theory
Baoding Liu @ Tsinghua University
A Basic Question
For any event, if there are multiple reasonable values that a measure may
take, what value is assigned to the event?
Maximum Uncertainty Principle (Liu1 , 2007)
For any event, if there are multiple reasonable values that a measure may
take, then the value as close to 0.5 as possible is assigned to the event.
1
Liu B, Uncertainty Theory, 2nd edn, Springer-Verlag, Berlin, 2007.
http://orsc.edu.cn/liu
http://orsc.edu.cn/liu/ut.pdf
http://orsc.edu.cn/liu/Lecture/Main.pdf
Uncertainty Theory
Baoding Liu @ Tsinghua University
Conditional Uncertain Measure (Liu2 , 2007)
Definition
The conditional uncertain measure of A given B is defined by

M{A ∩ B}
M{A ∩ B}


,
if
< 0.5


M{B}
M{B}


M{Ac ∩ B}
M{Ac ∩ B}
M{A|B} =
,
if
< 0.5
1
−


M{B}
M{B}




0.5,
otherwise
provided that M{B} > 0.
M{Ac ∩ B} M{A ∩ B}
M{A|B} ∈ 1 −
,
M{B}
M{B}
2
Liu B, Uncertainty Theory, 2nd edn, Springer-Verlag, Berlin, 2007.
http://orsc.edu.cn/liu
http://orsc.edu.cn/liu/ut.pdf
http://orsc.edu.cn/liu/Lecture/Main.pdf