modelling uncertainty
DESCRIPTION
Modelling uncertainty. Probability of an event. Classical method : If an experiment has n possible outcomes assign a probability of 1/ n to each experimental outcome. Relative frequency method : Probability is the relative frequency of the number of events satisfying the constraints. - PowerPoint PPT PresentationTRANSCRIPT
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Modelling uncertainty
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Probability of an event
• Classical method: If an experiment has n possible outcomes assign a probability of 1/n to each experimental outcome.
• Relative frequency method:Probability is the relative frequency of the number of events satisfying the constraints.
• Subjective method:Probability is a number characterising the likelihood of an event – degree of belief
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Axioms of the probability theory
Axiom I The probability value assigned to each experimental outcome must be between 0 and 1.
Axiom II The sum of all the experimental outcome probabilities must be 1.
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Conditional probability
denoted by P(A|B) expresses belief that event A is true assuming that event B is true (events A and B are dependent)
Definition
Let the probability of event B be positive. Conditional probability of event A under condition B is calculated as follows:
0P(B) re whe,P(B)
B)P(A,B)|P(A
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Joint probability
If events A1, A2,... Are mutually exclusive and cover the
sample space , and P(Ai) > 0 for i = 1, 2,... then for any
event B the following equality holds:
i
ii ))P(AA|P(BP(B)
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Bayes’ Theorem
If the events A1, A2,... fulfil the assumptions of the joint
probability theorem, and P(B) > 0, then for i =1, 2,... The
following equality holds:
iii
iii ))P(AA|P(B
))P(AA|P(BB)|P(A
Thomas Bayes (1701-1761)
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Bayes’ Theorem
Let us denote:
H – hipothesis
E – evidence
The Bayes’ rule has the form:
P(E)H)P(H)|P(E
E)|P(H
Prior probabilities
New information
Bayes’ theorem
Posterior probabilities
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Difficulties with joint probability distribution (tabular approach)
• the joint probability distribution has to be defined and stored in memory
• high computational effort required to calculate marginal and conditional probabilities
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B E A J M P(B,E,A,J,M)1 1 1 1 1 0,00011970001 1 1 1 0 0,00001330001 1 1 0 1 0,00005130001 1 1 0 0 0,00000570001 1 0 1 1 0,00000000501 1 0 1 0 0,00000009501 1 0 0 1 0,00000049501 1 0 0 0 0,00000940501 0 1 1 1 0,00580356001 0 1 1 0 0,00064484001 0 1 0 1 0,00248724001 0 1 0 0 0,00027636001 0 0 1 1 0,00000029401 0 0 1 0 0,00000558601 0 0 0 1 0,00002910601 0 0 0 0 0,00055301400 1 1 1 1 0,00361746000 1 1 1 0 0,00040194000 1 1 0 1 0,00155034000 1 1 0 0 0,00017226000 1 0 1 1 0,00000702900 1 0 1 0 0,00013355100 1 0 0 1 0,00069587100 1 0 0 0 0,01322154900 0 1 1 1 0,00061122600 0 1 1 0 0,00006791400 0 1 0 1 0,00026195400 0 1 0 0 0,00002910600 0 0 1 1 0,00048461490 0 0 1 0 0,00920768310 0 0 0 1 0,0479768751
n sample points
2n probabilities
P(B,M)
P(M)
M)P(B,M)|P(B
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Certainty factor
• Buchanan, Shortliffe 1975• Model developed for the rule expert system
MYCIN
If E then H
evidence (observation)
hipothesis
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Belief
othewise
1P(H) if
P(H)max{1,0}
P(H)P(H)}E),|max{P(H
1
E]MB[H,
• MB[H, E] – measure of the increase of belief that H is true based on observation E.
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Disbelief
• MD[H, E] – measure of the increase of disbelief that H is true based on observation E.
otherwise
0P(H) if
P(H)min{1,0}
P(H)P(H)}E),|min{P(H
1
E]MD[H,
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Certainty factor
E]MD[H,E]MB[H,E)CF(H,
CF [–1, 1]
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Interpretation of the certainty factor
Certainty factor is associated with a rule:
If evidence then hipothesis
and denotes the change in belief that H is true after observation E.
E HCF(H, E)
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Uncertainty propagation
E1
H
CF(H, E1)
E2 CF(H, E2)
Parallel rules
E1, E2 HCF(H, E1&E2)
otherwise )]E MB(H,[1-*)E MB(H,)E MB(H,
1)E&EMD(H, if
)E&EMB(H,
121
21
21
0
otherwise )]E MD(H,[1-*)E MD(H,)E MD(H,
1)E&EMB(H, if
)E&EMD(H,
121
21
21
0
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Uncertainty propagation
E1 HCF(E2, E1)
E2
CF(H, E2)
Serial rules
E1 HCF(H, E1)
otherwise )E)CF(H,E,CF(E
0)E,CF(E if )E)CF(H,E,CF(E)ECF(H,
212
122121
If CF(H,E2) is not defined, it is assumed to be 0.
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Certainty factor – probabilistic definition
Heckerman 1986
E)|P(HP(H)gdy E))|P(HP(H)(1
P(H)E)|P(H
P(H)E)|P(Hgdy P(H))E)(1|P(HP(H)E)|P(H
E)CF(H,
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Certainty measure
E HCF(H, E)
C(E) C(H)
0H)(EC(H)CF' if |}H)(ECF'|,|C(H)min{|1
H)(ECF'C(H)
0H)(ECF'C(H), if H)(EC(H))CF'(1C(H)
0H)(ECF'C(H), if H)(EC(H))CF'(1C(H)
(H)C'
C(E)}max{0,H)CF(EH)(ECF'
Grzymała-Busse 1991
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Example 1
C(s1 s2) = min(0,2; – 0,1) = – 0,1
CF’(h, s1 s2) = 0,4 * 0 = 0
s1h
CF(h, s1 s2) = 0,4
s2
C(s1) = 0,2
C(s2) = – 0,1C(h) = 0,3
C’(h) = 0,3 + (1– 0,3) * 0 = 0,3 + 0 = 0,3
0H)(ECF'C(H), if H)(EC(H))CF'(1C(H)(H)C'
C(E)}max{0,H)CF(EH)(ECF'
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Example 2
C(s1 s2) = min(0,2; 0,8) = 0,2
CF’(h, s1 s2) = 0,4 * 0,2 = 0,08
s1h
CF(h, s1 s2) = 0,4
s2
C(s1) = 0,2
C(s2) = 0,8C(h) = 0,3
C’(h) = 0,3 + (1– 0,3) * 0,08 = 0,3 + 0,7 * 0,08 = 0,356
0H)(ECF'C(H), if H)(EC(H))CF'(1C(H)(H)C'
C(E)}max{0,H)CF(EH)(ECF'
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Dempster-Shafer theory
Each hipothesis is characterised by two values: balief and plausibility.
It models not only belief, but also the amount of acquired information.
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Density probability function
ΘA
Θ
1m(A)
0]m[
0,12 :m
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Belief
Belief Bel [0,1] measures the value of acquired information supporting the belief that the considered set hipothesis is true.
ABm(B)Bel(A)
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Plausibility
Plausibility Pl [0,1] measures how much the belief that A is true is limited by evidence supporting A.
A)Bel(1Pl(A)
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Combining various sources of evidence
Assume two sources of evidence: X and Y represented by
respective subsets of : X1,...,Xm and Y1,...,Yn. Probability
density functions m1 and m2 are defined on X and Y
respectively. Combining observations from two sources a
new value m3(Z) is calculated for each subset of as
follows:
jYiX j2i1
ZjYiX j2i1
3 )(Y)m(Xm1
)(Y)m(Xm
(Z)m
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Example
={A, F, C, P} m1() = 1
Observation 1 m2({A, F, C}) = 0,6
m2() = 0,4
m1() = 1
m2({A, F, C}) = 0,6
m3({A, F, C}) = 0,6
m2() = 0,4
m3() = 0,4
A – allergyF – fluC – coldP - pneumonia
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Example
Observation 2 m4({F,C,P}) = 0,8
m4() = 0,2
m4({F,C,P}) = 0,8
m5({F,C}) = 0,48
m4() = 0,2
m5({A,F,C}) = 0,12
m3({A, F, C}) = 0,6 m3() = 0,4
m3({A,F,C}) = 0,6
m3() = 0,4 m5({F,C,P}) = 0,32 m5() = 0,08
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Example
Observation 3 m6({A}) = 0,75
m6() = 0,25
m6({A}) = 0,75
m7() = 0,36
m6() = 0,25
m7({F,C}) = 0,12
m7({A}) = 0,09 m7({A,F,C}) = 0,03
m5({F,C}) = 0,48 m5({A,F,C}) = 0,12
m5({F,C,P}) = 0,32 m5() = 0,08
m5({F,C}) = 0,48
m5({A,F,C}) = 0,12
m5({F,C,P}) = 0,32
m5() = 0,08
m7() = 0,24 m7({F,C,P}) = 0,08
m7({A}) = 0,06 m7() = 0,02
m7({A}) = 0,15
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Example
Observation 3 m6({A}) = 0,75
m6() = 0,25
m6({A}) = 0,75
m7() = 0,36
m6() = 0,25
m7({F,C}) = 0,12
m7({A}) = 0,09 m7({A,F,C}) = 0,03
m5({F,C}) = 0,48 m5({A,F,C}) = 0,12
m5({F,C,P}) = 0,32 m5() = 0,08
m5({F,C}) = 0,48
m5({A,F,C}) = 0,12
m5({F,C,P}) = 0,32
m5() = 0,08
m7() = 0,24 m7({F,C,P}) = 0,08
m7({A}) = 0,06 m7() = 0,02
m7() = 0,6
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Example
m7({F,C}) = 0,12
m7({A}) = 0,15
m7({A,F,C}) = 0,03
m7({F,C,P}) = 0,08
m7() = 0,02
m7({F,C}) = 0,3
m7({A}) = 0,375
m7({A,F,C}) = 0,075
m7({F,C,P}) = 0,2
m7() = 0,05
{A}: [0,375, 0,500]
{F}: [0, 0,625]
{C}: [0, 0,625]
{P}: [0, 0,250]
1 – 0,3 – 0,2
1 – 0,375
1 – 0,375 – 0,3 – 0,075
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Fuzzy sets (Zadeh)
Rough sets (Pawlak)
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Probabilistic reasoning
alarm
earthquakeburglary
John Mary
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Probabilistic reasoning
B – burglary
E – earthquake
A – alarm
J – John calls
M – Mary calls
Joint probability distribution – P(B,E,A,J,M)
?
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Joint probability distribution
B E A J M P(B,E,A,J,M)1 1 1 1 1 0,00011970001 1 1 1 0 0,00001330001 1 1 0 1 0,00005130001 1 1 0 0 0,00000570001 1 0 1 1 0,00000000501 1 0 1 0 0,00000009501 1 0 0 1 0,00000049501 1 0 0 0 0,00000940501 0 1 1 1 0,00580356001 0 1 1 0 0,00064484001 0 1 0 1 0,00248724001 0 1 0 0 0,00027636001 0 0 1 1 0,00000029401 0 0 1 0 0,00000558601 0 0 0 1 0,00002910601 0 0 0 0 0,00055301400 1 1 1 1 0,00361746000 1 1 1 0 0,00040194000 1 1 0 1 0,00155034000 1 1 0 0 0,00017226000 1 0 1 1 0,00000702900 1 0 1 0 0,00013355100 1 0 0 1 0,00069587100 1 0 0 0 0,01322154900 0 1 1 1 0,00061122600 0 1 1 0 0,00006791400 0 1 0 1 0,00026195400 0 1 0 0 0,00002910600 0 0 1 1 0,00048461490 0 0 1 0 0,00920768310 0 0 0 1 0,0479768751
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Probabilistic reasoning
What is the probability of a burglary if Mary called? P(B=y|M=y) ?
JE,A,
M)J,A,E,P(B,M)P(B,
Marginal probability:
Conditional probability:
0.1333130.055205370.0084917
0.0084917P(M)
B)P(M,M)|P(B
B M P(B,E,A,J,M)1 1 0,00849171 0 0,00150830 1 0,055205370 0 0,93479463
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Advantages of probabilistic reasoning
• Sound mathematical theory • On the basis of the joint probability distribution one can
reason about:– the reasons on the basis of the observed
consequences,– consequences on the basis of given evidence,– Any combination of the above ones.
• Clear semantics based on the interpretation of probability.
• Model can be taught with statistical data.
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Complexity of probabilistic reasoning
• in the „alarm” example– (25 – 1) = 31 values,– direct acces to unimportant information, e.g.
P(B=1,E=1,A=1,J=1,M=1) – calculating any practical value, e.g. P(B=1|M=1)
requires 29 elementary operations.
• in general– P(X1, ..., Xn) requires storing 2n-1 values
– difficult knowledge acquisition (not natural)– exponential complexity
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Bayes’ theorem
P(E)H)P(H)|P(E
E)|P(H
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Bayes’ theorem
BA
B depends on A
P(B)A)P(A)|P(B
B)|P(A
P(B|A)
A
A)P(A)|P(BP(B)
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The chain rule
P(X1,X2) = P(X1)P(X2|X1)
P(X1,X2,X3) = P(X1)P(X2|X1)P(X3|X1,X2)
................................................................
P(X1,X2,...,Xn) = P(X1)P(X2|X1)...P(Xn|X1,...,Xn-1)
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Conditional independence of variables in a domain
In any domain one can define a set of variables pa(Xi){X1, ..., Xi–1} such that Xi is independent of variables from the set {X1, ..., Xi–1} \ pa(Xi).
Thus
P(Xi|X1, ..., Xi – 1) = P(Xi|pa(Xi))
and
P(X1, ..., Xn) = P(Xi|pa(Xi))i=1
n
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Bayesian network
B1
A
B2 Bn
C1
.....
Bi directly influences A
Cm.....
P(A|B1, ..., Bn)
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Example
alarm
Mary callsJohn calls
earthquakeburglary
burglary earthquake P(alarm|burglary, earthquake)
true falsetrue true 0.950 0.050true false 0.940 0.060 false true 0.290 0.710false false 0.001 0.999
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Example
A
MJ
EB
P(B)
0.001P(E)
0.002
B E P(A)
T T 0.950T F 0.940F T 0.290F F 0.001
A P(M)
T 0.70F 0.01
A P(J)
T 0.90F 0.05
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Complexity of the representation
• Instead of 31 values it is enough to store 10.
• Easy construction of the model– Less parameters.– More intuitive parameters.
• Easy reasoning.
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Bayesian networks
Bayesian network is an acyclic directed graph which
• nodes represent formulas or variables in the considered domain,
• arcs represent dependence relation of variables, with related probability distributions.
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Bayesian networks
variable A with parent nodes pa(A) = {B1,...,Bn}
conditional probablity table P(A|B1,...,Bn) or P(A|pa(A))
if pa(A) = a priori probability equals P(A)
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Bayesian networks
B1
A
B2 BnB3 .....
pa(A)
P(A|B1, B2, ..., Bn)
Event Bi has no
predecesors (pa(Bi) = ) a priori probability P(Bi)
B1 ... Bn P(A|B1, Bn)
T T 0.18 T F 0.12 ................................. F F 0.28
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Local semantics of Bayesian network
• Only direct dependence relations between variables.
• Local conditional probability distribution.
• Assumption about conditional independence of variables not bounded in the graph.
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Global semantics of bayesian network
Joint probability distribution given implicite.
It can be calculated using the following rule:
i
n1iin1 )A,...,A|P(A)A,...,P(A
))P(AA|)...P(AA,...,A,A|P(A nn1n-n321
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Global semantics of bayesian network
Node numbering: node index is smaller than indices of its predecessors.
Finally:
Bayesian network is a complete probabilistic model.
))pa(A|P(A)A,...,A|P(A iin1ii
i
iin1 ))pa(A|P(A)A,...,P(A
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A2
Global probability distribution
B1
A1
B2 BnB3 .....
P(A1|B1, ...Bn) P(A2|B3, ...Bn)
pa(A1)
pa(A2)
B1 ... Bn A1 A2 P(A1,A2,B1, ...Bn)
T ... T T T T ... T T F ...................................................... F ... F F F
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Global probability distribution
A1
B2 BnB3 .....
A2
pa(A1)
pa(A2)
B1
P(A1|B1, ...Bn)
B1 ... Bn P(A1)
T ... T 0.25 T ... F .......................... F ... F
B1
B1 ... Bn A1 A2 P(A1,A2,B1, ...Bn)
T ... T T T 0.075 T ... T T F ...................................................... F ... F F F
B1
P(A2|B3, ...Bn)
B3 ... Bn P(A2)
T ... T 0.30 T ... F .......................... F ... F
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Reasoning in Bayesian networks
Updating evidence that a hipothesis H is true given some ecidence E, i.e. defining conditional probability distribution P(H|E).
Two types of reasoning:
•probability of a single hipothesis
•probability of all hipothesis.
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Example
A
MJ
EB
P(B)
0.001P(E)
0.002
B E P(A)
T T 0.950T F 0.940F T 0.290F F 0.001
A P(M)
T 0.70F 0.01
A P(J)
T 0.90F 0.05
John calls (J) and Mary calls (M). What is the probability that neither
burglary nor earthquake occurred if the alarm
rang?
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Example
A
MJ
EB
P(B)
0.001
P(E)
0.002
B E P(A)
T T 0.950T F 0.940F T 0.290F F 0.001
A P(M)
T 0.70F 0.01
A P(J)
T 0.90F 0.05
E)BAMP(J
E)B)P(E)P(B|A)P(A|A)P(M|P(J
i
iin1 ))pa(A|P(A)A,...,P(A
?E)BAMP(J
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Example
A
MJ
EB
P(B)
0.001
P(E)
0.002
B E P(A)
T T 0.950T F 0.940F T 0.290F F 0.001
A P(M)
T 0.70F 0.01
A P(J)
T 0.90F 0.05
?E)BAMP(J
E)BAMP(J E)B)P(E)P(B|A)P(A|A)P(M|P(J
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Example
A
MJ
EB
P(B)
0.001
P(E)
0.002
B E P(A)
T T 0.950T F 0.940F T 0.290F F 0.001
A P(M)
T 0.70F 0.01
A P(J)
T 0.90F 0.05
?E)BAMP(J
E)BAMP(J E)B)P(E)P(B|A)P(A|A)P(M|P(J
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Example
A
MJ
EB
P(B)
0.001
P(E)
0.002
B E P(A)
T T 0.950T F 0.940F T 0.290F F 0.001
A P(M)
T 0.70F 0.01
A P(J)
T 0.90F 0.05
?E)BAMP(J
E)BAMP(J E)B)P(E)P(B|A)P(A|A)P(M|P(J
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Example
A
MJ
EB
P(B)
0.001
P(E)
0.002
B E P(A)
T T 0.950T F 0.940F T 0.290F F 0.001
A P(M)
T 0.70F 0.01
A P(J)
T 0.90F 0.05
?E)BAMP(J
E)BAMP(J E)B)P(E)P(B|A)P(A|A)P(M|P(J
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Example
A
MJ
EB
P(B)
0.001
P(E)
0.002
B E P(A)
T T 0.950T F 0.940F T 0.290F F 0.001
A P(M)
T 0.70F 0.01
A P(J)
T 0.90F 0.05
?E)BAMP(J
E)BAMP(J E)B)P(E)P(B|A)P(A|A)P(M|P(J
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Example
A
MJ
EB
P(B)
0.001
P(E)
0.002
B E P(A)
T T 0.950T F 0.940F T 0.290F F 0.001
A P(M)
T 0.70F 0.01
A P(J)
T 0.90F 0.05
?E)BAMP(J
E)BAMP(J
0.00062 0.998*0.999*0.001*0.7*0.9 E)B)P(E)P(B|A)P(A|A)P(M|P(J
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Types of reasoning in Bayesian networks
B
J
Evidence B occurs and we qould like to update probability of hipothesis J.
Interpretation.
There was a burglary, what is the probability that John will call?
A
P(J|B) = P(J|A)P(A|B) = 0.9 * 0.95 = 0.86
P(B) = 0.001
A P(J)
T 0.90F 0.05
B P(A)
T 0.95F 0.01
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Types of reasoning in Bayesian networks
Wnioskowanie diagnostyczne
We observe J – what is the probability that B is true?
Diagnosis.
John calls. What is the probability of a burglary?
B
J
A
P(B|J) = P(J|B)*P(B)/P(J) =
(0,95*0,9*0,001)/(0,9+0,05) = 0,0009
P(B) = 0.001
A P(J)
T 0.90F 0.05
B P(A)
T 0.95F 0.01
diagnostic
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Types of reasoning in Bayesian networks
B E
We observe E. What is the probability that B is true?
Alarm rang, so P(B|A) = 0.376, but if earthuake is observed as well then P(B|A,E) = 0.03
A
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Types of reasoning in Bayesian networks
E
A
J
mixed
We observe E and J What is the probability of A.
John calls and we know that there was an earthquake. What is the probability that alarm rang?
P(A|J,E) = 0.03
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Multiply connected Bayesian network
B1
A2
B2 Bn
C1
.....
Cm.....
...A1 Ak
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Summary
• Models of uncertainty:
• Certainty factor, certainty measure
• Dempster-Shafer theory
• Bayesian networks
• Fuzzy sets
• Raough sets
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Summary
• Bayesian networks represent joint probability distribution.
• Reasoning in multiply connected BN is NP-hard.
• Exponential complexity may be avoided by:
• Constructing the net as a polytree
• Transforming a network to a polytree
• Approximate reasoning