Modelling uncertainty

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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(AB) 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
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Joint probability
If events A1, A2,... Are mutually exclusive and
cover the sample space ?, and P(Ai) gt 0 for i
1, 2,... then for any event B the following
equality holds
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Bayes Theorem
Thomas Bayes (1701-1761)
If the events A1, A2,... fulfil the assumptions
of the joint probability theorem, and P(B) gt 0,
then for i 1, 2,... The following equality holds
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Bayes Theorem
Prior probabilities
New information
Bayes theorem
Posterior probabilities
Let us denote H hipothesis E evidence The
Bayes rule has the form
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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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n sample points 2n probabilities
P(B,M)
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Wymagania odnosnie do modelu niepewnosci w
systemach regulowych

• W systemach wnioskowania logicznego regula
  postaci A ? B pozwala wywnioskowac B, gdy tylko
  zachodzi A, niezaleznie od innych faktów. W
  systemach probabilis-tycznych trzeba wziac pod
  uwage wszystkie dostepne przeslanki.
• Jezeli przeprowadzimy dowód jakiejs tezy, to tezy
  tej mozna uzyc w kolejnych dowodach bez potrzeby
  ponownego jej dowodzenia. W systemach
  probabilis-tycznych przeslanki uzyte do dowodu
  moga ulec zmianie.
• W logice prawdziwosc zdan zlozonych mozna
  wywnioskowac na podstawie wartosci logicznej
  termów. Wnioskowanie probabilistyczne nie
  zachowuje tej wlasnosci, chyba, ze nalozymy silne
  ograniczenia o niezaleznosci.

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Certainty factor

• Buchanan, Shortliffe 1975
• Model developed for the rule expert system MYCIN

If E then H
hipothesis
evidence (observation)
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Belief

• MBH, E measure of the increase of belief that
  H is true based on observation E.

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Disbelief

• MDH, E measure of the increase of disbelief
  that H is true based on observation E.

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Certainty factor
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.
CF(H, E)
E
H
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Uncertainty propagation
Parallel rules
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Uncertainty propagation
Serial rules
If CF(H,?E2) is not defined, it is assumed to be
0.
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Certainty factor probabilistic definition
Heckerman 1986
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Certainty measure
Grzymala-Busse 1991
C(H)
C(E)
CF(H, E)
E
H
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Example 1
C(s1 ? s2) min(0,2 0,1) 0,1
CF(h, s1 ? s2) 0,4 0 0
C(h) 0,3 (1 0,3) 0 0,3 0 0,3
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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
C(h) 0,3 (1 0,3) 0,08 0,3 0,7 0,08
0,356
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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
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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.
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Plausibility
Plausibility Pl ? 0,1 measures how much the
belief that A is true is limited by evidence
supporting ?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
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Example
A allergy F flu C cold P - pneumonia
m1(?) 1
? A, F, C, P
Observation 1
m2(A, F, C) 0,6
m2(?) 0,4
m2(A, F, C) 0,6
m2(?) 0,4
m1(?) 1
m3(A, F, C) 0,6
m3(?) 0,4
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Example
m3(A, F, C) 0,6
m3(?) 0,4
Observation 2
m4(F,C,P) 0,8
m4(?) 0,2
m4(F,C,P) 0,8
m4(?) 0,2
m5(F,C) 0,48
m5(A,F,C) 0,12
m3(A,F,C) 0,6
m3(?) 0,4
m5(F,C,P) 0,32
m5(?) 0,08
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Example
m5(F,C) 0,48
m5(A,F,C) 0,12
m5(F,C,P) 0,32
m5(?) 0,08
Observation 3
m6(A) 0,75
m6(?) 0,25
m6(A) 0,75
m6(?) 0,25
m7(?) 0,36
m7(F,C) 0,12
m5(F,C) 0,48
m7(A) 0,09
m7(A,F,C) 0,03
m5(A,F,C) 0,12
m7(?) 0,24
m7(F,C,P) 0,08
m5(F,C,P) 0,32
m7(A) 0,06
m7(?) 0,02
m5(?) 0,08
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Example
m5(F,C) 0,48
m5(A,F,C) 0,12
m5(F,C,P) 0,32
m5(?) 0,08
Observation 3
m6(A) 0,75
m6(?) 0,25
m7(?) 0,6
m6(A) 0,75
m6(?) 0,25
m7(?) 0,36
m7(F,C) 0,12
m5(F,C) 0,48
m7(A) 0,09
m7(A,F,C) 0,03
m5(A,F,C) 0,12
m7(?) 0,24
m7(F,C,P) 0,08
m5(F,C,P) 0,32
m7(A) 0,06
m7(?) 0,02
m5(?) 0,08
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Example
m7(A) 0,375
m7(A) 0,15
m7(F,C) 0,3
m7(F,C) 0,12
m7(A,F,C) 0,075
m7(A,F,C) 0,03
m7(F,C,P) 0,2
m7(F,C,P) 0,08
m7(?) 0,05
m7(?) 0,02
1 0,3 0,2
A 0,375, 0,500 F 0, 0,625 C 0,
0,625 P 0, 0,250
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
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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
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Probabilistic reasoning
What is the probability of a burglary if Mary
called? P(ByMy) ?
Marginal probability



Conditional probability
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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(B1,E1,A1,J1,M1)
• calculating any practical value, e.g. P(B1M1)
  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
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Bayes theorem
B depends on A
P(BA)
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The chain rule
P(X1,X2) P(X1)P(X2X1) P(X1,X2,X3)
P(X1)P(X2X1)P(X3X1,X2) .........................
....................................... P(X1,X2,..
.,Xn) P(X1)P(X2X1)...P(XnX1,...,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, ..., Xi1 such that Xi is
independent of variables from the set X1, ...,
Xi1 \ pa(Xi). Thus P(XiX1, ..., Xi 1)
P(Xipa(Xi)) and P(X1, ..., Xn) ? P(Xipa(Xi))
n
i1
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Bayesian network
P(AB1, ..., Bn)
Bi directly influences A
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Example
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Example
P(B) 0.001
P(E) 0.002
B E P(A) T T 0.950 T F 0.940 F T 0.290 F F
0.001
A P(J) T 0.90 F 0.05
A P(M) T 0.70 F 0.01
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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(AB1,...,Bn) or P(Apa(A)) if pa(A) ? a
priori probability equals P(A)
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Bayesian networks
pa(A)
P(AB1, B2, ..., Bn)
Event Bi has no predecesors (pa(Bi) ?) a
priori probability P(Bi)
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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
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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.
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Global probability distribution
pa(A2)
pa(A1)
A1
P(A2B3, ...Bn)
P(A1B1, ...Bn)
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Global probability distribution
pa(A2)
pa(A1)
A2
A1
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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(HE).
• Two types of reasoning
• probability of a single hipothesis
• probability of all hipothesis.

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Example
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





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Example

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Example

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Example

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Example

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Example

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Example
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Types of reasoning in Bayesian networks
Evidence B occurs and we qould like to update
probability of hipothesis J. Interpretation. Ther
e was a burglary, what is the probability that
John will call?
A
P(JB) P(JA)P(AB) 0.9 0.95 0.86
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Types of reasoning in Bayesian networks
B
P(B) 0.001
Wnioskowanie diagnostyczne We observe J what is
the probability that B is true? Diagnosis. John
calls. What is the probability of a burglary?
B P(A) T 0.95 F 0.01
A
J
A P(J) T 0.90 F 0.05
P(BJ) P(JB)P(B)/P(J) (0,950,90,001)/(0,9
0,05) 0,0009
diagnostic
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Types of reasoning in Bayesian networks
We observe E. What is the probability that B is
true? Alarm rang, so P(BA) 0.376, but if
earthuake is observed as well then P(BA,E) 0.03
A
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Types of reasoning in Bayesian networks
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(AJ,E) 0.03
mixed
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Types of reasoning in Bayesian networks
przyczynowe diagnostyczne
miedzy-przyczynowe mieszane
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Multiply connected Bayesian network
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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