BELIEF FUSION AND
BAYESIAN REASONING
WITH SUBJECTIVE LOGIC
FUSION 2014
Salamanca, 7 July 2014
Audun Jøsang, University of Oslo
http://folk.uio.no/josang/
Audun Jøsang
FUSION 2014
1
About me
• Prof. Audun Jøsang,
University of Oslo, 2008
• Research interests
– Information Security
– Belief Reasoning
• Bio
–
–
–
–
–
MSc Telecom, NTH, 1988
Engineer, Alcatel Telecom
MSc Security, London 1993
PhD Security, NTNU 1998
A.Prof. QUT, Australia, 2000
Audun Jøsang
FUSION 2014
Department of Informatics
@ University of Oslo
2
Tutorial overview
1. Representations of subjective opinions
2. Operators of subjective logic
3. Applications of subjective logic:
– Information fusion;
– Trust reasoning
– Bayesian reasoning
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Objective vs. Subjective Reality
(assumed)
(perceived)
Assumed
Perceived
objective reality
subjective
reality
≠
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4
Probability and Uncertainty
Frequentist:
Subjective:
• Certain when based on much
observation evidence
• Uncertain when based on
little observation evidence
• E.g.: Probability of heads
when flipping coin is 1/2
• Certain when dynamics of
situation are known
• Uncertain when dynamics
of situation are unknown
• E.g. Probability of Oswald
killed Kennedy is 1/2
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Characteristics and Formalisms
Assumed reality
Perceived reality
• Characteristics:
• Characteristics:
– Crisp, frequentist, quantum
• Formalisms:
• Formalisms:
– Frequentist probabilities
– Binary logic
– Quantum logic
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– Vague, fuzzy, uncertain
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–
–
–
–
Subjective probabilities
Fuzzy logic
Probabilistic logics
Subjective logic
7
Probabilistic and Subjective Logics
Binary Logic
+
Probabilistic Logic
Probability Calculus
+
Subjective logic
– Proposition structures
– Degrees of truth
– Uncertainty
Uncertainty
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Probabilistic Logic Examples
Binary Logic
Probabilistic logic
AND:
x∧y
OR:
x∨y
MP: { x→y, x } ⇒ y
p(x∧y) = p(x)p(y)
p(x∨y) = p(x) + p(y) − p(x)p(y)
p( y ) = p( x) p( y | x) + p( x ) p( y | x )
p( x | y) =
a( x) p( y | x)
a( x) p( y | x) + a( x ) p( y | x )
MT: { x→y, y } ⇒ x
a( x) p( y | x)
p( x | y ) =
a( x) p( y | x) + a( x ) p( y | x )
p( x) = p( y ) p( x | y ) + p( y ) p( x | y )
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a: base rate
9
Frames and opinions
X
Binary frame
x1
Binomial opinion
x2
X
X
n-ary frame
Multinomial
opinion
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x2
x1
n-ary frame
Hypernomial
opinion
x3
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x2
x1
x3
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Binomial Opinion
Multinomial Opinion
Hypernomial Opinion
Binary frame X
Focal element x
n-ary frame X
Focal elements x ∈ X
n-ary frame X
Focal elements x ∈ R (X)
X
Frame
X
x1
x2
x1
X
x2
x1
x2
x3
x3
Opinion
representation
Beta PDF
Dirichle PDF over X
PDF
representation
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Hyper Dirichlet PDF
over X
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A Frame and its Reduce Powerset
•
•
•
•
A frame X is a state space of distinct possibilities
Powerset P (X) = 2X , set of subsets, including {X, ∅}
Reduced powerset R (X) = P (X) \ {X, ∅}
R (X) = { x1, x2, x3, x4, x5, x6 } in example below
• Cardinality | X |
X
(= 3 in this example)
• |P (X)| =
x4
2|X|
(= 8 in this example)
x2
• |R (X)| = 2|X| − 2
x1
x6
x5
x3
(= 6 in this example)
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Binomial subjective opinions
• Belief masses on binary frames
–
–
–
–
bxA = b(x) is observer A’s belief in x
d xA = b(x ) is observer A’s disbelief in x
u xA = b( X ) is observer A’s uncertainty about x
a xA
is the base rate of x
Binomial opinion
ω xA = (bxA , d xA , u xA , a xA )
Binary frame
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X
x
x
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base rate of x
b + d +u =1
A
x
A
x
A
x
13
Barycentric Coordinate System
State space of
3 dimensions
θ1, θ2, θ3,
• E.g. probability additivity: p(θ1) + p(θ2) + p(θ3) = 1
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Opinions represented as point in a simplex
• Simplex class of geometric shapes:
– triangle is a 2D simplex
– tetrahedron is a 3D simplex
• Opinions represented as point inside a simplex.
• The equation Σbi + u = 1 represents a
barycentric coordinate system in a simplex.
u
axis
2D simplex is
3D Barycentric
b1
coordinate
system
b2
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axis
b2
u
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b1
axis
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ux
Binomial opinions
•
axis
Ordered quadruple:
ω x = (bx , d x , u x , a x )
Director
– bx : belief
– dx : disbelief
Opinion
ωX
– ux : uncertainty
– ax : base rate
•
bx + d x + u x = 1
Projector
dx
Expectation value
• Probability expectation value:
Example ωx =(0.2, 0.5, 0.3, 0.6),
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bx
axis
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Ex
ax
axis
Base rate
E (ω x ) = bx + a x u x
E(ωx) = 0.38
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What are base rates?
• In probability and statistics, base rate refers to
category probability unconditioned on evidence,
often referred to as prior probabilities.
• For example, if it were the case that 1% of the
public are "medical professionals" and 99% of
the public are not "medical professionals“, then
the base rates in this case are 1% and 99%,
respectively.
• E.g. when picking a random person, the prior
probability of being a medical professional is 1%
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Opinion types
Absolute opinion: bx=1 .
Equivalent to TRUE.
Dogmatic opinion: ux=0 .
Equivalent to probabilities.
Vacuous opinion: ux=1 .
Equivalent to UNDEFINED.
General uncertain opinion: ux≠0 .
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Beta PDF representation
Beta ( p | α , β ) =
Γ(α + β ) α −1
p (1 − p ) β −1
Γ(α )Γ( β )
α = r + Wa
β = s + W(1-a)
r: # observations of x
s: # observations of x
Beta Probability Density Function
a: base rate of x
W = 2: non-informative
prior weight
Example: r = 7,
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s = 1,
a = 0.5 (default),
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E(p) = 0.8
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Binomial Opinion ↔ Beta PDF
• (r,s,a) represents Beta PDF parameters.
• (b,d,u,a) represents binomial opinion.
 r = Wb / u

• Op → Beta:  s = Wd / u
b + d + u = 1

 b = r + sr+W

• Beta → Op: d = r + ss+W
u = W
r + s +W

W=2
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Online demo
http://folk.uio.no/josang/sl/
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Fuzzy verbal categories
Very unlikely
Unlikely
Somewhat unlikely
Chances about even
Somewhat likely
Likely
Very likely
Absolutely
Certainty Categories:
Absolutely not
Likelihood Categories:
9
8
7
6
5
4
3
2
1
Completely uncertain
E
9E 8E 7E 6E 5E 4E 3E 2E 1E
Very uncertain
D
9D 8D 7D 6D 5D 4D 3D 2D 1D
Uncertain
C
9C 8C 7C 6C 5C 4C 3C 2C 1C
Slightly uncertain
B
9B 8B 7B 6B 5B 4B 3B 2B 1B
Completely certain
A
9A 8A 7A 6A 5A 4A 3A 2A 1A
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Soliciting opinions from people
• People find it difficult to express opinions as
numerical values
• Fuzzy verbal categories are intuitively easier
• Opinions have 2-dimensional fuzzy categories
– Likelihood dimension
– Certainty dimension
• Suitable categories depend on application
–
–
–
–
Example shows 9 likelihoods and 5 certainties
1A corresponds to TRUE
9A corresponds to FALSE
High uncertainty most natural around medium likelihood
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Mapping fuzzy category to opinion
• Category mapped to corresponding field of triangle
• Mapping depends on base rate
• Non-existent categories depending on base-rates
base rate a = 1/3
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base rate a = 2/3
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Mapping categories to opinions
• Overlay category matrix with opinion triangle
• Matrix skewed as a function of base rate
• Not all categories map to opinions
– For a low base rate, it is impossible to describe an
event as highly likely and uncertain, but possible to
describe it as highly unlikely and uncertain.
– E.g. with regard to tuberculosis which has a low base
rate, it would be wrong to say that a patient is likely to
be infected, with high uncertainty. Similarly it would
be possible to say that the patient is probably not
infected, with high uncertainty
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n-ary frame of discernment
• Generalisation of binary state space
• Set of exclusive and exhaustive singletons.
• Example Frame: X={x1, x2, x3, x4}, | X | = 4.
X
x1
x2
x3
x4
• |R (X)| = 2|X| − 2 (= 14 in this example).
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Opinions over binary and n-ary frames
• Binary frames represent a
single state / proposition and
its complement.
• n-ary frames represent
multiple mutually exclusive
states / propositions
X
x
x
X
x1
x2
x3
x4
• Opinions over binary frames:
→ binomial opinions
• Opinions over n-ary frames:
→ multinomial opinions and hyper opinions
• Subjective logic operators for all types of opinions
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Multinomial Opinions
•
•
•
•
Frame:
X={x1…xk}
Uncertainty mass: u

Belief vector: b : {b( xi ) | i = 1 k}, u + Σb(xi) = 1

Base rates: a : {a ( xi ) | i = 1 k},
Σa(xi) = 1
• Multinomial opinion:
• Expectation:
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

ω = (b , u , a )

E( xi ) = b( xi ) + a ( xi )u
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Opinion tetrahedron (ternary frame)
uX - axis
Opinion
ωX
Projector
bx3 - axis

EX
Expectation value vector
point
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bx2 - axis

aX
bx1 - axis
Base rate vector
point
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Dirichlet PDF representation
 k

Γ ∑ α ( xi )  k
α ( xi )−1
 
i =1


Dir ( p | α ) = k
p ( xi )
∏
∏ Γ(α ( xi )) i =1
Σ p(xi) = 1
α(xi) = r(xi) + Wa(xi)
i =1
r (xi) : # observations of xi
Trinomial Dirichlet
Probability Density Function
a (xi) : base rate of xi
W = 2: non-informative
prior weight
Example:
– 6 red balls
– 1 yellow ball
– 1 black ball
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Multinomial Opinion ↔ Dirichlet PDF
 
• (r , a )
represents Dirichlet PDF parameters.
 
• (b , u , a ) represents multinomial opinion.
• Op → Dir:
• Dir → Op:
W=2
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Wb( xi )

r
x
(
)
=
 i
u


u + ∑ b( xi ) = 1
r ( xi )

=
(
)
b
x
 i W + r(x )
∑ i



W
 u=
W + ∑ r ( xi )

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Non-informative prior weight: W
• Value normally set to W = 2.
• When W is equal to the frame cardinality, then
the prior Dirichlet PDF is a uniform.
• Normally required that the prior Beta is uniform,
which dictates W = 2
• Beta PDF is a binomial Dirichlet PDF
• If requiring uniform prior Dirichlet PDF for large
multi-dimensional frames, then it would make
Dirichlet PDF insensitive to new observations,
which would be inadequate.
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Prior trinomial Dirichlet PDF, W = 2
Example:
Urn with balls of 3
different colours.
– t1: Red
– t2: Yellow
– t3: Black
Ternary a priori
probability density.
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Posterior trinomial Dirichlet PDF
A posteriori
probability density
after picking:
– 6 red balls (t1)
– 1 yellow ball (t2)
– 1 black ball (t3)
– W=2
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Posterior trinomial Dirichlet PDF
A posteriori
probability density
after picking:
– 20 red balls (t1)
– 20 yellow balls (t2)
– 20 black balls (t3)
– W=2
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Posterior trinomial Dirichlet PDF
A posteriori
probability density
after picking:
– 20 red balls (t1)
– 20 yellow balls (t2)
– 50 black balls (t3)
– W=2
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Hyper Opinions
•
•
•
•
•
Frame:
X={x1…xk}
Reduced powerset: R (X) = P (X) \ {X, ∅}
Uncertainty mass:
u

k
Belief vector: b : {b( xi ) | i = 1 (2 − 2)} , xi∈R (X)

Base rates: a : {a ( xi ) | i = 1 k},
Σa(xi) = 1
• Hyper opinion:
• Expectation:
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

ω = (b , u , a )

E( xi ) = a ( x j / xi )b( xi ) + a ( xi )u
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Hyper Dirichlet PDF
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Opinions v. Fuzzy membership functions
Fuzzy concept
Subjective
opinions
Crisp concept
Friendly
aircraft
Enemy
aircraft
Something
else
ω
250 cm
Fuzzy
membership
functions
Tall
Average
200 cm
150 cm
100 cm
Short
50 cm
0 cm
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Subjective Logic Operators
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Operator notation
• Possible attributes of opinions:
– Who: the belief owner (superscript)
– What: the proposition (subscript)
– Where: the frame (normally omitted)
Subject combination
Derived
opinion
ω
f SC ( A, B )
f PL ( x , y )∈ f FC ( X ,Y )
Propositional logic
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Argument opinions
= ω
Frame composition
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Subjective logic
operator
A
x∈ X
Propositions
ω
Subjects
B
y∈Y
Frames
41
Operator generalisation
• Subjective logic is a generalisation of binary
logic and probability calculus.
– Probability calculus i.c.o. dogmatic opinions
– Binary logic i.c.o. absolute opinions
• Includes uncertainty.
• Includes belief ownership
• Operator types:
– Classic operators, e.g. multiplication (AND) and
deduction (MODUS PONENS)
– Special operators: e.g. trust transitivity and consensus
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Homomorphism principle
• Assuming that corresponding probability
operator exists, then …
• Expectation value of opinion operator result is
equal to result of probability operator applied to
the expectation values of opinion arguments.
– e.g. E(ωx ·ωy) = E(ωx)· E(ωy) for multiplication
• Same for corresponding binary logic operators
– e.g. Let V(ωx) denote TRUE/FALSE valuation of
absolute opinions, then V(ωx ·ωy) = V(ωx) ∧ V(ωy)
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Subjective logic operators 1
Opinion operator name
Opinion
Logic
operator
operator
symbol
symbol
Addition
+
∪
UNION
Subtraction
-
\
DIFFERENCE
Complement
¬
x
NOT
Expectation
E(x)
n.a.
n.a.
Multiplication
·
Division
/
Comultiplication
п
Codivision
п
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∧
∧
∨
∨
Logic operator name
AND
UN-AND
OR
UN-OR
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Subjective logic operators 2
Opinion operator name
Opinion
Logic
operator
operator
symbol
symbol
Transitive discounting
⊗
:
TRANSITIVITY
Cumulative fusion
⊕
◊
n.a.
Averaging fusion
⊕
◊
n.a.
Constraint fusion

&
n.a.
cc

♥
n.a.
Consensus & Compromise
Conditional deduction
||
Conditional abduction
||
Logic operator name
DEDUCTION
(Modus Ponens)
ABDUCTION
(Modus Tollens)
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Addition
x
•
•
•
•
x∪y
Θ
y
z
A
A
A
Notation
ω x∪ y = ω x + ω y
Probability version: P(x∪y)=P(x) + P(y)
Commutative and associative.
No corresponding binary logic operator
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Subtraction
x\y
x
Θ
y
z
ω xA\ y = ω xA − ω yA
• Notation
• Probability version: P(x\y)=P(x) ‒ P(y)
• No corresponding binary logic operator
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Complement
• Notation:
• Involutive:
• Corresponds to NOT.
x
x
¬ω xA = ω xA
¬(¬ω ) = ω
A
x
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A
x
48
Cartesian product of frames
• Multiplication assumes a Cartesian product.
• Product set has Cardinality = |X|·|Y| .
• Coarsening needed as part of computation.
X
Y
x
y
x
×
X×Y
( x, y )
( x, y )
( x, y )
( x, y )
=
y
product:
coproduct:
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Binomial conjunction (multiplication)
•
•
•
•
Notation:
Probability version: p(x∧y)=p(x)·p(y)
Commutative and associative.
Corresponds to AND and probability product.
X
Y
x
y
∧
( x, y )
( x, y )
( x, y )
( x, y )
=
y
x
ω
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X×Y
A
x∧ y
= ω ⋅ω
A
x
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A
y
50
Binomial disjunction (comultiplication)
•
•
•
•
Notation:
Probability version: p(x∨y) = p(x) + p(y) – p(x)p(y)
Commutative and associative.
Corresponds to OR and probability coproduct.
X
Y
x
y
x
∨
X×Y
( x, y )
( x, y )
( x, y )
( x, y )
=
y
ω xA∨ y = ω xA ⋅ ω yA
п
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Multinomial conjunction (multiplication)
•
•
•
•
= ω ⋅ω
Notation: ω
Probability version: matrix multiplication
Commutative and associative
Produces hyper opinion
A
X
A
X ×Y
A
Y
part of 2X×Y
X
x
x
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Y
∧
( x, y )
( x, y )
( X , y)
( x, y )
( x, y )
( X , y)
( x, Y )
( x, Y )
( X ,Y )
y
=
y
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Non-distributivity of products
Multiplication is non-distributive on comultiplication
for opinions:
ω x ∧ ( y ∨ z ) ≠ ω ( x ∧ y )∨ ( x ∧ z )
and for probabilities
p ( x ∧ ( y ∨ z )) ≠ p (( x ∧ y ) ∨ ( x ∧ z ))
y
x
z
≠
x
y
x
z
Only applicable for binary logic: x ∧ (y∨z) = (x∧y) ∨ (x∧z)
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Algebraic properties
• Product:
E(ω x ∧ y ) = E(ω x )E(ω y )
• Coproduct:
E(ω x ∨ y ) = E(ω x ) + E(ω y ) − E(ω x )E(ω y )
• Complement: E (ω x ) = 1 − E (ω x )
• De Morgan 1: ω x ∧ y = ω x ∨ y
• De Morgan 2: ω x ∨ y = ω x ∧ y
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Cartesian quotient of frames
• Division assumes a pre-existing Cartesian product K
• Quotient set has Cardinality = |K|/|L|
• Coarsening needed as part of computation
X×Y = K
( x, y )
( x, y )
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( x, y )
( x, y )
X=L
Y = K/L
x
y
/
=
x
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y
55
Binomial un-conjunction (division)
A
A
A
ω
ω
ω
=
/
• Notation: k ∧ l
k
l
• Probability version: P( k∧l )=P( k )/P( l )
• Corresponds to UN-AND and probability
division
X
X×Y
( x, y ) = k
( x, y )
( x, y )
( x, y )
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∧
x =l
x
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Y
y = k ∧l
=
y
56
Binomial un-disjunction (codivision)
= ω ⋅ω
• Notation: ω
• Probability version: P( k∨l )=(P( k ) - P( l ))/(1 - P( l ))
• Corresponds to UN-OR and probability codivision
A
k
п
A
k ∨l
A
l
X
X×Y
( x, y )
( x, y )
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( x, y )
=k
∨
x =l
x
( x, y )
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Y
y =k ∨l
=
y
57
Truth table; Products and quotients
F
AND
product
x∧y
F
OR
coproduct
x∨y
F
UN-AND
quotient
x∧y
T or F
UN-OR
coquotient
x∨y
F
F
T
F
T
F
undefined
T
F
F
T
undefined
T
T
T
T
T
T
T or F
x
y
F
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Online demo of SL operators
http://folk.uio.no/josang/sl/
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Fusion in Subjective Logic
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Cumulative and Averaging Fusion
•
•
•
•
= ω ⊕ω
Notation: ω
Cumulative fusion: independent evidence
Averaging fusion: dependent evidence
Reduced to weighted average i.c.o. dogmatic opinions.
A◊B
x
A
ω
x
A
B
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ωBx
A
x
x
B
x
A◊B
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ω
A◊B
x
x
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Comparing Fusion and Conjunction
A
ω
x
A
Fusion
B ω Bx
ωxA
Conjunction
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A
ωAy
x
A◊B
x
ω xA◊B = ω xA ⊕ ω xB
x
y
x∧y
A
ω xA∧ y = ω xA ·ω yA
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Cumulative Fusion
• Accumulates evidence from different sources
• Symbol: ⊕
• Sum of Dirichlet evidence vectors
 
1. Convert opinions to Dir/Beta: ω → Dir ( p | r , α )

2. Add evidence vectors r to get cumulative Dir/Beta
 
3. Convert Dir/Beta to opinion Dir ( p | r , α ) → ω
• Commutative, Associative, Neutral element
• Applicable to situations where collected
evidence is independent
– E.g. observed over different time periods
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Averaging Fusion
• Average of evidence from different sources
• Symbol: ⊕
• Average of Dirichlet evidence vectors
 
1. Convert opinions to Dir/Beta: ω → Dir ( p | r , α )

2. Take average of evidence vectors r to produce
an average Dir/Beta
 
3. Convert Dir/Beta to opinion Dir ( p | r , α ) → ω
• Commutative, Idempotent, (not associative).
• Applicable to situations where collected
evidence is dependent
– E.g. same event observed by different observers
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Example: Reaching a verdict
•
•
•
•
J1, J2 , … Jn are n jury members.
“guilty” is a binary statement.
[J1, J2 , … Jn ] denotes the whole jury.
ωBRD is a politically defined threshold value for
“Beyond Reasonable Doubt”.
J1
J2
Jn
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[J1, J2 , … Jn]
“guilty”
ω
J1◊J 2 ◊◊Jn
"guilty"
“guilty”
> ωBRD ?
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Belief Constraint Fusion
•
•
•
•
•
=ω ω
Notation: ω
Commutative, Associative, Neutral element
No corresponding binary logic operator
Can not be applied for conflicting dogmatic opinions.
Extension of Dempster’s rule
A& B
x
A
ω
x
A
B
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ωBx
A
x 
x
B
x
A&B
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ωA&B
x
x
66
Example belief constraint fusion
• Alice, Bob and Clark want to go to the cinema together
• Options are films: “X”, “Y” and “Z”, i.e. frame Θ = {X, Y, Z}
Preferences of
Alice
ω
A
Θ
Bob
ω
B
Θ
Results of preference combination
Clark
Alice & Bob
ω
C
Θ
ω
A& B
Θ
Alice & Bob & Clark
ω
A& B & C
Θ
b(X)
=
0.99
0.00
0.00
0.00
N.A.
b(Y)
=
0.01
0.01
0.00
1.00
N.A.
b(Z)
=
0.00
0.99
0.00
0.00
N.A.
b(X∪Z) =
0.00
0.00
1.00
0.00
N.A.
• They can only agree on watching: “film Y”
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Comparing Averaging and Constraint Fusion
A
ω
x
A
Averaging
Fusion
B
ωBx
x
A◊B
ωxA◊B
x
• Jurors A and B reach a consensus about truth of x
Constraint
Fusion
A
ω
x
A
B
ωBx
x
A&B
A&B
ωx
x
• Agents A and B agree on whether x is a good choice
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Comparing cumulative and constraint fusion
• The values are from Zadeh’s example
• Alice and Bob have conflicting beliefs
• Options are: “X”, “Y” and “Z” ,
i,e, frame Θ = {X, Y,Z}
Opinions of
Alice
ωΘA
Bob
ω
B
Θ
Results of opinion fusion
Cumulative
ω
A◊B
Θ
Constraint
ω
A& B
Θ
b(X)
=
0.99
0.00
0.495
0.00
b(Y)
=
0.01
0.01
0.010
1.00
b(Z)
=
0.00
0.99
0.495
0.00
0.00
0.00
0.000
0.00
u(Θ) =
• Constraint fusion is equivalent to Dempster’s rule
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Consensus & Compromise Fusion
=ω ω
• Notation: ω
• Commutative, Semi-associative, Idempotent,
Neutral element
• No corresponding binary logic operator
• Produces consensus i.c.o. agreement.
• Produces compromise i.c.o. conflict
A♥ B
x
A
ω
x
A
B
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ωBx
A
cc
x
B
x
A♥B
ω
x
A♥B
x
x
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Selecting the right fusion operator (1)
• It is assumed that in case two belief arguments are in
total conflict, then no statement can be correct.
⇒ Constraining Fusion
• It is assumed that in case two belief arguments are in
total conflict, then all statements supported by the
arguments can be correct in a statistical sense.
⇒ Cumulative Fusion or Averaging Fusion
• It is assumed that in case two belief arguments are in
total conflict, then one of the supported statements is
correct, but it is uncertain which of them is correct.
⇒ Consensus & Compromise Fusion
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Selecting the right fusion operator (2)
• Idempotent belief fusion is assumed, i.e. a belief
argument fused with itself should always produce the
same belief argument.
⇒ Averaging Fusion or Consensus & Compromise Fusion
• Idempotent belief fusion is not assumed, i.e. a partially
uncertain belief argument fused with itself should produce
a fusion result with less uncertainty.
⇒ Cumulative Fusion or Constraining Fusion
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Exercise: Fusion of opinions
Scenario 1: Belief about single egg
Scenario 2: Predictive belief
• Genetic engineering of eggs
• Obs.1 of egg nr.1
– Male or Female (stochastic)
• Obs.2 of egg nr.2
– Mutation X or Y (stochastic)
• S-IA and S-IB observe gender
• S-II observes mutation
•
Determine
opinion about
gender (M,F)
and mutation
(X,Y) of egg
in Obs. 1
• Determine opinion about egg
gender (M,F) and mutation
(X,Y) in production process.
Observation 1
Sensor IA
Sensor IB
Averaging Fusion
⊕
Sensor II
Belief about single egg
⋅
Product
Cumulative
Fusion
Predictive belief
⊕
Observation 2
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Solution: Fusion of opinions
Belief about single egg
Predictive belief
• Binary frame {M,F}
• Quaternary frame
Θ={MX,MY,FX,FY}
ωMIA◊IB = ωMIA ⊕ ωMIB
ωFIA◊IB = ¬ωMIA◊IB
ωΘObs1◊Obs2 = ωΘObs1 ⊕ ωΘObs2
• Binary frame {X,Y}
ωXII
ωYII = ¬ωXII
• Quaternary frame
Θ={MX,MY,FX,FY}
Obs1
ωMX
= ωMIA◊IB ⋅ ωXII
ωΘObs1 = ((bMX , bMY , bFX , bFY ), u , (aMX , aMY , aFX , aFY ))
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Trust modelling
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Trust transitivity
Thanks to Bob’s advice,
Alice Alice trusts Eric to be a
good mechanic.
Indirect functional trust
4
Direct
referral trust
1
2
Bob has proven to Alice that
he is knowledgeable in matters
relating to car maintenance.
Bob
Direct
functional
Eric has proven to
trust
Bob that he is a
good mechanic.
3
Recommendation
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Eric
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Trust scope
• The trust scope defines the specific purpose(s)
of trust assumed in a given trust relationship.
• In other words, the trusted party is relied upon to
have certain qualities, and the scope defines the
trusting party’s view of what those qualities are.
• Aka: Trust purpose, trust context, subject matter
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Types of Trust
• Direct
• Indirect
Trust as a result of direct experience
Trust as a result of recommendations
(i.e. indirect knowledge)
• Functional Trusting entity x for scope σ
(e.g. “to be a good car mechanic”)
• Referral
Trusting x to recommend for scope σ
(e.g. “to be reliable at recommending
car mechanics)
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Functional trust derivation requirement
Alice
2
rec.
1
referral
trust
Bob
2
rec.
Claire
1
referral
trust
Eric
1
functional
trust
functional trust
3
•
Derivation of functional trust through a
transitive path, requires that the last trust
arc represents functional trust, and all
previous trust arcs represent referral trust.
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Trust scope consistency requirement
Alice
Bob
referral
trust σ1
Claire
referral
trust σ2
Eric
functional
trust σ3
functional trust σ4= σ1∩ σ2 ∩ σ3
•
A valid transitive trust path requires that
there exists a trust scope which is a
common subset of all trust scopes in the
path. The derived trust scope is then the
largest common subset.
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Trust network building blocks
Combination of serial and parallel trust paths
Bob
Alice
Eric
Claire
Trusting
party
1
df-trust
Trusted
party
David
Notation:
(implicit scope)
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derived if-trust
3
[A, E] = (( [A,B] : [B,C] ) ◊ ( [A,D] : [D,C] )) : [C,E]
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Additional aspects of trust
• Trust measure: µ
– Binary (e.g. “Trusted”, “Not trusted”)
– Discrete (strong-, weak-, trust or distrust)
– Continuous (percentage, probability, belief)
• Time: τ
– Time stamp when trust was assessed and expressed.
Very important as trust generally weakens with
temporal distance.
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Trust transitivity characteristics
Trust is diluted in a transitive chain.
A
B
C
rec.
rec.
trust
trust
D
trust
diluted trust
Computed with transitivity operator of SL
Graph notation:
[A, D] = [A, B] : [B, C] : [C, D]
Explicit notation: [A, D, ifσ] = [A, B, drσ] : [B, C, drσ] : [C, D, dfσ]
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Trust fusion characteristics
B
A
rec.
D
trust
trust
trust
C
trust
rec.
concentrated
diluted trust
trust
Strengthens trust
confidence
Computed with the fusion operator of subjective logic
Graph notation:
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[A, D] = ([A, B] : [B, D]) ◊ ([A, C] : [C, D])
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Indirect referral trust
3
B
1
2
A
D
E
C
1
3
Beware!
2
incorrect trust
4
Perceived
Reality:
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trust
rec.
([A, B] : [B, E]) ◊ ([A, C] : [C, E])
([A, B] : [B, D] : [D, E]) ◊ ([A, C] : [C, D] : [D, E])
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Hidden and perceived topologies
Perceived topology:
Hidden topology:
B
A
B
E
D
A
C
E
C
D
([A, B] : [B, E]) ◊ ([A, C] : [C, E])
≠ ([A, B] : [B, D] : [D, E]) ◊ ([A, C] : [C, D] : [D, E])
(D, E) is taken into account twice
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Correct indirect referral trust
1
1
B
A
D
E
C
1
correct trust
2
Perceived and real
topologies are equal:
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SAFE
trust
rec.
( ([A, B] : [B, D]) ◊ ([A, C] : [C, D]) ) : [D, E]
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How to model trust relationships?
Alice
•
•
•
•
Probabilities:
Min:
Max:
Average:
Bob
Claire
p(A:C) = p(A:B)⋅p(B:C)
T(A:C) = Min[ T(A:B), T(B:C) ]
T(A:C) = Max[ T(A:B), T(B:C) ]
T(A:C) = ( T(A:B) + T(B:C) )/2
• What is needed is a formalism that can express
and compute with uncertainty, i.e. “I don’t know”
• The answer is: Subjective Logic
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Trust transitivity in SL
•
•
•
•
= ω ⊗ω
Notation: ω
Associative and non-commutative.
Operator for transitive belief
No correspondence to logic or probability.
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A:B
x
A
B
B
x
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Trust notation with subjective logic
• Agent A trust agent B for trust scope σ
A
– Explicit notation: ω B (σ )
A
ω
– Implicit notation: B
(implicit trust scope)
• Example: ([A, B] : [B, D]) ◊ ([A, C] : [C, D])
– SL notation:
(ω ⊗ ω ) ⊕ (ω ⊗ ω )
A
B
B
D
A
C
C
D
B
A
D
C
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Example: Weighing testimonies
•
Computing beliefs about statements in court.
•
J is the judge.
•
W1, W2 , W3 are witnesses providing testimonies.
J
ω
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W1
W2
W3
statement x
( J :W1 ) ◊ ( J :W2 ) ◊ ( J :W3 )
x
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Trust network analysis with subjective logic
• Subjective logic can be used to analyse Directed
Series Parallel Graphs (DSPG)
• Complex networks must be simplified
B
Original graph:
(non-DSPG)
Simplified graph 1:
(DSPG graph)
A
C
E
C
E
D
B
A
D
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Building Directed Series-Parallel Graphs
• Repeatedly apply
– Series graph composition
– Parallel graph composition
Series graph composition:
A
A
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B
Parallel graph composition:
C
A
C
C
A
C
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Example DSPG composition
B
A
G
D
C
F
E
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PKI and trust transitivity
User E
CA B
CA A
User D
CA C
Trust in public keys (explicit through certificate chaining)
Trust in CA’s (implicitly expressed through policies)
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Computational trust with
subjective logic
http://folk.uio.no/josang/sl/
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96
Trust model exercise 1
Write the trust expressions corresponding to the trust networks.
Try to write both network notation and subjective logic notation.
C
B
A
F
1.a)
1.b)
E
D
A
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B
C
E
D
F
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Trust model exercise 2
• Write subjective logic expression corresponding to the certificate
network below.
B
D
A
C
ω
A
aut ( k D )
ω =ω
A
( rel ( B ) ∧ ( aut ( k B ))
ω =ω
A
( rel ( C ) ∧ ( aut ( k C ))
A
B
A
C
=
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Trust model exercise 3
Draw the trust network corresponding to the following expression:
(((ω BA ⊗ (ω DB ⊗ ω FD ) ⊕ (ω EB ⊗ ω FE )) ⊕ (ωCA ⊗ ω FC )) ⊗ ωGF ) ⊕ ωGA
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Solutions to trust model exercises
• 1a:
ω FA = (ω BA ⊗ ω CB ⊗ ω FC ) ⊕ (ω DA ⊗ ω ED ⊗ ω FE )
• 1b:
ω FA = (((ω BA ⊗ ω DB ) ⊕ (ωCA ⊗ ω DC )) ⊗ ω FD ) ⊕ (ω EA ⊗ ω FE )
• 2:
B
A
A
C
A
A
A
((
)
((
)
)
ω
ω
ω
ω
ωaut
ω
ω
⊕
⋅
⊗
=
⋅
⊗
aut ( k ) )
rel ( C )
aut ( k )
aut ( k )
(k )
rel ( B )
aut ( k )
D
B
D
C
D
D
• 3:
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ω
A
G
=
B
E
F
C
A
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G
100
Bayesian belief reasoning
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Conditional deduction
A
A
A
A
ω
=
ω

ω
ω
( y| x , y| x )
• Notation: y|| x
x
• Probability: p(y||x) = p(x)·p(y|x) + p(x)·p(y|x)
• Corresponds to MODUS PONENS and
conditional inference.
• Ternary operator
ω xA
A
ω
A
y|x
ω Ay|x
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x
x→y
x→y
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A
ωAy
x
y
102
Conditional abduction
• Notation:
• Corresponds to MODUS TOLLENS and reverse
conditional inference.
• Quaternary operator
ω
A
ω xA
ω Ax|y
ω Ax|y
a(y)
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A
y ||x
= ω  (ω , ω , a ( y ))
A
x
x
y→x
y→x
y
A
x| y
A
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A
x| y
ω Ay
x
y
103
Deductive vs. abductive reasoning
Deductive Reasoning
(reasoning with causal evidence)
Likelihood of hypothesis,
when the evidence is
true; and when false.
p(y|x), p(y|¬x)
Evidence, x
a(y)
a(x)
p(x)
Hypothesis, y
p(y) ?
p(x|y), p(x|¬y)
Abductive Reasoning
(reasoning with derivative evidence)
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Likelihood of evidence,
when the hypothesis is
true; and when false.
105
The Base Rate Fallacy
• The base rate fallacy is an error that occurs
when p(y||x), the conditional probability of some
hypothesis y given some evidence x, is
assessed without taking account of the "base
rate" of y, often as a result of wrongly assuming
equality between the two inverse conditionals:
p(y|x) = p(x|y).
• The correct type of reasoning where the
conditional p(y|x) is correctly derived, is
commonly referred to as abduction.
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Deduction with subjective logic
Y
X
y1
y2
⋅⋅⋅
ωY | x1
x2
ωY | x2
.
.
.
ωX
.
.
.
ωY | xk
xk
Conditionals ωY | X
x1
yl
Parent
=
ωY || X
Child
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Deduction visualisation
• Evidence pyramid is mapped inside hypothesis
pyramid as a function of the conditionals.
• Conclusion opinion is linearly mapped
Opinions on parent frame X
Opinions on child frame Y
u
ωY || X
X
ω X
u
Y
ωY |x
ωY || X
ωX
ωY |x
ωY |x
bx2
2
1
3
by2
bx3
bx1
by3
by
1
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Deduction – online operator demo
http://folk.uio.no/josang/sl/
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Abduction with subjective logic
Y
X
x1
y2
⋅⋅⋅
yl
Conditionals ωX | Y
x2
.
.
.
y1
ωX
.
ωX | y1 ωX | y2
.
ωX | yl
.
xk
Parent
=
ωY || X
Child
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Abduction Visualisation
uX
X opinion
triangle
ω X ||Yˆ
ωX |y
ωX |y
bx2
3
ωYˆ
2
Conditional
uY
Y opinion
triangle
ωX|Y
ωX |y
by2
1
bx3
bx1
by3
by1
Conditional opinion
Inversion
uX
ωY || Xˆ
ω X̂
X opinion
triangle
ωX
Conditional
bx2
ωY ||X
ωY|X
ωY |x
3
bx3
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bx1
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by3
uY
Y opinion
triangle
ωY |x
2
ωY |x
by2
1
by1
111
Example: Medical reasoning
• A medical test has a reliability determined by the
pharmaceutical company in terms of true
positive rate and false positive rate.
• ω(positive | infected) and ω(positive | not infected)
• GP needs to determine ω(infected) , i.e. opinion
about wether the patient has the disease.
• GP derives ω(infected | positive) and ω(infected | negative)
• Finally compute diagnosis ω(infected || test result)
• Medical reasoning with subjective logic works
even in the presence of high uncertainty
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Abduction – Online operator demo
http://folk.uio.no/josang/sl/
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Deduction and abduction notation
• Binomial deduction ω y|| x
• Multinomial deduction
• Binomial abduction
(ω y| x , ω y| x )
ωY || X = ω X
ωY | X
ω y|| x = ω x (ω x| y , ω x| y , a y )
• Multinomial abduction
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= ωx
ωY || X = ω X
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
(ω X |Y , aY )
114
Bayesian logic
• Subjective logic represents a calculus for Beta
and Dirichlet PDFs
• Analytically correct for 1st moment, i.e.
expectation value.
• Approximation for 2nd moment (i.e. variance)
• Analytic or numeric combination of PDFs give
high computational complexity
• Subjective logic gives very low computational
complexity
• Bayesian logic
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Bayesian network representation
Node R
ωR
Node X
ωX || R
Node S
ωS
ωX || S
⊕
Node Z
ωX || (R,S,T)
ωZ || X
Node T
ωT
⊕
ωX || T
ωZ || ( X,Y)
Node Y
ωY
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ωZ || Y
116
Exercise: Bayesian networks
1. Draw Bayesian network corresponding to:
ω Z = ω Z ||Y ⊕ ω Z ||Y
ωY = ωY || X ⊕ ωY || X 2
ωY = ωY ||X
1
1
1
2
2
2
1
1
3
2. Write SL expressions corresponding to
Bayesian network on previous slide
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Solution 1 – Bayesian network
Node X1
ωX1
Node Y1
ωY1 || X1
Node Z
⊕
ωY1
ωZ || Y1
Node X2
ωX2
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⊕
ωY1 || X2
Node X3
Node Y2
ωX3
ωY2 || X3
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ωZ
ωZ || Y2
118
Solution 2 – Bayesian network
ω Z = ω Z || X ⊕ ω Z ||Y
ω X = ω X ||R ⊕ ω X ||S ⊕ ω X ||T
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Final remarks on Subjective Logic
• Compatible with
– Binary logic
– Probabilistic logic
– Includes degrees of uncertainty
• Characteristics
–
–
–
–
Fast computation
Always correct probability expectation value
Sometimes approximate variance values
Analysis of situations with significant uncertainty,
• Belief fusion
• Bayesian networks
• Trust networks
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References
• Papers and online demo at: http://folk.uio.no/josang/
• Some relevant papers:
– Subjective logic (draft book 2014)
– Inverting Conditionals Opinions in Subjective Logic (2014)
– Determining Model Correctness for Situations of Belief Fusion (2013)
– Interpretation and Fusion of Hyper Opinions in Subjective Logic (2012)
– Preference Combination using Subjective Logic (2011)
– Cumulative and Averaging Fusion of Beliefs (2010)
– Conditional Reasoning with Subjective Logic (2008)
– Simplification and Analysis of Transitive Trust Networks (2006)
– Analysis of Competing Hypotheses using Subjective Logic (2005)
– Conditional Deduction Under Uncertainty (2005)
– Multiplication and Co-multiplication of Beliefs (2004)
– A Logic for Uncertain Probabilities (2001)
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