[PPT] - Composite Trust Composite Trust Composite Trust A formal PowerPoint Presentation

SLIDE 1

Composite Trust Composite Trust Composite Trust

A formal derivation of conjunction A formal derivation of conjunction A formal derivation of conjunction and disjunction of trust opinions. and disjunction of trust opinions. and disjunction of trust opinions.

Tim Muller and Patrick Schweitzer Tim Muller and Patrick Schweitzer

Composite Trust 1 Tim Muller and Patrick Schweitzer Composite Trust 1 Tim Muller and Patrick Schweitzer

SLIDE 2

Motivating Example Motivating Example Motivating Example Motivating Example

Trust opinion on a seller :

Trust opinion on a seller B:
Trust opinion on a seller B:

– Based on 5 successful and 1 failed transactions. – Based on 5 successful and 1 failed transactions.

Trust opinion on a delivery agency :

Trust opinion on a delivery agency C:
Trust opinion on a delivery agency C:

– Based on 4 successful and 2 failed transactions. – Based on 4 successful and 2 failed transactions.

User contemplates a purchase from via .

User contemplates a purchase from B via C.
User contemplates a purchase from B via C.

– Succeeds iff B and C individually succeed. – Succeeds iff B and C individually succeed.

Tim Muller and Patrick Schweitzer 2 Composite Trust Tim Muller and Patrick Schweitzer 2 Composite Trust

SLIDE 3

Motivating Example Motivating Example Motivating Example Motivating Example

Trust opinion on a seller :

Trust opinion on a seller B:
Trust opinion on a seller B:

– Based on 5 successful and 1 failed transactions. – Based on 5 successful and 1 failed transactions.

Trust opinion on a delivery agency :

Trust opinion on a delivery agency C:
Trust opinion on a delivery agency C:

– Based on 4 successful and 2 failed transactions. – Based on 4 successful and 2 failed transactions.

User contemplates a purchase from via .

User contemplates a purchase from B via C.
User contemplates a purchase from B via C.

– Succeeds iff B and C individually succeed. – Succeeds iff B and C individually succeed.

What is the trust opinion on  ?

What is the trust opinion on BC?
What is the trust opinion on BC?

– 9 successes and 3 failures? – 9 successes and 3 failures?

Tim Muller and Patrick Schweitzer 2 Composite Trust Tim Muller and Patrick Schweitzer 2 Composite Trust

SLIDE 4

Motivating Example Motivating Example Motivating Example Motivating Example

Trust opinion on a seller :

Trust opinion on a seller B:
Trust opinion on a seller B:

– Based on 5 successful and 1 failed transactions. – Based on 5 successful and 1 failed transactions.

Trust opinion on a delivery agency :

Trust opinion on a delivery agency C:
Trust opinion on a delivery agency C:

– Based on 4 successful and 2 failed transactions. – Based on 4 successful and 2 failed transactions.

User contemplates a purchase from via .

User contemplates a purchase from B via C.
User contemplates a purchase from B via C.

– Succeeds iff B and C individually succeed. – Succeeds iff B and C individually succeed.

What is the trust opinion on  ?

What is the trust opinion on BC?
What is the trust opinion on BC?

– 4 successes and 2 failures? – 4 successes and 2 failures?

Tim Muller and Patrick Schweitzer 2 Composite Trust Tim Muller and Patrick Schweitzer 2 Composite Trust

SLIDE 5

Motivating Example Motivating Example Motivating Example Motivating Example

Trust opinion on a seller :

Trust opinion on a seller B:
Trust opinion on a seller B:

– Based on 5 successful and 1 failed transactions. – Based on 5 successful and 1 failed transactions.

Trust opinion on a delivery agency :

Trust opinion on a delivery agency C:
Trust opinion on a delivery agency C:

– Based on 4 successful and 2 failed transactions. – Based on 4 successful and 2 failed transactions.

User contemplates a purchase from via .

User contemplates a purchase from B via C.
User contemplates a purchase from B via C.

– Succeeds iff B and C individually succeed. – Succeeds iff B and C individually succeed.

What is the trust opinion on  ?

What is the trust opinion on BC?
What is the trust opinion on BC?

– Is a trust opinion even s successes and f failures? – Is a trust opinion even s successes and f failures?

Tim Muller and Patrick Schweitzer 2 Composite Trust Tim Muller and Patrick Schweitzer 2 Composite Trust

SLIDE 6

Motivating Example Motivating Example Motivating Example Motivating Example

Trust opinion on a seller :

Simple trust opinion

Trust opinion on a seller B:

Simple trust opinion

Trust opinion on a seller B:

– Based on 5 successful and 1 failed transactions. – Based on 5 successful and 1 failed transactions.

Trust opinion on a delivery agency :

Trust opinion on a delivery agency C:
Trust opinion on a delivery agency C:

– Based on 4 successful and 2 failed transactions. – Based on 4 successful and 2 failed transactions.

User contemplates a purchase from via .

User contemplates a purchase from B via C.
User contemplates a purchase from B via C.

– Succeeds iff B and C individually succeed. – Succeeds iff B and C individually succeed.

What is the trust opinion on  ?

What is the trust opinion on BC?
What is the trust opinion on BC?

– Is a trust opinion even s successes and f failures? – Is a trust opinion even s successes and f failures?

Composite trust opinion Composite trust opinion Composite trust opinion

Tim Muller and Patrick Schweitzer 2 Composite Trust Tim Muller and Patrick Schweitzer 2 Composite Trust

SLIDE 7

Trust Opinions and Probability Trust Opinions and Probability Trust Opinions and Probability Trust Opinions and Probability

Targets have an unknown integrity parameter.

Targets have an unknown integrity parameter.
Targets have an unknown integrity parameter.

– The subject estimates integrity. – The subject estimates integrity.

Subject constructs a probability density

Subject constructs a probability density
Subject constructs a probability density

function over targets integrity. function over targets integrity. function over targets integrity.

– Using knowledge of subject – Using knowledge of subject – Using knowledge of subject

Tim Muller and Patrick Schweitzer 3 Composite Trust Tim Muller and Patrick Schweitzer 3 Composite Trust

SLIDE 8

Trust Opinions and Probability Trust Opinions and Probability Trust Opinions and Probability Trust Opinions and Probability

Targets have an unknown integrity parameter.

Targets have an unknown integrity parameter.
Targets have an unknown integrity parameter.

– The subject estimates integrity. – The subject estimates integrity.

Subject constructs a probability density

Subject constructs a probability density
Subject constructs a probability density

function over targets integrity. function over targets integrity. function over targets integrity.

– Using knowledge of subject – Using knowledge of subject – Using knowledge of subject

nsity ity density

bability densi
bability de
bability de

Probabil Integrity parameter Probabil Integrity parameter

Tim Muller and Patrick Schweitzer 3 Composite Trust Tim Muller and Patrick Schweitzer 3 Composite Trust

SLIDE 9

Trust Opinions and Probability Trust Opinions and Probability Trust Opinions and Probability Trust Opinions and Probability

Targets have an unknown integrity parameter.

Targets have an unknown integrity parameter.
Targets have an unknown integrity parameter.

– The subject estimates integrity. – The subject estimates integrity.

Subject constructs a probability density

Subject constructs a probability density
Subject constructs a probability density

function over targets integrity. function over targets integrity. function over targets integrity.

– Using knowledge of subject

Trust opinions

– Using knowledge of subject

Trust opinions

– Using knowledge of subject

Tim Muller and Patrick Schweitzer 3 Composite Trust Tim Muller and Patrick Schweitzer 3 Composite Trust

SLIDE 10

Assumptions on Simple Trust Opinions Assumptions on Simple Trust Opinions Assumptions on Simple Trust Opinions Assumptions on Simple Trust Opinions

I. No prior knowledge about integrity of agents. I. No prior knowledge about integrity of agents. I. No prior knowledge about integrity of agents.

II. Probability of success is (solely) determined by
II. Probability of success is (solely) determined by
II. Probability of success is (solely) determined by

the integrity parameter. the integrity parameter.

III. Outcomes in past interactions are (solely)
III. Outcomes in past interactions are (solely)

determined by the integrity parameter. determined by the integrity parameter. determined by the integrity parameter.

IV. Unrelated events have no connection.
IV. Unrelated events have no connection.
IV. Unrelated events have no connection.

Tim Muller and Patrick Schweitzer 4 Composite Trust Tim Muller and Patrick Schweitzer 4 Composite Trust

SLIDE 11

Assumptions on Simple Trust Opinions Assumptions on Simple Trust Opinions Assumptions on Simple Trust Opinions Assumptions on Simple Trust Opinions

I. No prior knowledge about integrity of agents. I. No prior knowledge about integrity of agents. I. No prior knowledge about integrity of agents.

– Can be extended to encompass prior knowledge. – Can be extended to encompass prior knowledge. – Can be extended to encompass prior knowledge.

II. Probability of success is (solely) determined by
II. Probability of success is (solely) determined by
II. Probability of success is (solely) determined by

the integrity parameter. the integrity parameter.

– Can be extended to deal with external influences. – Can be extended to deal with external influences.

III. Outcomes in past interactions are (solely)
III. Outcomes in past interactions are (solely)

determined by the integrity parameter. determined by the integrity parameter. determined by the integrity parameter.

– Can be extended to incorporate time, state or context. – Can be extended to incorporate time, state or context. – Can be extended to incorporate time, state or context.

IV. Unrelated events have no connection.
IV. Unrelated events have no connection.
IV. Unrelated events have no connection.

Tim Muller and Patrick Schweitzer 4 Composite Trust Tim Muller and Patrick Schweitzer 4 Composite Trust

SLIDE 12

The Beta Model The Beta Model The Beta Model The Beta Model

Beta distribution (PDF):

1

Beta distribution (PDF):

1 1

1 ( ; , ) (1 ) f x x x

 

 

   

Beta distribution (PDF):

β( ; ,

) (1 ) ( , ) f x x x         

Simple trust opinions are beta distributions

( , )   

Simple trust opinions are beta distributions

(Lemma 1). (Lemma 1). (Lemma 1).

If the subject observes s, f from the target, his
If the subject observes s, f from the target, his

trust opinion is .

 

trust opinion is .

β( ;

1, 1) f x s f  

trust opinion is .

β( ;

1, 1) f x s f  

Tim Muller and Patrick Schweitzer 5 Composite Trust Tim Muller and Patrick Schweitzer 5 Composite Trust

SLIDE 13

Problem Statement Problem Statement Problem Statement Problem Statement

Recall:
Recall:
Recall:

– Status of interaction may depend on two agents. – Status of interaction may depend on two agents. – What is the trust opinion on BC? – What is the trust opinion on BC? – What is the trust opinion on BC? – Targets have an internal integrity parameter. – Targets have an internal integrity parameter. – A trust opinion is a probability density function over – A trust opinion is a probability density function over the targets integrity. – A trust opinion is a probability density function over the targets integrity. the targets integrity. – Simple trust opinions can be derived from – Simple trust opinions can be derived from assumptions. assumptions.

Can we derive composite trust opinions from the
Can we derive composite trust opinions from the
Can we derive composite trust opinions from the

assumptions? assumptions?

– If not, which additional assumptions do we require? – If not, which additional assumptions do we require? – If not, which additional assumptions do we require?

Tim Muller and Patrick Schweitzer 6 Composite Trust Tim Muller and Patrick Schweitzer 6 Composite Trust

SLIDE 14

Problem Statement Problem Statement Problem Statement Problem Statement

Recall:
Recall:
Recall:

– Status of interaction may depend on two agents. – Status of interaction may depend on two agents. – What is the trust opinion on BC? – What is the trust opinion on BC? – What is the trust opinion on BC? – Targets have an internal integrity parameter. – Targets have an internal integrity parameter. – A trust opinion is a probability density function over – A trust opinion is a probability density function over the targets integrity. – A trust opinion is a probability density function over the targets integrity. the targets integrity. – Simple trust opinions can be derived from – Simple trust opinions can be derived from assumptions. assumptions.

Can we derive composite trust opinions from the
Can we derive composite trust opinions from the
Can we derive composite trust opinions from the

assumptions? assumptions?

– If not, which additional assumptions do we require? – If not, which additional assumptions do we require? – If not, which additional assumptions do we require?

Tim Muller and Patrick Schweitzer 6 Composite Trust Tim Muller and Patrick Schweitzer 6 Composite Trust

SLIDE 15

Problem Statement Problem Statement Problem Statement Problem Statement

Recall:
Recall:
Recall:

– Status of interaction may depend on two agents. – Status of interaction may depend on two agents. – What is the trust opinion on BC? – What is the trust opinion on BC? – What is the trust opinion on BC? – Targets have an internal integrity parameter. – Targets have an internal integrity parameter. – A trust opinion is a probability density function over – A trust opinion is a probability density function over the targets integrity. – A trust opinion is a probability density function over the targets integrity. the targets integrity. – Simple trust opinions can be derived from – Simple trust opinions can be derived from assumptions. assumptions.

Can we derive composite trust opinions from the
Can we derive composite trust opinions from the
Can we derive composite trust opinions from the

assumptions? assumptions?

– If not, which additional assumptions do we require? – If not, which additional assumptions do we require? – If not, which additional assumptions do we require?

Tim Muller and Patrick Schweitzer 6 Composite Trust Tim Muller and Patrick Schweitzer 6 Composite Trust

SLIDE 16

Problem Statement Problem Statement Problem Statement Problem Statement

Recall:
Recall:
Recall:

– Status of interaction may depend on two agents. – Status of interaction may depend on two agents. – What is the trust opinion on BC? – What is the trust opinion on BC? – What is the trust opinion on BC? – Targets have an internal integrity parameter. – Targets have an internal integrity parameter. – A trust opinion is a probability density function over – A trust opinion is a probability density function over the targets integrity. – A trust opinion is a probability density function over the targets integrity. the targets integrity. – Simple trust opinions can be derived from – Simple trust opinions can be derived from assumptions. assumptions.

Can we derive composite trust opinions from the
Can we derive composite trust opinions from the
Can we derive composite trust opinions from the

assumptions? assumptions?

– If not, which additional assumptions do we require? – If not, which additional assumptions do we require? – If not, which additional assumptions do we require?

Tim Muller and Patrick Schweitzer 6 Composite Trust Tim Muller and Patrick Schweitzer 6 Composite Trust

SLIDE 17

Random Variables Random Variables Random Variables Random Variables

represents the outcome of the

represents the outcome of the

S F

: { , } E  

represents the outcome of the

interaction with target T.

S F

: { , }

T

E  

interaction with target T. interaction with target T.

represents the integrity parameter
represents the integrity parameter

 

: 0,1

T

R  

represents the integrity parameter
f the target T.

 

: 0,1

T

R  

f the target T.
f the target T.
represents the past interactions
represents the past interactions

:

A B

O     

represents the past interactions

between agents A and B.

:

B

O     

between agents A and B. between agents A and B.

Tim Muller and Patrick Schweitzer 7 Composite Trust Tim Muller and Patrick Schweitzer 7 Composite Trust

SLIDE 18

Random Variables Random Variables Random Variables Random Variables

represents the outcome of the

represents the outcome of the

S F

: { , } E  

represents the outcome of the

interaction with target T.

S F

: { , }

T

E  

interaction with target T. interaction with target T.

represents the integrity parameter
represents the integrity parameter

 

: 0,1

T

R  

represents the integrity parameter
f the target T.

 

: 0,1

T

R  

f the target T.
f the target T.
represents the past interactions
represents the past interactions

:

A B

O     

represents the past interactions

between agents A and B.

:

B

O     

between agents A and B. between agents A and B.

Trust opinion is

 

Trust opinion is

 

|

R

f x 

Trust opinion is

Lemma 1:

 

   

|

T

R

f x 

Lemma 1:

 

β

| ( , ) ; 1, 1

A R B

f x O m n f x m n    

Lemma 1:

 

β

| ( , ) ; 1, 1

B

R B

f x O m n f x m n    

Tim Muller and Patrick Schweitzer 7 Composite Trust Tim Muller and Patrick Schweitzer 7 Composite Trust

SLIDE 19

Formulating Composite Trust Opinions Formulating Composite Trust Opinions Formulating Composite Trust Opinions Formulating Composite Trust Opinions

Extend the shape of simple trust opinions:

Extend the shape of simple trust opinions:
Extend the shape of simple trust opinions:

 

–

 

| ( , ), ( ', ')

A A R B C

f x O m n O m n  

–

 

| ( , ), ( ', ')

B C

R B C

f x O m n O m n



 

Tim Muller and Patrick Schweitzer 8 Composite Trust Tim Muller and Patrick Schweitzer 8 Composite Trust

SLIDE 20

Formulating Composite Trust Opinions Formulating Composite Trust Opinions Formulating Composite Trust Opinions Formulating Composite Trust Opinions

Extend the shape of simple trust opinions:

Extend the shape of simple trust opinions:
Extend the shape of simple trust opinions:

 

–

 

| ( , ), ( ', ')

A A R B C

f x O m n O m n  

–

Use the following assumptions:

 

| ( , ), ( ', ')

B C

R B C

f x O m n O m n



 

Use the following assumptions:
Use the following assumptions:

I. No prior knowledge about integrity of agents. I. No prior knowledge about integrity of agents. II. Probability of success is (solely) determined by II. Probability of success is (solely) determined by II. Probability of success is (solely) determined by the integrity parameter. the integrity parameter.

III. Outcomes in past interactions are (solely)
III. Outcomes in past interactions are (solely)

determined by the integrity parameter. determined by the integrity parameter. determined by the integrity parameter.

IV. Unrelated events have no connection.
IV. Unrelated events have no connection.
IV. Unrelated events have no connection.

Tim Muller and Patrick Schweitzer 8 Composite Trust Tim Muller and Patrick Schweitzer 8 Composite Trust

SLIDE 21

Formulating Composite Trust Opinions Formulating Composite Trust Opinions Formulating Composite Trust Opinions Formulating Composite Trust Opinions

Extend the shape of simple trust opinions:

Extend the shape of simple trust opinions:
Extend the shape of simple trust opinions:

 

–

 

| ( , ), ( ', ')

A A R B C

f x O m n O m n  

–

Use the following assumptions:

 

| ( , ), ( ', ')

B C

R B C

f x O m n O m n



 

Use the following assumptions:
Use the following assumptions:

I. No prior knowledge about integrity of agents.

B

R

I. No prior knowledge about integrity of agents. II. Probability of success is (solely) determined by

B

R

II. Probability of success is (solely) determined by II. Probability of success is (solely) determined by the integrity parameter.

E R 

the integrity parameter.

T T

E R 

III. Outcomes in past interactions are (solely)

T T

III. Outcomes in past interactions are (solely)

determined by the integrity parameter.

A 

determined by the integrity parameter.

A B B

O R 

determined by the integrity parameter.

IV. Unrelated events have no connection.

B B

O R 

IV. Unrelated events have no connection.
IV. Unrelated events have no connection.

Tim Muller and Patrick Schweitzer 8 Composite Trust Tim Muller and Patrick Schweitzer 8 Composite Trust

SLIDE 22

Formulating Composite Trust Opinions Formulating Composite Trust Opinions Formulating Composite Trust Opinions Formulating Composite Trust Opinions

Extend the shape of simple trust opinions:

Extend the shape of simple trust opinions:
Extend the shape of simple trust opinions:

 

–

 

| ( , ), ( ', ')

A A R B C

f x O m n O m n  

–

Stuck; no applicable assumptions:

 

| ( , ), ( ', ')

B C

R B C

f x O m n O m n



 

Stuck; no applicable assumptions:
Stuck; no applicable assumptions:

I. No prior knowledge about integrity of agents.

B

R

I. No prior knowledge about integrity of agents. II. Probability of success is (solely) determined by

B

R

II. Probability of success is (solely) determined by II. Probability of success is (solely) determined by the integrity parameter.

E R 

the integrity parameter.

T T

E R 

III. Outcomes in past interactions are (solely)

T T

III. Outcomes in past interactions are (solely)

determined by the integrity parameter.

A 

determined by the integrity parameter.

A B B

O R 

determined by the integrity parameter.

IV. Unrelated events have no connection.

B B

O R 

IV. Unrelated events have no connection.
IV. Unrelated events have no connection.

Tim Muller and Patrick Schweitzer 8 Composite Trust Tim Muller and Patrick Schweitzer 8 Composite Trust

SLIDE 23

Formulating Composite Trust Opinions Formulating Composite Trust Opinions Formulating Composite Trust Opinions Formulating Composite Trust Opinions

Extend the shape of simple trust opinions:

Extend the shape of simple trust opinions:
Extend the shape of simple trust opinions:

 

–

 

| ( , ), ( ', ')

A A R B c

f x O m n O m n  

–

Stuck; no applicable assumptions:

 

| ( , ), ( ', ')

B C

R B c

f x O m n O m n



 

Stuck; no applicable assumptions:
Stuck; no applicable assumptions:

– No assumptions about success and failure of – No assumptions about success and failure of interactions with composite targets ( ).

E

interactions with composite targets ( ).

T

E

interactions with composite targets ( ). – No assumptions about the integrity of composite

T

E

– No assumptions about the integrity of composite targets ( ).

T

R

targets ( ).

T

R

Tim Muller and Patrick Schweitzer 8 Composite Trust Tim Muller and Patrick Schweitzer 8 Composite Trust

SLIDE 24

Additional Assumptions Additional Assumptions Additional Assumptions Additional Assumptions

Assume:

Assume:
Assume:
V. The integrity of a composite target is completely
V. The integrity of a composite target is completely

determined by the integrity of its parts ( ).

R R 

determined by the integrity of its parts ( ).

A T

R R 

determined by the integrity of its parts ( ).

VI. The interaction with ST is a success iff the

A T

R R 

VI. The interaction with ST is a success iff the

ST interaction with S and the interaction with T is a interaction with S and the interaction with T is a success ( ).

, E E E 

success ( ).

,

A B A B

E E E  

success ( ).

VII. The interaction with ST is a success iff the

,

A B A B

E E E  

VII. The interaction with ST is a success iff the

interaction with S or the interaction with T is a interaction with S or the interaction with T is a success ( ).

, E E E 

success ( ).

,

A B A B

E E E  

success ( ).

A B A B 



Tim Muller and Patrick Schweitzer 9 Composite Trust Tim Muller and Patrick Schweitzer 9 Composite Trust

SLIDE 25

Additional Assumptions Additional Assumptions Additional Assumptions Additional Assumptions

Assume:

Not defined how!

Assume:

Not defined how!

Assume:
V. The integrity of a composite target is completely
V. The integrity of a composite target is completely

determined by the integrity of its parts ( ).

R R 

determined by the integrity of its parts ( ).

A T

R R 

determined by the integrity of its parts ( ).

VI. The interaction with ST is a success iff the

A T

R R 

VI. The interaction with ST is a success iff the

ST interaction with S and the interaction with T is a interaction with S and the interaction with T is a success ( ).

, E E E 

success ( ).

,

A B A B

E E E  

success ( ).

VII. The interaction with ST is a success iff the

,

A B A B

E E E  

VII. The interaction with ST is a success iff the

interaction with S or the interaction with T is a interaction with S or the interaction with T is a success ( ).

, E E E 

success ( ).

,

A B A B

E E E  

success ( ).

A B A B 



Tim Muller and Patrick Schweitzer 9 Composite Trust Tim Muller and Patrick Schweitzer 9 Composite Trust

SLIDE 26

Composite Trust Opinions Composite Trust Opinions Composite Trust Opinions Composite Trust Opinions

Lemma 2: If and are disjoint, then

Lemma 2: If S and T are disjoint, then R

R R  

Lemma 2: If S and T are disjoint, then

S T S T

R R R

 



Tim Muller and Patrick Schweitzer 10 Composite Trust Tim Muller and Patrick Schweitzer 10 Composite Trust

SLIDE 27

Composite Trust Opinions Composite Trust Opinions Composite Trust Opinions Composite Trust Opinions

Lemma 2: If and are disjoint, then

Lemma 2: If S and T are disjoint, then R

R R  

Lemma 2: If S and T are disjoint, then

S T S T

R R R

 



Corollary 2: If S and T are disjoint, then
Corollary 2: If S and T are disjoint, then

     

R R R  E E E 

     

S T S T

R R R



 E E E 

Tim Muller and Patrick Schweitzer 10 Composite Trust Tim Muller and Patrick Schweitzer 10 Composite Trust

SLIDE 28

Composite Trust Opinions Composite Trust Opinions Composite Trust Opinions Composite Trust Opinions

Lemma 2: If and are disjoint, then

Lemma 2: If S and T are disjoint, then R

R R  

Lemma 2: If S and T are disjoint, then

S T S T

R R R

 



Corollary 2: If S and T are disjoint, then
Corollary 2: If S and T are disjoint, then

     

R R R  E E E 

     

S T S T

R R R



 E E E 

Corollary 3: Conjunction and disjunction of



Corollary 3: Conjunction and disjunction of
Corollary 3: Conjunction and disjunction of

independent trust opinions is independent trust opinions is commutative and associative. commutative and associative. commutative and associative.

Tim Muller and Patrick Schweitzer 10 Composite Trust Tim Muller and Patrick Schweitzer 10 Composite Trust

SLIDE 29

Composite Trust Opinions Composite Trust Opinions Composite Trust Opinions Composite Trust Opinions

Lemma 2: If and are disjoint, then

Lemma 2: If S and T are disjoint, then R

R R  

Lemma 2: If S and T are disjoint, then

S T S T

R R R

 



Theorem 4: If S and T are disjoint, then
Theorem 4: If S and T are disjoint, then

1

 

   

1 1

| | | d x f x f f y y         



   

1

| | | d

S T S T

R R R x

x f x f f y y x y   



       



x x

y  



Tim Muller and Patrick Schweitzer 10 Composite Trust Tim Muller and Patrick Schweitzer 10 Composite Trust

SLIDE 30

Composite Trust Opinions Composite Trust Opinions Composite Trust Opinions Composite Trust Opinions

Lemma 2: If and are disjoint, then

Lemma 2: If S and T are disjoint, then R

R R  

Lemma 2: If S and T are disjoint, then

S T S T

R R R

 



Theorem 4: If S and T are disjoint, then
Theorem 4: If S and T are disjoint, then

1

 

   

1 1

| | | d x f x f f y y         



   

1

| | | d

S T S T

R R R x

x f x f f y y x y   



       



– Corollary 1: A subject can compute a trust opinion

x x

y  



– Corollary 1: A subject can compute a trust opinion for every well-formed target. for every well-formed target. for every well-formed target.

Tim Muller and Patrick Schweitzer 10 Composite Trust Tim Muller and Patrick Schweitzer 10 Composite Trust

SLIDE 31

Example of a Computation Example of a Computation Example of a Computation Example of a Computation

Subject calculated for target  .

Subject A calculated for target BC.
Subject A calculated for target BC.
Interactions with B: 5 successes, 1 failure.
Interactions with B: 5 successes, 1 failure.
Interactions with C: 4 successes, 2 failures.
Interactions with C: 4 successes, 2 failures.

Tim Muller and Patrick Schweitzer 11 Composite Trust Tim Muller and Patrick Schweitzer 11 Composite Trust

SLIDE 32

Example of a Computation Example of a Computation Example of a Computation Example of a Computation

Subject calculated for target  .

Subject A calculated for target BC.
Subject A calculated for target BC.
Interactions with B: 5 successes, 1 failure.
Interactions with B: 5 successes, 1 failure.
Interactions with C: 4 successes, 2 failures.
Interactions with C: 4 successes, 2 failures.

 

| (5,1), (4,2)

A A

f x O O  

 

| (5,1), (4,2)

B C

A A R B C

f x O O



 

 

B C

R B C



Tim Muller and Patrick Schweitzer 11 Composite Trust Tim Muller and Patrick Schweitzer 11 Composite Trust

SLIDE 33

Example of a Computation Example of a Computation Example of a Computation Example of a Computation

Subject calculated for target  .

Subject A calculated for target BC.
Subject A calculated for target BC.
Interactions with B: 5 successes, 1 failure.
Interactions with B: 5 successes, 1 failure.
Interactions with C: 4 successes, 2 failures.
Interactions with C: 4 successes, 2 failures.

 

| (5,1), (4,2)

A A

f x O O  

 

| (5,1), (4,2)

B C

A A R B C

f x O O



 

 

B C

R B C



1 1

x  

 

1 1

| (5,1), (4,2) | (5,1), (4,2) d

A A A A

x f O O f y O O y          



 

1

| (5,1), (4,2) | (5,1), (4,2) d

B C

A A A A R B C R B C x

x f O O f y O O y x y          



x x

y  



Tim Muller and Patrick Schweitzer 11 Composite Trust Tim Muller and Patrick Schweitzer 11 Composite Trust

SLIDE 34

Example of a Computation Example of a Computation Example of a Computation Example of a Computation

Subject calculated for target  .

Subject A calculated for target BC.
Subject A calculated for target BC.
Interactions with B: 5 successes, 1 failure.
Interactions with B: 5 successes, 1 failure.
Interactions with C: 4 successes, 2 failures.
Interactions with C: 4 successes, 2 failures.

 

| (5,1), (4,2)

A A

f x O O  

 

| (5,1), (4,2)

B C

A A R B C

f x O O



 

 

B C

R B C



1 1

x  

 

1 1

| (5,1), (4,2) | (5,1), (4,2) d

A A A A

x f O O f y O O y          



 

1

| (5,1), (4,2) | (5,1), (4,2) d

B C

A A A A R B C R B C x

x f O O f y O O y x y          



x x

y  



 

4 2 2

2205 1 4 5 (4 2 )log( ) x x x x x x      

 

2205 1 4 5 (4 2 )log( ) x x x x x x      

Tim Muller and Patrick Schweitzer 11 Composite Trust Tim Muller and Patrick Schweitzer 11 Composite Trust

SLIDE 35

Example (ctd.) Example (ctd.) Example (ctd.) Example (ctd.)

B C BC

Concentration of BC more than B (5,1) and

B C BC

Concentration of BC more than B (5,1) and
Concentration of BC more than B (5,1) and

less than C (4,2). less than C (4,2). less than C (4,2).

– Formally using entropy: -0.86, -0.62 and -0.67 bits. – Formally using entropy: -0.86, -0.62 and -0.67 bits.

Swapping successes and failures: -2.08 bits.

Swapping successes and failures: -2.08 bits.
Swapping successes and failures: -2.08 bits.

– Failures carry more information for conjunction. – Failures carry more information for conjunction. – Failures carry more information for conjunction.

Tim Muller and Patrick Schweitzer 12 Composite Trust Tim Muller and Patrick Schweitzer 12 Composite Trust

SLIDE 36

Example (ctd.) Example (ctd.) Example (ctd.) Example (ctd.)

B C BC

Concentration of BC more than B (5,1) and

B C BC

Concentration of BC more than B (5,1) and
Concentration of BC more than B (5,1) and

less than C (4,2). less than C (4,2). less than C (4,2).

– Formally using entropy: -0.86, -0.62 and -0.67 bits. – Formally using entropy: -0.86, -0.62 and -0.67 bits.

Swapping successes and failures: -2.08 bits.

Swapping successes and failures: -2.08 bits.
Swapping successes and failures: -2.08 bits.

– Failures carry more information for conjunction. – Failures carry more information for conjunction. – Failures carry more information for conjunction.

Tim Muller and Patrick Schweitzer 12 Composite Trust Tim Muller and Patrick Schweitzer 12 Composite Trust

SLIDE 37

Example (ctd.) Example (ctd.) Example (ctd.) Example (ctd.)

B C BC

Concentration of BC more than B (5,1) and

B C BC

Concentration of BC more than B (5,1) and
Concentration of BC more than B (5,1) and

less than C (4,2). less than C (4,2). less than C (4,2).

– Formally using entropy: -0.86, -0.62 and -0.67 bits. – Formally using entropy: -0.86, -0.62 and -0.67 bits.

Swapping successes and failures: -2.08 bits.

Swapping successes and failures: -2.08 bits.
Swapping successes and failures: -2.08 bits.

– Failures carry more information for conjunction. – Failures carry more information for conjunction. – Failures carry more information for conjunction.

Tim Muller and Patrick Schweitzer 12 Composite Trust Tim Muller and Patrick Schweitzer 12 Composite Trust

SLIDE 38

Beta Model Revisited Beta Model Revisited Beta Model Revisited Beta Model Revisited

In the Beta Model, trust opinions are

In the Beta Model, trust opinions are
In the Beta Model, trust opinions are

represented by beta distributions. represented by beta distributions. represented by beta distributions.

– Can composite trust opinions be represented as – Can composite trust opinions be represented as – Can composite trust opinions be represented as beta distributions? beta distributions?

Tim Muller and Patrick Schweitzer 13 Composite Trust Tim Muller and Patrick Schweitzer 13 Composite Trust

SLIDE 39

Beta Model Revisited Beta Model Revisited Beta Model Revisited Beta Model Revisited

In the Beta Model, trust opinions are

In the Beta Model, trust opinions are
In the Beta Model, trust opinions are

represented by beta distributions. represented by beta distributions. represented by beta distributions.

– Can composite trust opinions be represented as – Can composite trust opinions be represented as – Can composite trust opinions be represented as beta distributions? beta distributions?

Theorem 5: A composite trust opinion need
Theorem 5: A composite trust opinion need
Theorem 5: A composite trust opinion need

not be representable by a beta not be representable by a beta distribution. distribution. distribution.

Tim Muller and Patrick Schweitzer 13 Composite Trust Tim Muller and Patrick Schweitzer 13 Composite Trust

SLIDE 40

Implications of Theorem 5 Implications of Theorem 5 Implications of Theorem 5 Implications of Theorem 5

Models where composite trust opinions

Models where composite trust opinions
Models where composite trust opinions

correspond to beta distributions must either: correspond to beta distributions must either: correspond to beta distributions must either:

– Violate one of the assumptions of the Beta Model. – Violate one of the assumptions of the Beta Model. – Violate one of the assumptions of the Beta Model. – Violate one of the three extra assumptions. – Violate one of the three extra assumptions. – Violate one of the three extra assumptions.

Tim Muller and Patrick Schweitzer 14 Composite Trust Tim Muller and Patrick Schweitzer 14 Composite Trust

SLIDE 41

Implications of Theorem 5 Implications of Theorem 5 Implications of Theorem 5 Implications of Theorem 5

Models where composite trust opinions

Models where composite trust opinions
Models where composite trust opinions

correspond to beta distributions must either: correspond to beta distributions must either: correspond to beta distributions must either:

– Violate one of the assumptions of the Beta Model. – Violate one of the assumptions of the Beta Model. – Violate one of the assumptions of the Beta Model.

Their support for the representing simple trust opinions
Their support for the representing simple trust opinions

as beta distributions comes from these assumptions. as beta distributions comes from these assumptions.

– Violate one of the three extra assumptions. – Violate one of the three extra assumptions. – Violate one of the three extra assumptions.

Tim Muller and Patrick Schweitzer 14 Composite Trust Tim Muller and Patrick Schweitzer 14 Composite Trust

SLIDE 42

Implications of Theorem 5 Implications of Theorem 5 Implications of Theorem 5 Implications of Theorem 5

Models where composite trust opinions

Models where composite trust opinions
Models where composite trust opinions

correspond to beta distributions must either: correspond to beta distributions must either: correspond to beta distributions must either:

– Violate one of the assumptions of the Beta Model. – Violate one of the assumptions of the Beta Model. – Violate one of the assumptions of the Beta Model.

Their support for the representing simple trust opinions
Their support for the representing simple trust opinions

as beta distributions comes from these assumptions. as beta distributions comes from these assumptions.

– Violate one of the three extra assumptions. – Violate one of the three extra assumptions. – Violate one of the three extra assumptions.

The integrity of a composite target is not completely
The integrity of a composite target is not completely

determined by the integrity of its parts. determined by the integrity of its parts. determined by the integrity of its parts.

Tim Muller and Patrick Schweitzer 14 Composite Trust Tim Muller and Patrick Schweitzer 14 Composite Trust

SLIDE 43

Implications of Theorem 5 Implications of Theorem 5 Implications of Theorem 5 Implications of Theorem 5

Models where composite trust opinions

Models where composite trust opinions
Models where composite trust opinions

correspond to beta distributions must either: correspond to beta distributions must either: correspond to beta distributions must either:

– Violate one of the assumptions of the Beta Model. – Violate one of the assumptions of the Beta Model. – Violate one of the assumptions of the Beta Model.

Their support for the representing simple trust opinions
Their support for the representing simple trust opinions

as beta distributions comes from these assumptions. as beta distributions comes from these assumptions.

– Violate one of the three extra assumptions. – Violate one of the three extra assumptions. – Violate one of the three extra assumptions.

The interaction with ST is a success while the
The interaction with ST is a success while the

interaction with S or the interaction with T is not a interaction with S or the interaction with T is not a interaction with S or the interaction with T is not a success, or vice versa. success, or vice versa.

Tim Muller and Patrick Schweitzer 14 Composite Trust Tim Muller and Patrick Schweitzer 14 Composite Trust

SLIDE 44

Implications of Theorem 5 Implications of Theorem 5 Implications of Theorem 5 Implications of Theorem 5

Models where composite trust opinions

Models where composite trust opinions
Models where composite trust opinions

correspond to beta distributions must either: correspond to beta distributions must either: correspond to beta distributions must either:

– Violate one of the assumptions of the Beta Model. – Violate one of the assumptions of the Beta Model. – Violate one of the assumptions of the Beta Model.

Their support for the representing simple trust opinions
Their support for the representing simple trust opinions

as beta distributions comes from these assumptions. as beta distributions comes from these assumptions.

– Violate one of the three extra assumptions. – Violate one of the three extra assumptions. – Violate one of the three extra assumptions.

The interaction with ST is a success while neither the
The interaction with ST is a success while neither the

interaction with S nor the interaction with T is a interaction with S nor the interaction with T is a interaction with S nor the interaction with T is a success, or vice versa. success, or vice versa.

Tim Muller and Patrick Schweitzer 14 Composite Trust Tim Muller and Patrick Schweitzer 14 Composite Trust

SLIDE 45

Subjective Logic Subjective Logic Subjective Logic Subjective Logic

BC as derived above. BC in Subjective Logic. BC as derived above. BC in Subjective Logic.

Conjunction of B and C in Subjective logic
Conjunction of B and C in Subjective logic
Conjunction of B and C in Subjective logic

gives a completely different result. gives a completely different result.

– SL is far flatter; representing more uncertainty. – SL is far flatter; representing more uncertainty. – SL has expected value above 0.5, should be below. – SL has expected value above 0.5, should be below.

Tim Muller and Patrick Schweitzer 15 Composite Trust Tim Muller and Patrick Schweitzer 15 Composite Trust

SLIDE 46

Improving Subjective Logic Improving Subjective Logic Improving Subjective Logic Improving Subjective Logic

Mapping between trust opinions and beta

Mapping between trust opinions and beta
Mapping between trust opinions and beta

distributions is the core idea. distributions is the core idea. distributions is the core idea.

– Models like SL are too effective to disregard. – Models like SL are too effective to disregard. – Models like SL are too effective to disregard.

Difference between correct distribution and
Difference between correct distribution and
Difference between correct distribution and

beta distribution cannot (generally) be 0. beta distribution cannot (generally) be 0. beta distribution cannot (generally) be 0.

– There are beta distributions with the same – There are beta distributions with the same – There are beta distributions with the same expected value and variance as the correct one. expected value and variance as the correct one.

Tim Muller and Patrick Schweitzer 16 Composite Trust Tim Muller and Patrick Schweitzer 16 Composite Trust

SLIDE 47

Conclusion Conclusion Conclusion Conclusion

Representing trust opinions as distributions over
Representing trust opinions as distributions over
Representing trust opinions as distributions over

unknown integrity parameters is useful. unknown integrity parameters is useful.

By adding some assumptions to the Beta Model,
By adding some assumptions to the Beta Model,

composite trust opinions can be formally derived. composite trust opinions can be formally derived. composite trust opinions can be formally derived.

– The subject can derive a trust opinion about any well- – The subject can derive a trust opinion about any well- formed target. formed target. formed target. – Our extra assumptions are reasonable. – Our extra assumptions are reasonable.

Not all trust opinions can be represented by beta

Not all trust opinions can be represented by beta
Not all trust opinions can be represented by beta

distributions. distributions. distributions.

– Popular models assume they can. – Popular models assume they can.

Tim Muller and Patrick Schweitzer 17 Composite Trust Tim Muller and Patrick Schweitzer 17 Composite Trust

SLIDE 48

Questions Questions Questions Questions

Tim Muller and Patrick Schweitzer 24 Composite Trust Tim Muller and Patrick Schweitzer 24 Composite Trust