Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Imprecise Probabilistic Graphical ModelsCredal Networks and Other Guys

Alessandro Antonucci & Cassio de Campos & Giorgio Corani

{alessandro,cassio,giorgio}@idsia.ch

SISPTA School on Imprecise Probability

Durham, September 5, 2010

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Imprecise Probability Group @ IDSIA

IDSIA = Dalle Molle Institute for AI

Imprecise Probability Group(1 professor, 4 researchers, 1 phd)

Theory of imprecise probability

Probabilistic graphical models

Data mining and classification

Observations modelling (missing

data)

Data fusion and filtering

Applications to environmental

modelling, military decision making,

risk analysis, bioinformatics, biology,

tracking, vision, . . .

IDSIA

1990

2000

Marco

Alex

Cassio

Giorgio

Alessio

Denis

Zaffalon

Antonucc

i

de Camp

os

Corani

Benavoli

Mauà

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Imprecise Probability Group @ IDSIA

IDSIA = Dalle Molle Institute for AI

Imprecise Probability Group(1 professor, 4 researchers, 1 phd)

Theory of imprecise probability

Probabilistic graphical models

Data mining and classification

Observations modelling (missing

data)

Data fusion and filtering

Applications to environmental

modelling, military decision making,

risk analysis, bioinformatics, biology,

tracking, vision, . . .

IDSIA

1990

2000

Marco

Alex

Cassio

Giorgio

Alessio

Denis

Zaffalon

Antonucc

i

de Camp

os

Corani

Benavoli

Mauà

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Imprecise Probability Group @ IDSIA

IDSIA = Dalle Molle Institute for AI

Imprecise Probability Group(1 professor, 4 researchers, 1 phd)

Theory of imprecise probability

Probabilistic graphical models

Data mining and classification

Observations modelling (missing

data)

Data fusion and filtering

Applications to environmental

modelling, military decision making,

risk analysis, bioinformatics, biology,

tracking, vision, . . .

IDSIA

1990

2000

Marco

Alex

Cassio

Giorgio

Alessio

Denis

Zaffalon

Antonucc

i

de Camp

os

Corani

Benavoli

Mauà

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Imprecise Probability Group @ IDSIA

IDSIA = Dalle Molle Institute for AI

Imprecise Probability Group(1 professor, 4 researchers, 1 phd)

Theory of imprecise probability

Probabilistic graphical models

Data mining and classification

Observations modelling (missing

data)

Data fusion and filtering

Applications to environmental

modelling, military decision making,

risk analysis, bioinformatics, biology,

tracking, vision, . . .

IDSIA

1990

2000

Marco

Alex

Cassio

Giorgio

Alessio

Denis

Zaffalon

Antonucc

i

de Camp

os

Corani

Benavoli

Mauà

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Imprecise Probability Group @ IDSIA

IDSIA = Dalle Molle Institute for AI

Imprecise Probability Group(1 professor, 4 researchers, 1 phd)

Theory of imprecise probability

Probabilistic graphical models

Data mining and classification

Observations modelling (missing

data)

Data fusion and filtering

Applications to environmental

modelling, military decision making,

risk analysis, bioinformatics, biology,

tracking, vision, . . .

IDSIA

1990

2000

Marco

Alex

Cassio

Giorgio

Alessio

Denis

Zaffalon

Antonucc

i

de Camp

os

Corani

Benavoli

Mauà

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Imprecise Probability Group @ IDSIA

IDSIA = Dalle Molle Institute for AI

Imprecise Probability Group(1 professor, 4 researchers, 1 phd)

Theory of imprecise probability

Probabilistic graphical models

Data mining and classification

Observations modelling (missing

data)

Data fusion and filtering

Applications to environmental

modelling, military decision making,

risk analysis, bioinformatics, biology,

tracking, vision, . . .

IDSIA

1990

2000

Marco

Alex

Cassio

Giorgio

Alessio

Denis

Zaffalon

Antonucc

i

de Camp

os

Corani

Benavoli

Mauà

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Probabilistic Graphical Modelsaka Decomposable Multivariate Probabilistic Models

(whose decomposability is induced by independence )

X1 X2 X3

X4 X5

X6 X7 X8

global modelφ(X1,X2,X3,X4,X5,X6,X7,X8)local model

φ(X1,X2,X4)local modelφ(X2,X3,X5)

local modelφ(X4,X6,X7)

local modelφ(X5,X7,X8)

φ(X1, X2, X3, X4, X5, X6, X7, X8) = φ(X1, X2, X4) ⊗ φ(X2, X3, X5) ⊗ φ(X4, X6, X7) ⊗ φ(X5, X7, X8)

undirected graphs

precise/imprecise Markov random fields

directed graphs

Bayesian/credal networks

mixed graphs

chain graphs

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Probabilistic Graphical Modelsaka Decomposable Multivariate Probabilistic Models

(whose decomposability is induced by independence )

X1 X2 X3

X4 X5

X6 X7 X8

global modelφ(X1,X2,X3,X4,X5,X6,X7,X8)local model

φ(X1,X2,X4)local modelφ(X2,X3,X5)

local modelφ(X4,X6,X7)

local modelφ(X5,X7,X8)

φ(X1, X2, X3, X4, X5, X6, X7, X8) = φ(X1, X2, X4) ⊗ φ(X2, X3, X5) ⊗ φ(X4, X6, X7) ⊗ φ(X5, X7, X8)

undirected graphs

precise/imprecise Markov random fields

directed graphs

Bayesian/credal networks

mixed graphs

chain graphs

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Probabilistic Graphical Modelsaka Decomposable Multivariate Probabilistic Models

(whose decomposability is induced by independence )

X1 X2 X3

X4 X5

X6 X7 X8

global modelφ(X1,X2,X3,X4,X5,X6,X7,X8)

local modelφ(X1,X2,X4)

local modelφ(X2,X3,X5)

local modelφ(X4,X6,X7)

local modelφ(X5,X7,X8)

φ(X1, X2, X3, X4, X5, X6, X7, X8) = φ(X1, X2, X4) ⊗ φ(X2, X3, X5) ⊗ φ(X4, X6, X7) ⊗ φ(X5, X7, X8)

undirected graphs

precise/imprecise Markov random fields

directed graphs

Bayesian/credal networks

mixed graphs

chain graphs

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Probabilistic Graphical Modelsaka Decomposable Multivariate Probabilistic Models

(whose decomposability is induced by independence )

X1 X2 X3

X4 X5

X6 X7 X8

global modelφ(X1,X2,X3,X4,X5,X6,X7,X8)

local modelφ(X1,X2,X4)

local modelφ(X2,X3,X5)

local modelφ(X4,X6,X7)

local modelφ(X5,X7,X8)

φ(X1, X2, X3, X4, X5, X6, X7, X8) = φ(X1, X2, X4) ⊗ φ(X2, X3, X5) ⊗ φ(X4, X6, X7) ⊗ φ(X5, X7, X8)

undirected graphs

precise/imprecise Markov random fields

directed graphs

Bayesian/credal networks

mixed graphs

chain graphs

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Probabilistic Graphical Modelsaka Decomposable Multivariate Probabilistic Models

(whose decomposability is induced by independence )

X1 X2 X3

X4 X5

X6 X7 X8

global modelφ(X1,X2,X3,X4,X5,X6,X7,X8)local model

φ(X1,X2,X4)local modelφ(X2,X3,X5)

local modelφ(X4,X6,X7)

local modelφ(X5,X7,X8)

φ(X1, X2, X3, X4, X5, X6, X7, X8) = φ(X1, X2, X4) ⊗ φ(X2, X3, X5) ⊗ φ(X4, X6, X7) ⊗ φ(X5, X7, X8)

undirected graphs

precise/imprecise Markov random fields

directed graphs

Bayesian/credal networks

mixed graphs

chain graphs

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Probabilistic Graphical Modelsaka Decomposable Multivariate Probabilistic Models

(whose decomposability is induced by independence )

X1 X2 X3

X4 X5

X6 X7 X8

global modelφ(X1,X2,X3,X4,X5,X6,X7,X8)local model

φ(X1,X2,X4)local modelφ(X2,X3,X5)

local modelφ(X4,X6,X7)

local modelφ(X5,X7,X8)

φ(X1, X2, X3, X4, X5, X6, X7, X8) = φ(X1, X2, X4) ⊗ φ(X2, X3, X5) ⊗ φ(X4, X6, X7) ⊗ φ(X5, X7, X8)

undirected graphs

precise/imprecise Markov random fields

directed graphs

Bayesian/credal networks

mixed graphs

chain graphs

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Probabilistic Graphical Modelsaka Decomposable Multivariate Probabilistic Models

(whose decomposability is induced by independence )

X1 X2 X3

X4 X5

X6 X7 X8

global modelφ(X1,X2,X3,X4,X5,X6,X7,X8)local model

φ(X1,X2,X4)local modelφ(X2,X3,X5)

local modelφ(X4,X6,X7)

local modelφ(X5,X7,X8)

φ(X1, X2, X3, X4, X5, X6, X7, X8) = φ(X1, X2, X4) ⊗ φ(X2, X3, X5) ⊗ φ(X4, X6, X7) ⊗ φ(X5, X7, X8)

undirected graphs

precise/imprecise Markov random fields

directed graphs

Bayesian/credal networks

mixed graphs

chain graphs

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

[Exe #1] Fault trees (Vesely et al, 1981)

brake fails = [ pads ∨ ( sensor ∧ controller ∧ actuator ) ]

devices failures are independent

padsfails

ORgate

ANDgate

sensorfails

controllerfails

actuatorfails

brakefails

111

0

?

1

1

1

1.8.8

.2

?

.64

= .2× .64 + .8× .64 + .2× .36 =.712

.712

1[.8,1][.8,1]

[0,.2]

?

[.64,1]

[.64,1]

[.64,1]

with [.7, 1] insteadP(brake fails)∈[.49,1]

Indecision!

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

[Exe #1] Fault trees (Vesely et al, 1981)

brake fails = [ pads ∨ ( sensor ∧ controller ∧ actuator ) ]

devices failures are independent

padsfails

ORgate

ANDgate

sensorfails

controllerfails

actuatorfails

brakefails

111

0

?

1

1

1

1.8.8

.2

?

.64

= .2× .64 + .8× .64 + .2× .36 =.712

.712

1[.8,1][.8,1]

[0,.2]

?

[.64,1]

[.64,1]

[.64,1]

with [.7, 1] insteadP(brake fails)∈[.49,1]

Indecision!

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

[Exe #1] Fault trees (Vesely et al, 1981)

brake fails = [ pads ∨ ( sensor ∧ controller ∧ actuator ) ]

devices failures are independent

padsfails

ORgate

ANDgate

sensorfails

controllerfails

actuatorfails

brakefails

111

0

?

1

1

1

1.8.8

.2

?

.64

= .2× .64 + .8× .64 + .2× .36 =.712

.712

1[.8,1][.8,1]

[0,.2]

?

[.64,1]

[.64,1]

[.64,1]

with [.7, 1] insteadP(brake fails)∈[.49,1]

Indecision!

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

[Exe #1] Fault trees (Vesely et al, 1981)

brake fails = [ pads ∨ ( sensor ∧ controller ∧ actuator ) ]

devices failures are independent

padsfails

ORgate

ANDgate

sensorfails

controllerfails

actuatorfails

brakefails

111

0

?

1

1

1

1.8.8

.2

?

.64

= .2× .64 + .8× .64 + .2× .36 =.712

.712

1[.8,1][.8,1]

[0,.2]

?

[.64,1]

[.64,1]

[.64,1]

with [.7, 1] insteadP(brake fails)∈[.49,1]

Indecision!

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

[Exe #1] Fault trees (Vesely et al, 1981)

brake fails = [ pads ∨ ( sensor ∧ controller ∧ actuator ) ]

devices failures are independent

padsfails

ORgate

ANDgate

sensorfails

controllerfails

actuatorfails

brakefails

111

0

?

1

1

1

1.8.8

.2

?

.64

= .2× .64 + .8× .64 + .2× .36 =.712

.712

1[.8,1][.8,1]

[0,.2]

?

[.64,1]

[.64,1]

[.64,1]

with [.7, 1] insteadP(brake fails)∈[.49,1]

Indecision!

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

[Exe #1] Fault trees (Vesely et al, 1981)

brake fails = [ pads ∨ ( sensor ∧ controller ∧ actuator ) ]

devices failures are independent

padsfails

ORgate

ANDgate

sensorfails

controllerfails

actuatorfails

brakefails

111

0

?

1

1

1

1.8.8

.2

?

.64

= .2× .64 + .8× .64 + .2× .36 =.712

.712

1[.8,1][.8,1]

[0,.2]

?

[.64,1]

[.64,1]

[.64,1]

with [.7, 1] insteadP(brake fails)∈[.49,1]

Indecision!

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

[Exe #1] Fault trees (Vesely et al, 1981)

brake fails = [ pads ∨ ( sensor ∧ controller ∧ actuator ) ]

devices failures are independent

padsfails

ORgate

ANDgate

sensorfails

controllerfails

actuatorfails

brakefails

111

0

?

1

1

1

1.8.8

.2

?

.64

= .2× .64 + .8× .64 + .2× .36 =.712

.712

1[.8,1][.8,1]

[0,.2]

?

[.64,1]

[.64,1]

[.64,1]

with [.7, 1] insteadP(brake fails)∈[.49,1]

Indecision!

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

[Exe #1] Fault trees (Vesely et al, 1981)

brake fails = [ pads ∨ ( sensor ∧ controller ∧ actuator ) ]

devices failures are independent

padsfails

ORgate

ANDgate

sensorfails

controllerfails

actuatorfails

brakefails

111

0

?

1

1

1

1.8.8

.2

?

.64

= .2× .64 + .8× .64 + .2× .36 =.712

.712

1[.8,1][.8,1]

[0,.2]

?

[.64,1]

[.64,1]

[.64,1]

with [.7, 1] insteadP(brake fails)∈[.49,1]

Indecision!

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

[Exe #1] Fault trees (Vesely et al, 1981)

brake fails = [ pads ∨ ( sensor ∧ controller ∧ actuator ) ]

devices failures are independent

padsfails

ORgate

ANDgate

sensorfails

controllerfails

actuatorfails

brakefails

111

0

?

1

1

1

1.8.8

.2

?

.64

= .2× .64 + .8× .64 + .2× .36 =.712

.712

1[.8,1][.8,1]

[0,.2]

?

[.64,1]

[.64,1]

[.64,1]

with [.7, 1] insteadP(brake fails)∈[.49,1]

Indecision!

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

[Exe #1] Fault trees (Vesely et al, 1981)

brake fails = [ pads ∨ ( sensor ∧ controller ∧ actuator ) ]

devices failures are independent

padsfails

ORgate

ANDgate

sensorfails

controllerfails

actuatorfails

brakefails

111

0

?

1

1

1

1.8.8

.2

?

.64

= .2× .64 + .8× .64 + .2× .36 =.712

.712

1[.8,1][.8,1]

[0,.2]

?

[.64,1]

[.64,1]

[.64,1]

with [.7, 1] insteadP(brake fails)∈[.49,1]

Indecision!

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

[Exe #1] Fault trees (Vesely et al, 1981)

brake fails = [ pads ∨ ( sensor ∧ controller ∧ actuator ) ]

devices failures are independent

padsfails

ORgate

ANDgate

sensorfails

controllerfails

actuatorfails

brakefails

111

0

?

1

1

1

1.8.8

.2

?

.64

= .2× .64 + .8× .64 + .2× .36 =.712

.712

1[.8,1][.8,1]

[0,.2]

?

[.64,1]

[.64,1]

[.64,1]

with [.7, 1] insteadP(brake fails)∈[.49,1]

Indecision!

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

[Exe #1] Fault trees (Vesely et al, 1981)

brake fails = [ pads ∨ ( sensor ∧ controller ∧ actuator ) ]

devices failures are independent

padsfails

ORgate

ANDgate

sensorfails

controllerfails

actuatorfails

brakefails

111

0

?

1

1

1

1.8.8

.2

?

.64

= .2× .64 + .8× .64 + .2× .36 =.712

.712

1[.8,1][.8,1]

[0,.2]

?

[.64,1]

[.64,1]

[.64,1]

with [.7, 1] insteadP(brake fails)∈[.49,1]

Indecision!

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

[Exe #1] Fault trees (Vesely et al, 1981)

brake fails = [ pads ∨ ( sensor ∧ controller ∧ actuator ) ]

devices failures are independent

padsfails

ORgate

ANDgate

sensorfails

controllerfails

actuatorfails

brakefails

111

0

?

1

1

1

1.8.8

.2

?

.64

= .2× .64 + .8× .64 + .2× .36 =.712

.712

1[.8,1][.8,1]

[0,.2]

?

[.64,1]

[.64,1]

[.64,1]

with [.7, 1] insteadP(brake fails)∈[.49,1]

Indecision!

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

[Exe #1] Fault trees (Vesely et al, 1981)

brake fails = [ pads ∨ ( sensor ∧ controller ∧ actuator ) ]

devices failures are independent

padsfails

ORgate

ANDgate

sensorfails

controllerfails

actuatorfails

brakefails

111

0

?

1

1

1

1.8.8

.2

?

.64

= .2× .64 + .8× .64 + .2× .36 =.712

.712

1[.8,1][.8,1]

[0,.2]

?

[.64,1]

[.64,1]

[.64,1]

with [.7, 1] insteadP(brake fails)∈[.49,1]

Indecision!

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

[Exe #1] Fault trees (Vesely et al, 1981)

brake fails = [ pads ∨ ( sensor ∧ controller ∧ actuator ) ]

devices failures are independent

padsfails

ORgate

ANDgate

sensorfails

controllerfails

actuatorfails

brakefails

111

0

?

1

1

1

1.8.8

.2

?

.64

= .2× .64 + .8× .64 + .2× .36 =.712

.712

1[.8,1][.8,1]

[0,.2]

?

[.64,1]

[.64,1]

[.64,1]

with [.7, 1] insteadP(brake fails)∈[.49,1]

Indecision!

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Outline

Motivations for imprecise probability

Credal sets (basic concepts and operations)

Independence relations

Credal networks

Modelling observations/missingness

Decision making

Inference algorithms

Other probabilistic graphical models

Conclusions

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Three different levels of knowledge

FIFA’10 final match between Holland and SpainResult of Holland after the regular time? Win, draw or loss?

DETERMINISM

The Dutch goalkeeper isunbeatable and Holland

always makes a goal

Holland (certainly) wins

P(Win)P(Draw)P(Loss)

=

100

UNCERTAINTY

Win is two times moreprobable than draw, and

this being three timesmore probable than loss

P(Win)P(Draw)P(Loss)

=

.6.3.1

IMPRECISION

Win is more probablethan draw, and this is

more probable than loss

P(Win) > P(Draw)P(Draw) > P(Loss)

P(Win)P(Draw)P(Loss)

=

[ α3 + β +

γ2

α3 +

γ2

α3

]∀α, β, γ such thatα > 0, β > 0, γ > 0,α+ β + γ = 1

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Three different levels of knowledge

FIFA’10 final match between Holland and SpainResult of Holland after the regular time? Win, draw or loss?

DETERMINISM

The Dutch goalkeeper isunbeatable and Holland

always makes a goal

Holland (certainly) wins

P(Win)P(Draw)P(Loss)

=

100

UNCERTAINTY

Win is two times moreprobable than draw, and

this being three timesmore probable than loss

P(Win)P(Draw)P(Loss)

=

.6.3.1

IMPRECISION

Win is more probablethan draw, and this is

more probable than loss

P(Win) > P(Draw)P(Draw) > P(Loss)

P(Win)P(Draw)P(Loss)

=

[ α3 + β +

γ2

α3 +

γ2

α3

]∀α, β, γ such thatα > 0, β > 0, γ > 0,α+ β + γ = 1

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Three different levels of knowledge

FIFA’10 final match between Holland and SpainResult of Holland after the regular time? Win, draw or loss?

DETERMINISM

The Dutch goalkeeper isunbeatable and Holland

always makes a goal

Holland (certainly) wins

P(Win)P(Draw)P(Loss)

=

100

UNCERTAINTY

Win is two times moreprobable than draw, and

this being three timesmore probable than loss

P(Win)P(Draw)P(Loss)

=

.6.3.1

IMPRECISION

Win is more probablethan draw, and this is

more probable than loss

P(Win) > P(Draw)P(Draw) > P(Loss)

P(Win)P(Draw)P(Loss)

=

[ α3 + β +

γ2

α3 +

γ2

α3

]∀α, β, γ such thatα > 0, β > 0, γ > 0,α+ β + γ = 1

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Three different levels of knowledge

FIFA’10 final match between Holland and SpainResult of Holland after the regular time? Win, draw or loss?

DETERMINISM

The Dutch goalkeeper isunbeatable and Holland

always makes a goal

Holland (certainly) wins

P(Win)P(Draw)P(Loss)

=

100

UNCERTAINTY

Win is two times moreprobable than draw, and

this being three timesmore probable than loss

P(Win)P(Draw)P(Loss)

=

.6.3.1

IMPRECISION

Win is more probablethan draw, and this is

more probable than loss

P(Win) > P(Draw)P(Draw) > P(Loss)

P(Win)P(Draw)P(Loss)

=

[ α3 + β +

γ2

α3 +

γ2

α3

]∀α, β, γ such thatα > 0, β > 0, γ > 0,α+ β + γ = 1

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Three different levels of knowledge (ii)

DETERMINISM UNCERTAINTY IMPRECISION

INFORMATIVENESS

EXPRESSIVENESS

Limit when learning fromlarge (complete) data sets

Propositional(Boolean) Logic

Bayesian Theoryof Probability

Walley’s Theory ofCoherent LowerPrevisions

Natural Embedding (de Cooman)

Only special cases

Small/incomplete dataExpert’s (qualitative) knowledgeVague/incomplete observations

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Three different levels of knowledge (ii)

DETERMINISM UNCERTAINTY IMPRECISION

INFORMATIVENESS

EXPRESSIVENESS

Limit when learning fromlarge (complete) data sets

Propositional(Boolean) Logic

Bayesian Theoryof Probability

Walley’s Theory ofCoherent LowerPrevisions

Natural Embedding (de Cooman)

Only special cases

Small/incomplete dataExpert’s (qualitative) knowledgeVague/incomplete observations

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Three different levels of knowledge (ii)

DETERMINISM UNCERTAINTY IMPRECISION

INFORMATIVENESS

EXPRESSIVENESS

Limit when learning fromlarge (complete) data sets

Propositional(Boolean) Logic

Bayesian Theoryof Probability

Walley’s Theory ofCoherent LowerPrevisions

Natural Embedding (de Cooman)

Only special cases

Small/incomplete dataExpert’s (qualitative) knowledgeVague/incomplete observations

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Three different levels of knowledge (ii)

DETERMINISM UNCERTAINTY IMPRECISION

INFORMATIVENESS

EXPRESSIVENESS

Limit when learning fromlarge (complete) data sets

Propositional(Boolean) Logic

Bayesian Theoryof Probability

Walley’s Theory ofCoherent LowerPrevisions

Natural Embedding (de Cooman)

Only special cases

Small/incomplete dataExpert’s (qualitative) knowledgeVague/incomplete observations

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Three different levels of knowledge (ii)

DETERMINISM UNCERTAINTY IMPRECISION

INFORMATIVENESS

EXPRESSIVENESS

Limit when learning fromlarge (complete) data sets

Propositional(Boolean) Logic

Bayesian Theoryof Probability

Walley’s Theory ofCoherent LowerPrevisions

Natural Embedding (de Cooman)

Only special cases

Small/incomplete dataExpert’s (qualitative) knowledgeVague/incomplete observations

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Three different levels of knowledge (ii)

DETERMINISM UNCERTAINTY IMPRECISION

INFORMATIVENESS

EXPRESSIVENESS

Limit when learning fromlarge (complete) data sets

Propositional(Boolean) Logic

Bayesian Theoryof Probability

Walley’s Theory ofCoherent LowerPrevisions

Natural Embedding (de Cooman)

Only special cases

Small/incomplete dataExpert’s (qualitative) knowledgeVague/incomplete observations

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Three different levels of knowledge (ii)

DETERMINISM UNCERTAINTY IMPRECISION

INFORMATIVENESS

EXPRESSIVENESS

Limit when learning fromlarge (complete) data sets

Propositional(Boolean) Logic

Bayesian Theoryof Probability

Walley’s Theory ofCoherent LowerPrevisions

Natural Embedding (de Cooman)

Only special cases

Small/incomplete dataExpert’s (qualitative) knowledgeVague/incomplete observations

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Three different levels of knowledge (ii)

DETERMINISM UNCERTAINTY IMPRECISION

INFORMATIVENESS

EXPRESSIVENESS

Limit when learning fromlarge (complete) data sets

Propositional(Boolean) Logic

Bayesian Theoryof Probability

Walley’s Theory ofCoherent LowerPrevisions

Natural Embedding (de Cooman)

Only special cases

Small/incomplete dataExpert’s (qualitative) knowledgeVague/incomplete observations

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Three different levels of knowledge (ii)

DETERMINISM UNCERTAINTY IMPRECISION

INFORMATIVENESS

EXPRESSIVENESS

Limit when learning fromlarge (complete) data sets

Propositional(Boolean) Logic

Bayesian Theoryof Probability

Walley’s Theory ofCoherent LowerPrevisions

Natural Embedding (de Cooman)

Only special cases

Small/incomplete dataExpert’s (qualitative) knowledgeVague/incomplete observations

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

From CLPs to Credal Sets

probability distributionp : X → R{

p(x) ≥ 0∀x ∈ X∑x∈X p(x) = 1

coherent linear previsionP : L(X )→ R{

P(f + g) = P(f ) + P(g)P(f ) ≥ inf f , ∀f , g ∈ L(X )

P(f ) =∑

x∈X p(x)f (x)

P(x) = P(I{x})

Bayesian / precise

Modelling knowledge about X , taking values in X

Credal / imprecise

set of distributionsK (X) = { p(X) | constraints }

coherent lower previsionP : L(X )→ R P(f ) ≥ min f ∀f ∈ L(X )P(λf ) = λP(f )∀λ > 0P(f + g) ≥ P(f ) + P(g)∃ set M of linear previsions s.t.

P(f ) = infP∈M P(f ) ∀f ∈ L(X )lower envelope Theorem

(a set and its convex closure have the same lower envelope!)

convex closed

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

From CLPs to Credal Sets

probability distributionp : X → R{

p(x) ≥ 0∀x ∈ X∑x∈X p(x) = 1

coherent linear previsionP : L(X )→ R{

P(f + g) = P(f ) + P(g)P(f ) ≥ inf f , ∀f , g ∈ L(X )

P(f ) =∑

x∈X p(x)f (x)

P(x) = P(I{x})

Bayesian / preciseModelling knowledge about X , taking values in X

Credal / imprecise

set of distributionsK (X) = { p(X) | constraints }

coherent lower previsionP : L(X )→ R P(f ) ≥ min f ∀f ∈ L(X )P(λf ) = λP(f )∀λ > 0P(f + g) ≥ P(f ) + P(g)∃ set M of linear previsions s.t.

P(f ) = infP∈M P(f ) ∀f ∈ L(X )lower envelope Theorem

(a set and its convex closure have the same lower envelope!)

convex closed

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

From CLPs to Credal Sets

probability distributionp : X → R{

p(x) ≥ 0∀x ∈ X∑x∈X p(x) = 1

coherent linear previsionP : L(X )→ R{

P(f + g) = P(f ) + P(g)P(f ) ≥ inf f , ∀f , g ∈ L(X )

P(f ) =∑

x∈X p(x)f (x)

P(x) = P(I{x})

Bayesian / preciseModelling knowledge about X , taking values in X

Credal / imprecise

set of distributionsK (X) = { p(X) | constraints }

coherent lower previsionP : L(X )→ R P(f ) ≥ min f ∀f ∈ L(X )P(λf ) = λP(f )∀λ > 0P(f + g) ≥ P(f ) + P(g)∃ set M of linear previsions s.t.

P(f ) = infP∈M P(f ) ∀f ∈ L(X )lower envelope Theorem

(a set and its convex closure have the same lower envelope!)

convex closed

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

From CLPs to Credal Sets

probability distributionp : X → R{

p(x) ≥ 0∀x ∈ X∑x∈X p(x) = 1

coherent linear previsionP : L(X )→ R{

P(f + g) = P(f ) + P(g)P(f ) ≥ inf f , ∀f , g ∈ L(X )

P(f ) =∑

x∈X p(x)f (x)

P(x) = P(I{x})

Bayesian / preciseModelling knowledge about X , taking values in X

Credal / imprecise

set of distributionsK (X) = { p(X) | constraints }

coherent lower previsionP : L(X )→ R P(f ) ≥ min f ∀f ∈ L(X )P(λf ) = λP(f )∀λ > 0P(f + g) ≥ P(f ) + P(g)

∃ set M of linear previsions s.t.P(f ) = infP∈M P(f ) ∀f ∈ L(X )

lower envelope Theorem(a set and its convex closure have the same lower envelope!)

convex closed

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

From CLPs to Credal Sets

probability distributionp : X → R{

p(x) ≥ 0∀x ∈ X∑x∈X p(x) = 1

coherent linear previsionP : L(X )→ R{

P(f + g) = P(f ) + P(g)P(f ) ≥ inf f , ∀f , g ∈ L(X )

P(f ) =∑

x∈X p(x)f (x)

P(x) = P(I{x})

Bayesian / preciseModelling knowledge about X , taking values in X

Credal / imprecise

set of distributionsK (X) = { p(X) | constraints }

coherent lower previsionP : L(X )→ R P(f ) ≥ min f ∀f ∈ L(X )P(λf ) = λP(f )∀λ > 0P(f + g) ≥ P(f ) + P(g)∃ set M of linear previsions s.t.

P(f ) = infP∈M P(f ) ∀f ∈ L(X )

lower envelope Theorem(a set and its convex closure have the same lower envelope!)

convex closed

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

From CLPs to Credal Sets

probability distributionp : X → R{

p(x) ≥ 0∀x ∈ X∑x∈X p(x) = 1

coherent linear previsionP : L(X )→ R{

P(f + g) = P(f ) + P(g)P(f ) ≥ inf f , ∀f , g ∈ L(X )

P(f ) =∑

x∈X p(x)f (x)

P(x) = P(I{x})

Bayesian / preciseModelling knowledge about X , taking values in X

Credal / imprecise

set of distributionsK (X) = { p(X) | constraints }

coherent lower previsionP : L(X )→ R P(f ) ≥ min f ∀f ∈ L(X )P(λf ) = λP(f )∀λ > 0P(f + g) ≥ P(f ) + P(g)∃ set M of linear previsions s.t.

P(f ) = infP∈M P(f ) ∀f ∈ L(X )lower envelope Theorem

(a set and its convex closure have the same lower envelope!)

convex closed

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

From CLPs to Credal Sets

probability distributionp : X → R{

p(x) ≥ 0∀x ∈ X∑x∈X p(x) = 1

coherent linear previsionP : L(X )→ R{

P(f + g) = P(f ) + P(g)P(f ) ≥ inf f , ∀f , g ∈ L(X )

P(f ) =∑

x∈X p(x)f (x)

P(x) = P(I{x})

Bayesian / preciseModelling knowledge about X , taking values in X

Credal / imprecise

set of distributionsK (X) = { p(X) | constraints }

coherent lower previsionP : L(X )→ R P(f ) ≥ min f ∀f ∈ L(X )P(λf ) = λP(f )∀λ > 0P(f + g) ≥ P(f ) + P(g)∃ set M of linear previsions s.t.

P(f ) = infP∈M P(f ) ∀f ∈ L(X )lower envelope Theorem

(a set and its convex closure have the same lower envelope!)

convex closed

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Credal sets (Levi, 1980)

A closed convex set K (X ) of probability mass functions

Equivalently described by its extreme points ext[K (X )]

Focus on categorical variables (|X | < +∞)and finitely generated CSs (polytopes, |ext[K (X )]| < +∞)

V-representation

enumerate the

extreme points

H-representation

linear constraints

on the probabilities

LRS software

(Avis & Fukuda)

Given a function (gamble) f (X ), lower expectation:

E [f (X )] := minP(X)∈K (X)∑

x∈X P(x)f (x)

LP task: the optimum is on an extreme!

E [f (X )] = minP(X)∈ext[K (X)]∑

x∈X P(x)f (x)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Credal sets (Levi, 1980)

A closed convex set K (X ) of probability mass functions

Equivalently described by its extreme points ext[K (X )]

Focus on categorical variables (|X | < +∞)and finitely generated CSs (polytopes, |ext[K (X )]| < +∞)

V-representation

enumerate the

extreme points

H-representation

linear constraints

on the probabilities

LRS software

(Avis & Fukuda)

Given a function (gamble) f (X ), lower expectation:

E [f (X )] := minP(X)∈K (X)∑

x∈X P(x)f (x)

LP task: the optimum is on an extreme!

E [f (X )] = minP(X)∈ext[K (X)]∑

x∈X P(x)f (x)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Credal sets (Levi, 1980)

A closed convex set K (X ) of probability mass functions

Equivalently described by its extreme points ext[K (X )]

Focus on categorical variables (|X | < +∞)and finitely generated CSs (polytopes, |ext[K (X )]| < +∞)

V-representation

enumerate the

extreme points

H-representation

linear constraints

on the probabilities

LRS software

(Avis & Fukuda)

Given a function (gamble) f (X ), lower expectation:

E [f (X )] := minP(X)∈K (X)∑

x∈X P(x)f (x)

LP task: the optimum is on an extreme!

E [f (X )] = minP(X)∈ext[K (X)]∑

x∈X P(x)f (x)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Credal sets (Levi, 1980)

A closed convex set K (X ) of probability mass functions

Equivalently described by its extreme points ext[K (X )]

Focus on categorical variables (|X | < +∞)and finitely generated CSs (polytopes, |ext[K (X )]| < +∞)

V-representation

enumerate the

extreme points

H-representation

linear constraints

on the probabilities

LRS software

(Avis & Fukuda)

Given a function (gamble) f (X ), lower expectation:

E [f (X )] := minP(X)∈K (X)∑

x∈X P(x)f (x)

LP task: the optimum is on an extreme!

E [f (X )] = minP(X)∈ext[K (X)]∑

x∈X P(x)f (x)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Credal sets (Levi, 1980)

A closed convex set K (X ) of probability mass functions

Equivalently described by its extreme points ext[K (X )]

Focus on categorical variables (|X | < +∞)and finitely generated CSs (polytopes, |ext[K (X )]| < +∞)

V-representation

enumerate the

extreme points

H-representation

linear constraints

on the probabilities

LRS software

(Avis & Fukuda)

Given a function (gamble) f (X ), lower expectation:

E [f (X )] := minP(X)∈K (X)∑

x∈X P(x)f (x)

LP task: the optimum is on an extreme!

E [f (X )] = minP(X)∈ext[K (X)]∑

x∈X P(x)f (x)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Credal sets (Levi, 1980)

A closed convex set K (X ) of probability mass functions

Equivalently described by its extreme points ext[K (X )]

Focus on categorical variables (|X | < +∞)and finitely generated CSs (polytopes, |ext[K (X )]| < +∞)

V-representation

enumerate the

extreme points

H-representation

linear constraints

on the probabilities

LRS software

(Avis & Fukuda)

Given a function (gamble) f (X ), lower expectation:

E [f (X )] := minP(X)∈K (X)∑

x∈X P(x)f (x)

LP task: the optimum is on an extreme!

E [f (X )] = minP(X)∈ext[K (X)]∑

x∈X P(x)f (x)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Credal sets (Levi, 1980)

A closed convex set K (X ) of probability mass functions

Equivalently described by its extreme points ext[K (X )]

Focus on categorical variables (|X | < +∞)and finitely generated CSs (polytopes, |ext[K (X )]| < +∞)

V-representation

enumerate the

extreme points

H-representation

linear constraints

on the probabilities

LRS software

(Avis & Fukuda)

Given a function (gamble) f (X ), lower expectation:

E [f (X )] := minP(X)∈K (X)∑

x∈X P(x)f (x)

LP task: the optimum is on an extreme!

E [f (X )] = minP(X)∈ext[K (X)]∑

x∈X P(x)f (x)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Credal Sets over Boolean Variables

Boolean X , values in X = {x ,¬x}Determinism ≡ degenerate mass fE.g., X = x ⇐⇒ P(X) =

[10

]Uncertainty ≡ prob mass functionP(X) =

[p

1− p

]with p ∈ [0, 1]

Imprecision credal seton the probability simplex

K (X) ≡{

P(X) =[

p1− p

] ∣∣∣.4 ≤ p ≤ .7}A CS over a Boolean variable cannothave more than two vertices!

ext[K (X)] ={[

.7

.3

],

[.4.6

]}

P(x)

P(¬x)

P(X ) =[

10

]P(X ) =[

.7

.3

]P(X ) =[

.4

.6

]

.4 .7

.6

.3

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Credal Sets over Boolean Variables

Boolean X , values in X = {x ,¬x}

Determinism ≡ degenerate mass fE.g., X = x ⇐⇒ P(X) =

[10

]Uncertainty ≡ prob mass functionP(X) =

[p

1− p

]with p ∈ [0, 1]

Imprecision credal seton the probability simplex

K (X) ≡{

P(X) =[

p1− p

] ∣∣∣.4 ≤ p ≤ .7}A CS over a Boolean variable cannothave more than two vertices!

ext[K (X)] ={[

.7

.3

],

[.4.6

]}

P(x)

P(¬x)

P(X ) =[

10

]P(X ) =[

.7

.3

]P(X ) =[

.4

.6

]

.4 .7

.6

.3

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Credal Sets over Boolean Variables

Boolean X , values in X = {x ,¬x}Determinism ≡ degenerate mass fE.g., X = x ⇐⇒ P(X) =

[10

]

Uncertainty ≡ prob mass functionP(X) =

[p

1− p

]with p ∈ [0, 1]

Imprecision credal seton the probability simplex

K (X) ≡{

P(X) =[

p1− p

] ∣∣∣.4 ≤ p ≤ .7}A CS over a Boolean variable cannothave more than two vertices!

ext[K (X)] ={[

.7

.3

],

[.4.6

]}

P(x)

P(¬x)

P(X ) =[

10

]P(X ) =[

.7

.3

]P(X ) =[

.4

.6

]

.4 .7

.6

.3

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Credal Sets over Boolean Variables

Boolean X , values in X = {x ,¬x}Determinism ≡ degenerate mass fE.g., X = x ⇐⇒ P(X) =

[10

]

Uncertainty ≡ prob mass functionP(X) =

[p

1− p

]with p ∈ [0, 1]

Imprecision credal seton the probability simplex

K (X) ≡{

P(X) =[

p1− p

] ∣∣∣.4 ≤ p ≤ .7}A CS over a Boolean variable cannothave more than two vertices!

ext[K (X)] ={[

.7

.3

],

[.4.6

]}

P(x)

P(¬x)

P(X ) =[

10

]

P(X ) =[

.7

.3

]P(X ) =[

.4

.6

]

.4 .7

.6

.3

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Credal Sets over Boolean Variables

Boolean X , values in X = {x ,¬x}Determinism ≡ degenerate mass fE.g., X = x ⇐⇒ P(X) =

[10

]Uncertainty ≡ prob mass functionP(X) =

[p

1− p

]with p ∈ [0, 1]

Imprecision credal seton the probability simplex

K (X) ≡{

P(X) =[

p1− p

] ∣∣∣.4 ≤ p ≤ .7}A CS over a Boolean variable cannothave more than two vertices!

ext[K (X)] ={[

.7

.3

],

[.4.6

]}

P(x)

P(¬x)

P(X ) =[

10

]P(X ) =[

.7

.3

]P(X ) =[

.4

.6

]

.4 .7

.6

.3

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Credal Sets over Boolean Variables

Boolean X , values in X = {x ,¬x}Determinism ≡ degenerate mass fE.g., X = x ⇐⇒ P(X) =

[10

]Uncertainty ≡ prob mass functionP(X) =

[p

1− p

]with p ∈ [0, 1]

Imprecision credal seton the probability simplex

K (X) ≡{

P(X) =[

p1− p

] ∣∣∣.4 ≤ p ≤ .7}A CS over a Boolean variable cannothave more than two vertices!

ext[K (X)] ={[

.7

.3

],

[.4.6

]}

P(x)

P(¬x)

P(X ) =[

10

]

P(X ) =[

.7

.3

]

P(X ) =[

.4

.6

]

.4 .7

.6

.3

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Credal Sets over Boolean Variables

Boolean X , values in X = {x ,¬x}Determinism ≡ degenerate mass fE.g., X = x ⇐⇒ P(X) =

[10

]Uncertainty ≡ prob mass functionP(X) =

[p

1− p

]with p ∈ [0, 1]

Imprecision credal seton the probability simplex

K (X) ≡{

P(X) =[

p1− p

] ∣∣∣.4 ≤ p ≤ .7}A CS over a Boolean variable cannothave more than two vertices!

ext[K (X)] ={[

.7

.3

],

[.4.6

]}

P(x)

P(¬x)

P(X ) =[

10

]P(X ) =[

.7

.3

]

P(X ) =[

.4

.6

]

.4 .7

.6

.3

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Credal Sets over Boolean Variables

Boolean X , values in X = {x ,¬x}Determinism ≡ degenerate mass fE.g., X = x ⇐⇒ P(X) =

[10

]Uncertainty ≡ prob mass functionP(X) =

[p

1− p

]with p ∈ [0, 1]

Imprecision credal seton the probability simplex

K (X) ≡{

P(X) =[

p1− p

] ∣∣∣.4 ≤ p ≤ .7}A CS over a Boolean variable cannothave more than two vertices!

ext[K (X)] ={[

.7

.3

],

[.4.6

]}

P(x)

P(¬x)

P(X ) =[

10

]P(X ) =[

.7

.3

]P(X ) =[

.4

.6

]

.4 .7

.6

.3

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Credal Sets over Boolean Variables

Boolean X , values in X = {x ,¬x}Determinism ≡ degenerate mass fE.g., X = x ⇐⇒ P(X) =

[10

]Uncertainty ≡ prob mass functionP(X) =

[p

1− p

]with p ∈ [0, 1]

Imprecision credal seton the probability simplex

K (X) ≡{

P(X) =[

p1− p

] ∣∣∣.4 ≤ p ≤ .7}

A CS over a Boolean variable cannothave more than two vertices!

ext[K (X)] ={[

.7

.3

],

[.4.6

]}

P(x)

P(¬x)

P(X ) =[

10

]P(X ) =[

.7

.3

]P(X ) =[

.4

.6

]

.4 .7

.6

.3

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Credal Sets over Boolean Variables

Boolean X , values in X = {x ,¬x}Determinism ≡ degenerate mass fE.g., X = x ⇐⇒ P(X) =

[10

]Uncertainty ≡ prob mass functionP(X) =

[p

1− p

]with p ∈ [0, 1]

Imprecision credal seton the probability simplex

K (X) ≡{

P(X) =[

p1− p

] ∣∣∣.4 ≤ p ≤ .7}A CS over a Boolean variable cannothave more than two vertices!

ext[K (X)] ={[

.7

.3

],

[.4.6

]}

P(x)

P(¬x)

P(X ) =[

10

]P(X ) =[

.7

.3

]P(X ) =[

.4

.6

]

.4 .7

.6

.3

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Geometric Representation of CSs (ternary variables)

Ternary X (e.g., X = {win,draw,loss})

P(X ) ≡ point in the space (simplex)No bounds to |ext[K (X )]|Modelling ignorance

Uniform models indifferenceVacuous credal set

Expert qualitative knowledgeWin is more probable than draw,which more probable than loss

Learning from small datasetsLearning with multiple priors(e.g., IDM with s = 2)

Learning from incomplete dataConsidering all the possibleexplanation of the missing data

P(win)

P(draw)

P(loss)

P(X) =

.6.3.1

P(X)K (X)

P0(x) = 1|ΩX |

P0(X)K0(X)

K0(X)={P(X)

∣∣∣∣ ∑x P(x) = 1,P(x) ≥ 0}

(Walley, 1991)

From natural language tolinear contraints on probabilities

extremely probable P(x) ≥ 0.98very high probability P(x) ≥ 0.9

highly probable P(x) ≥ 0.85very probable P(x) ≥ 0.75

has a very good chance P(x) ≥ 0.65quite probable P(x) ≥ 0.6

P(x) ≥ 0.5has a good chance 0.4 ≤ P(x) ≤ 0.85

is improbable (unlikely) P(x) ≤ 0.5is somewhat unlikely P(x) ≤ 0.4

is very unlikely P(x) ≤ 0.25has little chance P(x) ≤ 0.2

is highly improbable P(x) ≤ 0.15is has very low probability P(x) ≤ 0.1

is extremely unlikely P(x) ≤ 0.02

P(win)

P(draw)

P(loss)

Previous matches:Holland 4 wins,Draws 1,

Spain 3 wins

1957: Spain vs. Holland 5 − 11973: Holland vs. Spain 3 − 21980: Spain vs. Holland 1 − 01983: Spain vs. Holland 1 − 01983: Holland vs. Spain 2 − 11987: Spain vs. Holland 1 − 12000: Spain vs. Holland 1 − 22001: Holland vs. Spain 1 − 02005: Spain vs. Holland ∗ − ∗ (missing)2008: Holland vs. Spain ∗ − ∗ (missing)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Geometric Representation of CSs (ternary variables)

Ternary X (e.g., X = {win,draw,loss})

P(X ) ≡ point in the space (simplex)No bounds to |ext[K (X )]|Modelling ignorance

Uniform models indifferenceVacuous credal set

Expert qualitative knowledgeWin is more probable than draw,which more probable than loss

Learning from small datasetsLearning with multiple priors(e.g., IDM with s = 2)

Learning from incomplete dataConsidering all the possibleexplanation of the missing data

P(win)

P(draw)

P(loss)

P(X) =

.6.3.1

P(X)K (X)

P0(x) = 1|ΩX |

P0(X)K0(X)

K0(X)={P(X)

∣∣∣∣ ∑x P(x) = 1,P(x) ≥ 0}

(Walley, 1991)

From natural language tolinear contraints on probabilities

extremely probable P(x) ≥ 0.98very high probability P(x) ≥ 0.9

highly probable P(x) ≥ 0.85very probable P(x) ≥ 0.75

has a very good chance P(x) ≥ 0.65quite probable P(x) ≥ 0.6

P(x) ≥ 0.5has a good chance 0.4 ≤ P(x) ≤ 0.85

is improbable (unlikely) P(x) ≤ 0.5is somewhat unlikely P(x) ≤ 0.4

is very unlikely P(x) ≤ 0.25has little chance P(x) ≤ 0.2

is highly improbable P(x) ≤ 0.15is has very low probability P(x) ≤ 0.1

is extremely unlikely P(x) ≤ 0.02

P(win)

P(draw)

P(loss)

Previous matches:Holland 4 wins,Draws 1,

Spain 3 wins

1957: Spain vs. Holland 5 − 11973: Holland vs. Spain 3 − 21980: Spain vs. Holland 1 − 01983: Spain vs. Holland 1 − 01983: Holland vs. Spain 2 − 11987: Spain vs. Holland 1 − 12000: Spain vs. Holland 1 − 22001: Holland vs. Spain 1 − 02005: Spain vs. Holland ∗ − ∗ (missing)2008: Holland vs. Spain ∗ − ∗ (missing)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Geometric Representation of CSs (ternary variables)

Ternary X (e.g., X = {win,draw,loss})

P(X ) ≡ point in the space (simplex)No bounds to |ext[K (X )]|Modelling ignorance

Uniform models indifferenceVacuous credal set

Expert qualitative knowledgeWin is more probable than draw,which more probable than loss

Learning from small datasetsLearning with multiple priors(e.g., IDM with s = 2)

Learning from incomplete dataConsidering all the possibleexplanation of the missing data

P(win)

P(draw)

P(loss)

P(X) =

.6.3.1

P(X)

K (X)

P0(x) = 1|ΩX |

P0(X)K0(X)

K0(X)={P(X)

∣∣∣∣ ∑x P(x) = 1,P(x) ≥ 0}

(Walley, 1991)

From natural language tolinear contraints on probabilities

extremely probable P(x) ≥ 0.98very high probability P(x) ≥ 0.9

highly probable P(x) ≥ 0.85very probable P(x) ≥ 0.75

has a very good chance P(x) ≥ 0.65quite probable P(x) ≥ 0.6

P(x) ≥ 0.5has a good chance 0.4 ≤ P(x) ≤ 0.85

is improbable (unlikely) P(x) ≤ 0.5is somewhat unlikely P(x) ≤ 0.4

is very unlikely P(x) ≤ 0.25has little chance P(x) ≤ 0.2

is highly improbable P(x) ≤ 0.15is has very low probability P(x) ≤ 0.1

is extremely unlikely P(x) ≤ 0.02

P(win)

P(draw)

P(loss)

Previous matches:Holland 4 wins,Draws 1,

Spain 3 wins

1957: Spain vs. Holland 5 − 11973: Holland vs. Spain 3 − 21980: Spain vs. Holland 1 − 01983: Spain vs. Holland 1 − 01983: Holland vs. Spain 2 − 11987: Spain vs. Holland 1 − 12000: Spain vs. Holland 1 − 22001: Holland vs. Spain 1 − 02005: Spain vs. Holland ∗ − ∗ (missing)2008: Holland vs. Spain ∗ − ∗ (missing)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Geometric Representation of CSs (ternary variables)

Ternary X (e.g., X = {win,draw,loss})

P(X ) ≡ point in the space (simplex)No bounds to |ext[K (X )]|Modelling ignorance

Uniform models indifferenceVacuous credal set

Expert qualitative knowledgeWin is more probable than draw,which more probable than loss

Learning from small datasetsLearning with multiple priors(e.g., IDM with s = 2)

Learning from incomplete dataConsidering all the possibleexplanation of the missing data

P(win)

P(draw)

P(loss)

P(X) =

.6.3.1

P(X)

K (X)

P0(x) = 1|ΩX |

P0(X)K0(X)

K0(X)={P(X)

∣∣∣∣ ∑x P(x) = 1,P(x) ≥ 0}

(Walley, 1991)

From natural language tolinear contraints on probabilities

extremely probable P(x) ≥ 0.98very high probability P(x) ≥ 0.9

highly probable P(x) ≥ 0.85very probable P(x) ≥ 0.75

has a very good chance P(x) ≥ 0.65quite probable P(x) ≥ 0.6

P(x) ≥ 0.5has a good chance 0.4 ≤ P(x) ≤ 0.85

is improbable (unlikely) P(x) ≤ 0.5is somewhat unlikely P(x) ≤ 0.4

is very unlikely P(x) ≤ 0.25has little chance P(x) ≤ 0.2

is highly improbable P(x) ≤ 0.15is has very low probability P(x) ≤ 0.1

is extremely unlikely P(x) ≤ 0.02

P(win)

P(draw)

P(loss)

Previous matches:Holland 4 wins,Draws 1,

Spain 3 wins

1957: Spain vs. Holland 5 − 11973: Holland vs. Spain 3 − 21980: Spain vs. Holland 1 − 01983: Spain vs. Holland 1 − 01983: Holland vs. Spain 2 − 11987: Spain vs. Holland 1 − 12000: Spain vs. Holland 1 − 22001: Holland vs. Spain 1 − 02005: Spain vs. Holland ∗ − ∗ (missing)2008: Holland vs. Spain ∗ − ∗ (missing)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Geometric Representation of CSs (ternary variables)

Ternary X (e.g., X = {win,draw,loss})

P(X ) ≡ point in the space (simplex)No bounds to |ext[K (X )]|Modelling ignorance

Uniform models indifferenceVacuous credal set

Expert qualitative knowledgeWin is more probable than draw,which more probable than loss

Learning from small datasetsLearning with multiple priors(e.g., IDM with s = 2)

Learning from incomplete dataConsidering all the possibleexplanation of the missing data

P(win)

P(draw)

P(loss)

P(X) =

.6.3.1

P(X)K (X)

P0(x) = 1|ΩX |

P0(X)K0(X)

K0(X)={P(X)

∣∣∣∣ ∑x P(x) = 1,P(x) ≥ 0}

(Walley, 1991)

From natural language tolinear contraints on probabilities

extremely probable P(x) ≥ 0.98very high probability P(x) ≥ 0.9

highly probable P(x) ≥ 0.85very probable P(x) ≥ 0.75

has a very good chance P(x) ≥ 0.65quite probable P(x) ≥ 0.6

P(x) ≥ 0.5has a good chance 0.4 ≤ P(x) ≤ 0.85

is improbable (unlikely) P(x) ≤ 0.5is somewhat unlikely P(x) ≤ 0.4

is very unlikely P(x) ≤ 0.25has little chance P(x) ≤ 0.2

is highly improbable P(x) ≤ 0.15is has very low probability P(x) ≤ 0.1

is extremely unlikely P(x) ≤ 0.02

P(win)

P(draw)

P(loss)

Previous matches:Holland 4 wins,Draws 1,

Spain 3 wins

1957: Spain vs. Holland 5 − 11973: Holland vs. Spain 3 − 21980: Spain vs. Holland 1 − 01983: Spain vs. Holland 1 − 01983: Holland vs. Spain 2 − 11987: Spain vs. Holland 1 − 12000: Spain vs. Holland 1 − 22001: Holland vs. Spain 1 − 02005: Spain vs. Holland ∗ − ∗ (missing)2008: Holland vs. Spain ∗ − ∗ (missing)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Geometric Representation of CSs (ternary variables)

Ternary X (e.g., X = {win,draw,loss})

P(X ) ≡ point in the space (simplex)No bounds to |ext[K (X )]|Modelling ignorance

Uniform models indifferenceVacuous credal set

Expert qualitative knowledgeWin is more probable than draw,which more probable than loss

Learning from small datasetsLearning with multiple priors(e.g., IDM with s = 2)

Learning from incomplete dataConsidering all the possibleexplanation of the missing data

P(win)

P(draw)

P(loss)

P(X) =

.6.3.1

P(X)K (X)

P0(x) = 1|ΩX |

P0(X)

K0(X)

K0(X)={P(X)

∣∣∣∣ ∑x P(x) = 1,P(x) ≥ 0}

(Walley, 1991)

From natural language tolinear contraints on probabilities

extremely probable P(x) ≥ 0.98very high probability P(x) ≥ 0.9

highly probable P(x) ≥ 0.85very probable P(x) ≥ 0.75

has a very good chance P(x) ≥ 0.65quite probable P(x) ≥ 0.6

P(x) ≥ 0.5has a good chance 0.4 ≤ P(x) ≤ 0.85

is improbable (unlikely) P(x) ≤ 0.5is somewhat unlikely P(x) ≤ 0.4

is very unlikely P(x) ≤ 0.25has little chance P(x) ≤ 0.2

is highly improbable P(x) ≤ 0.15is has very low probability P(x) ≤ 0.1

is extremely unlikely P(x) ≤ 0.02

P(win)

P(draw)

P(loss)

Previous matches:Holland 4 wins,Draws 1,

Spain 3 wins

1957: Spain vs. Holland 5 − 11973: Holland vs. Spain 3 − 21980: Spain vs. Holland 1 − 01983: Spain vs. Holland 1 − 01983: Holland vs. Spain 2 − 11987: Spain vs. Holland 1 − 12000: Spain vs. Holland 1 − 22001: Holland vs. Spain 1 − 02005: Spain vs. Holland ∗ − ∗ (missing)2008: Holland vs. Spain ∗ − ∗ (missing)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Geometric Representation of CSs (ternary variables)

Ternary X (e.g., X = {win,draw,loss})

P(X ) ≡ point in the space (simplex)No bounds to |ext[K (X )]|Modelling ignorance

Uniform models indifferenceVacuous credal set

Expert qualitative knowledgeWin is more probable than draw,which more probable than loss

Learning from small datasetsLearning with multiple priors(e.g., IDM with s = 2)

Learning from incomplete dataConsidering all the possibleexplanation of the missing data

P(win)

P(draw)

P(loss)

P(X) =

.6.3.1

P(X)K (X)

P0(x) = 1|ΩX |

P0(X)

K0(X)

K0(X)={P(X)

∣∣∣∣ ∑x P(x) = 1,P(x) ≥ 0}

(Walley, 1991)

From natural language tolinear contraints on probabilities

extremely probable P(x) ≥ 0.98very high probability P(x) ≥ 0.9

highly probable P(x) ≥ 0.85very probable P(x) ≥ 0.75

has a very good chance P(x) ≥ 0.65quite probable P(x) ≥ 0.6

P(x) ≥ 0.5has a good chance 0.4 ≤ P(x) ≤ 0.85

is improbable (unlikely) P(x) ≤ 0.5is somewhat unlikely P(x) ≤ 0.4

is very unlikely P(x) ≤ 0.25has little chance P(x) ≤ 0.2

is highly improbable P(x) ≤ 0.15is has very low probability P(x) ≤ 0.1

is extremely unlikely P(x) ≤ 0.02

P(win)

P(draw)

P(loss)

Previous matches:Holland 4 wins,Draws 1,

Spain 3 wins

1957: Spain vs. Holland 5 − 11973: Holland vs. Spain 3 − 21980: Spain vs. Holland 1 − 01983: Spain vs. Holland 1 − 01983: Holland vs. Spain 2 − 11987: Spain vs. Holland 1 − 12000: Spain vs. Holland 1 − 22001: Holland vs. Spain 1 − 02005: Spain vs. Holland ∗ − ∗ (missing)2008: Holland vs. Spain ∗ − ∗ (missing)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Geometric Representation of CSs (ternary variables)

Ternary X (e.g., X = {win,draw,loss})

P(X ) ≡ point in the space (simplex)No bounds to |ext[K (X )]|Modelling ignorance

Uniform models indifferenceVacuous credal set

Expert qualitative knowledgeWin is more probable than draw,which more probable than loss

Learning from small datasetsLearning with multiple priors(e.g., IDM with s = 2)

Learning from incomplete dataConsidering all the possibleexplanation of the missing data

P(win)

P(draw)

P(loss)

P(X) =

.6.3.1

P(X)K (X)

P0(x) = 1|ΩX |

P0(X)K0(X)

K0(X)={P(X)

∣∣∣∣ ∑x P(x) = 1,P(x) ≥ 0}

(Walley, 1991)

From natural language tolinear contraints on probabilities

extremely probable P(x) ≥ 0.98very high probability P(x) ≥ 0.9

highly probable P(x) ≥ 0.85very probable P(x) ≥ 0.75

has a very good chance P(x) ≥ 0.65quite probable P(x) ≥ 0.6

P(x) ≥ 0.5has a good chance 0.4 ≤ P(x) ≤ 0.85

is improbable (unlikely) P(x) ≤ 0.5is somewhat unlikely P(x) ≤ 0.4

is very unlikely P(x) ≤ 0.25has little chance P(x) ≤ 0.2

is highly improbable P(x) ≤ 0.15is has very low probability P(x) ≤ 0.1

is extremely unlikely P(x) ≤ 0.02

P(win)

P(draw)

P(loss)

Previous matches:Holland 4 wins,Draws 1,

Spain 3 wins

1957: Spain vs. Holland 5 − 11973: Holland vs. Spain 3 − 21980: Spain vs. Holland 1 − 01983: Spain vs. Holland 1 − 01983: Holland vs. Spain 2 − 11987: Spain vs. Holland 1 − 12000: Spain vs. Holland 1 − 22001: Holland vs. Spain 1 − 02005: Spain vs. Holland ∗ − ∗ (missing)2008: Holland vs. Spain ∗ − ∗ (missing)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Geometric Representation of CSs (ternary variables)

Ternary X (e.g., X = {win,draw,loss})

P(X ) ≡ point in the space (simplex)No bounds to |ext[K (X )]|Modelling ignorance

Uniform models indifferenceVacuous credal set

Expert qualitative knowledgeWin is more probable than draw,which more probable than loss

Learning from small datasetsLearning with multiple priors(e.g., IDM with s = 2)

Learning from incomplete dataConsidering all the possibleexplanation of the missing data

P(win)

P(draw)

P(loss)

P(X) =

.6.3.1

P(X)K (X)

P0(x) = 1|ΩX |

P0(X)K0(X)

K0(X)={P(X)

∣∣∣∣ ∑x P(x) = 1,P(x) ≥ 0}

(Walley, 1991)

From natural language tolinear contraints on probabilities

extremely probable P(x) ≥ 0.98very high probability P(x) ≥ 0.9

highly probable P(x) ≥ 0.85very probable P(x) ≥ 0.75

has a very good chance P(x) ≥ 0.65quite probable P(x) ≥ 0.6

P(x) ≥ 0.5has a good chance 0.4 ≤ P(x) ≤ 0.85

is improbable (unlikely) P(x) ≤ 0.5is somewhat unlikely P(x) ≤ 0.4

is very unlikely P(x) ≤ 0.25has little chance P(x) ≤ 0.2

is highly improbable P(x) ≤ 0.15is has very low probability P(x) ≤ 0.1

is extremely unlikely P(x) ≤ 0.02

P(win)

P(draw)

P(loss)

Previous matches:Holland 4 wins,Draws 1,

Spain 3 wins

1957: Spain vs. Holland 5 − 11973: Holland vs. Spain 3 − 21980: Spain vs. Holland 1 − 01983: Spain vs. Holland 1 − 01983: Holland vs. Spain 2 − 11987: Spain vs. Holland 1 − 12000: Spain vs. Holland 1 − 22001: Holland vs. Spain 1 − 02005: Spain vs. Holland ∗ − ∗ (missing)2008: Holland vs. Spain ∗ − ∗ (missing)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Geometric Representation of CSs (ternary variables)

Ternary X (e.g., X = {win,draw,loss})

P(X ) ≡ point in the space (simplex)No bounds to |ext[K (X )]|Modelling ignorance

Uniform models indifferenceVacuous credal set

Expert qualitative knowledgeWin is more probable than draw,which more probable than loss

Learning from small datasetsLearning with multiple priors(e.g., IDM with s = 2)

Learning from incomplete dataConsidering all the possibleexplanation of the missing data

P(win)

P(draw)

P(loss)

P(X) =

.6.3.1

P(X)K (X)

P0(x) = 1|ΩX |

P0(X)K0(X)

K0(X)={P(X)

∣∣∣∣ ∑x P(x) = 1,P(x) ≥ 0}

(Walley, 1991)

From natural language tolinear contraints on probabilities

extremely probable P(x) ≥ 0.98very high probability P(x) ≥ 0.9

highly probable P(x) ≥ 0.85very probable P(x) ≥ 0.75

has a very good chance P(x) ≥ 0.65quite probable P(x) ≥ 0.6

P(x) ≥ 0.5has a good chance 0.4 ≤ P(x) ≤ 0.85

is improbable (unlikely) P(x) ≤ 0.5is somewhat unlikely P(x) ≤ 0.4

is very unlikely P(x) ≤ 0.25has little chance P(x) ≤ 0.2

is highly improbable P(x) ≤ 0.15is has very low probability P(x) ≤ 0.1

is extremely unlikely P(x) ≤ 0.02

P(win)

P(draw)

P(loss)

Previous matches:Holland 4 wins,Draws 1,

Spain 3 wins

1957: Spain vs. Holland 5 − 11973: Holland vs. Spain 3 − 21980: Spain vs. Holland 1 − 01983: Spain vs. Holland 1 − 01983: Holland vs. Spain 2 − 11987: Spain vs. Holland 1 − 12000: Spain vs. Holland 1 − 22001: Holland vs. Spain 1 − 02005: Spain vs. Holland ∗ − ∗ (missing)2008: Holland vs. Spain ∗ − ∗ (missing)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Geometric Representation of CSs (ternary variables)

Ternary X (e.g., X = {win,draw,loss})

P(X ) ≡ point in the space (simplex)No bounds to |ext[K (X )]|Modelling ignorance

Uniform models indifferenceVacuous credal set

Expert qualitative knowledgeWin is more probable than draw,which more probable than loss

Learning from small datasetsLearning with multiple priors(e.g., IDM with s = 2)

Learning from incomplete dataConsidering all the possibleexplanation of the missing data

P(win)

P(draw)

P(loss)

P(X) =

.6.3.1

P(X)K (X)

P0(x) = 1|ΩX |

P0(X)K0(X)

K0(X)={P(X)

∣∣∣∣ ∑x P(x) = 1,P(x) ≥ 0}

(Walley, 1991)

From natural language tolinear contraints on probabilities

extremely probable P(x) ≥ 0.98very high probability P(x) ≥ 0.9

highly probable P(x) ≥ 0.85very probable P(x) ≥ 0.75

has a very good chance P(x) ≥ 0.65quite probable P(x) ≥ 0.6

P(x) ≥ 0.5has a good chance 0.4 ≤ P(x) ≤ 0.85

is improbable (unlikely) P(x) ≤ 0.5is somewhat unlikely P(x) ≤ 0.4

is very unlikely P(x) ≤ 0.25has little chance P(x) ≤ 0.2

is highly improbable P(x) ≤ 0.15is has very low probability P(x) ≤ 0.1

is extremely unlikely P(x) ≤ 0.02

P(win)

P(draw)

P(loss)

Previous matches:Holland 4 wins,Draws 1,

Spain 3 wins

1957: Spain vs. Holland 5 − 11973: Holland vs. Spain 3 − 21980: Spain vs. Holland 1 − 01983: Spain vs. Holland 1 − 01983: Holland vs. Spain 2 − 11987: Spain vs. Holland 1 − 12000: Spain vs. Holland 1 − 22001: Holland vs. Spain 1 − 02005: Spain vs. Holland ∗ − ∗ (missing)2008: Holland vs. Spain ∗ − ∗ (missing)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Geometric Representation of CSs (ternary variables)

Ternary X (e.g., X = {win,draw,loss})

P(X ) ≡ point in the space (simplex)No bounds to |ext[K (X )]|Modelling ignorance

Uniform models indifferenceVacuous credal set

Expert qualitative knowledgeWin is more probable than draw,which more probable than loss

Learning from small datasetsLearning with multiple priors(e.g., IDM with s = 2)

Learning from incomplete dataConsidering all the possibleexplanation of the missing data

P(win)

P(draw)

P(loss)

P(X) =

.6.3.1

P(X)K (X)

P0(x) = 1|ΩX |

P0(X)K0(X)

K0(X)={P(X)

∣∣∣∣ ∑x P(x) = 1,P(x) ≥ 0}

(Walley, 1991)

From natural language tolinear contraints on probabilities

extremely probable P(x) ≥ 0.98very high probability P(x) ≥ 0.9

highly probable P(x) ≥ 0.85very probable P(x) ≥ 0.75

has a very good chance P(x) ≥ 0.65quite probable P(x) ≥ 0.6

P(x) ≥ 0.5has a good chance 0.4 ≤ P(x) ≤ 0.85

is improbable (unlikely) P(x) ≤ 0.5is somewhat unlikely P(x) ≤ 0.4

is very unlikely P(x) ≤ 0.25has little chance P(x) ≤ 0.2

is highly improbable P(x) ≤ 0.15is has very low probability P(x) ≤ 0.1

is extremely unlikely P(x) ≤ 0.02

P(win)

P(draw)

P(loss)

Previous matches:Holland 4 wins,Draws 1,

Spain 3 wins

1957: Spain vs. Holland 5 − 11973: Holland vs. Spain 3 − 21980: Spain vs. Holland 1 − 01983: Spain vs. Holland 1 − 01983: Holland vs. Spain 2 − 11987: Spain vs. Holland 1 − 12000: Spain vs. Holland 1 − 22001: Holland vs. Spain 1 − 02005: Spain vs. Holland ∗ − ∗ (missing)2008: Holland vs. Spain ∗ − ∗ (missing)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Geometric Representation of CSs (ternary variables)

Ternary X (e.g., X = {win,draw,loss})

P(X ) ≡ point in the space (simplex)No bounds to |ext[K (X )]|Modelling ignorance

Uniform models indifferenceVacuous credal set

Expert qualitative knowledgeWin is more probable than draw,which more probable than loss

Learning from small datasetsLearning with multiple priors(e.g., IDM with s = 2)

Learning from incomplete dataConsidering all the possibleexplanation of the missing data

P(win)

P(draw)

P(loss)

P(X) =

.6.3.1

P(X)K (X)

P0(x) = 1|ΩX |

P0(X)K0(X)

K0(X)={P(X)

∣∣∣∣ ∑x P(x) = 1,P(x) ≥ 0}

(Walley, 1991)

From natural language tolinear contraints on probabilities

extremely probable P(x) ≥ 0.98very high probability P(x) ≥ 0.9

highly probable P(x) ≥ 0.85very probable P(x) ≥ 0.75

has a very good chance P(x) ≥ 0.65quite probable P(x) ≥ 0.6

P(x) ≥ 0.5has a good chance 0.4 ≤ P(x) ≤ 0.85

is improbable (unlikely) P(x) ≤ 0.5is somewhat unlikely P(x) ≤ 0.4

is very unlikely P(x) ≤ 0.25has little chance P(x) ≤ 0.2

is highly improbable P(x) ≤ 0.15is has very low probability P(x) ≤ 0.1

is extremely unlikely P(x) ≤ 0.02

P(win)

P(draw)

P(loss)

Previous matches:Holland 4 wins,Draws 1,

Spain 3 wins

1957: Spain vs. Holland 5 − 11973: Holland vs. Spain 3 − 21980: Spain vs. Holland 1 − 01983: Spain vs. Holland 1 − 01983: Holland vs. Spain 2 − 11987: Spain vs. Holland 1 − 12000: Spain vs. Holland 1 − 22001: Holland vs. Spain 1 − 02005: Spain vs. Holland ∗ − ∗ (missing)2008: Holland vs. Spain ∗ − ∗ (missing)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Basic operations with credal sets

Joint

PRECISEMass functions

P(X ,Y )

IMPRECISECredal sets

K (X ,Y )

MarginalizationP(X ) s.t.

p(x) =∑

y p(x , y)

K (X ) ={P(X)

∣∣∣∣ P(x) = ∑y P(x , y)P(X ,Y ) ∈ K (X ,Y )}

ConditioningP(X |y) s.t.

p(x |y) = P(x,y)∑y P(x,y)

K (X |y) ={P(X |y)

∣∣∣∣∣ P(x |y) = P(x,y)∑y P(x,y)P(X ,Y ) ∈ K (X ,Y )}

Combination P(x , y) = P(x |y)P(y) K (X |Y ) ⊗ K (Y ) =P(X ,Y )∣∣∣∣∣∣

P(x , y)=P(x |y)P(y)P(X |y) ∈ K (X |y)P(Y ) ∈ K (Y )

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Basic operations with credal sets

Joint

PRECISEMass functions

P(X ,Y )

IMPRECISECredal sets

K (X ,Y )

MarginalizationP(X ) s.t.

p(x) =∑

y p(x , y)

K (X ) ={P(X)

∣∣∣∣ P(x) = ∑y P(x , y)P(X ,Y ) ∈ K (X ,Y )}

ConditioningP(X |y) s.t.

p(x |y) = P(x,y)∑y P(x,y)

K (X |y) ={P(X |y)

∣∣∣∣∣ P(x |y) = P(x,y)∑y P(x,y)P(X ,Y ) ∈ K (X ,Y )}

Combination P(x , y) = P(x |y)P(y) K (X |Y ) ⊗ K (Y ) =P(X ,Y )∣∣∣∣∣∣

P(x , y)=P(x |y)P(y)P(X |y) ∈ K (X |y)P(Y ) ∈ K (Y )

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Basic operations with credal sets

Joint

PRECISEMass functions

P(X ,Y )

IMPRECISECredal sets

K (X ,Y )

MarginalizationP(X ) s.t.

p(x) =∑

y p(x , y)

K (X ) ={P(X)

∣∣∣∣ P(x) = ∑y P(x , y)P(X ,Y ) ∈ K (X ,Y )}

ConditioningP(X |y) s.t.

p(x |y) = P(x,y)∑y P(x,y)

K (X |y) ={P(X |y)

∣∣∣∣∣ P(x |y) = P(x,y)∑y P(x,y)P(X ,Y ) ∈ K (X ,Y )}

Combination P(x , y) = P(x |y)P(y) K (X |Y ) ⊗ K (Y ) =P(X ,Y )∣∣∣∣∣∣

P(x , y)=P(x |y)P(y)P(X |y) ∈ K (X |y)P(Y ) ∈ K (Y )

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Basic operations with credal sets

Joint

PRECISEMass functions

P(X ,Y )

IMPRECISECredal sets

K (X ,Y )

MarginalizationP(X ) s.t.

p(x) =∑

y p(x , y)

K (X ) ={P(X)

∣∣∣∣ P(x) = ∑y P(x , y)P(X ,Y ) ∈ K (X ,Y )}

ConditioningP(X |y) s.t.

p(x |y) = P(x,y)∑y P(x,y)

K (X |y) ={P(X |y)

∣∣∣∣∣ P(x |y) = P(x,y)∑y P(x,y)P(X ,Y ) ∈ K (X ,Y )}

Combination P(x , y) = P(x |y)P(y) K (X |Y ) ⊗ K (Y ) =P(X ,Y )∣∣∣∣∣∣

P(x , y)=P(x |y)P(y)P(X |y) ∈ K (X |y)P(Y ) ∈ K (Y )

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Basic operations with credal sets

Joint

PRECISEMass functions

P(X ,Y )

IMPRECISECredal sets

K (X ,Y )

MarginalizationP(X ) s.t.

p(x) =∑

y p(x , y)

K (X ) ={P(X)

∣∣∣∣ P(x) = ∑y P(x , y)P(X ,Y ) ∈ K (X ,Y )}

ConditioningP(X |y) s.t.

p(x |y) = P(x,y)∑y P(x,y)

K (X |y) ={P(X |y)

∣∣∣∣∣ P(x |y) = P(x,y)∑y P(x,y)P(X ,Y ) ∈ K (X ,Y )}

Combination P(x , y) = P(x |y)P(y) K (X |Y ) ⊗ K (Y ) =P(X ,Y )∣∣∣∣∣∣

P(x , y)=P(x |y)P(y)P(X |y) ∈ K (X |y)P(Y ) ∈ K (Y )

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Basic operations with credal sets

Joint

PRECISEMass functions

P(X ,Y )

IMPRECISECredal sets

K (X ,Y )

MarginalizationP(X ) s.t.

p(x) =∑

y p(x , y)

K (X ) ={P(X)

∣∣∣∣ P(x) = ∑y P(x , y)P(X ,Y ) ∈ K (X ,Y )}

ConditioningP(X |y) s.t.

p(x |y) = P(x,y)∑y P(x,y)

K (X |y) ={P(X |y)

∣∣∣∣∣ P(x |y) = P(x,y)∑y P(x,y)P(X ,Y ) ∈ K (X ,Y )}

Combination P(x , y) = P(x |y)P(y) K (X |Y ) ⊗ K (Y ) =P(X ,Y )∣∣∣∣∣∣

P(x , y)=P(x |y)P(y)P(X |y) ∈ K (X |y)P(Y ) ∈ K (Y )

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Basic operations with credal sets

Joint

PRECISEMass functions

P(X ,Y )

IMPRECISECredal sets

K (X ,Y )

MarginalizationP(X ) s.t.

p(x) =∑

y p(x , y)

K (X ) ={P(X)

∣∣∣∣ P(x) = ∑y P(x , y)P(X ,Y ) ∈ K (X ,Y )}

ConditioningP(X |y) s.t.

p(x |y) = P(x,y)∑y P(x,y)

K (X |y) ={P(X |y)

∣∣∣∣∣ P(x |y) = P(x,y)∑y P(x,y)P(X ,Y ) ∈ K (X ,Y )}

Combination P(x , y) = P(x |y)P(y) K (X |Y ) ⊗ K (Y ) =P(X ,Y )∣∣∣∣∣∣

P(x , y)=P(x |y)P(y)P(X |y) ∈ K (X |y)P(Y ) ∈ K (Y )

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Basic operations with credal sets

Joint

PRECISEMass functions

P(X ,Y )

IMPRECISECredal sets

K (X ,Y )

MarginalizationP(X ) s.t.

p(x) =∑

y p(x , y)

K (X ) ={P(X)

∣∣∣∣ P(x) = ∑y P(x , y)P(X ,Y ) ∈ K (X ,Y )}

ConditioningP(X |y) s.t.

p(x |y) = P(x,y)∑y P(x,y)

K (X |y) ={P(X |y)

∣∣∣∣∣ P(x |y) = P(x,y)∑y P(x,y)P(X ,Y ) ∈ K (X ,Y )}

Combination P(x , y) = P(x |y)P(y) K (X |Y ) ⊗ K (Y ) =P(X ,Y )∣∣∣∣∣∣

P(x , y)=P(x |y)P(y)P(X |y) ∈ K (X |y)P(Y ) ∈ K (Y )

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Basic operations with credal sets

Joint

PRECISEMass functions

P(X ,Y )

IMPRECISECredal sets

K (X ,Y )

MarginalizationP(X ) s.t.

p(x) =∑

y p(x , y)

K (X ) ={P(X)

∣∣∣∣ P(x) = ∑y P(x , y)P(X ,Y ) ∈ K (X ,Y )}

ConditioningP(X |y) s.t.

p(x |y) = P(x,y)∑y P(x,y)

K (X |y) ={P(X |y)

∣∣∣∣∣ P(x |y) = P(x,y)∑y P(x,y)P(X ,Y ) ∈ K (X ,Y )}

Combination P(x , y) = P(x |y)P(y) K (X |Y ) ⊗ K (Y ) =P(X ,Y )∣∣∣∣∣∣

P(x , y)=P(x |y)P(y)P(X |y) ∈ K (X |y)P(Y ) ∈ K (Y )

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Basic operations with credal sets (vertices)

Joint

Marginalization

Conditioning

Combination

IMPRECISECredal sets

K (X ,Y )

K (X ) ={P(X)

∣∣∣∣ P(x) = ∑y P(x , y)P(X ,Y ) ∈ K (X ,Y )}

K (X |y) ={P(X |y)

∣∣∣∣∣ P(x |y) = P(x,y)∑y P(x,y)P(X ,Y ) ∈ K (X ,Y )}

K (X |Y ) ⊗ K (Y ) =P(X ,Y )∣∣∣∣∣∣

P(x , y)=P(x |y)P(y)P(X |y) ∈ K (X |y)P(Y ) ∈ K (Y )

IMPRECISEExtremes

= CH{

Pj (X ,Y )}nv

j=1

= CH{P(X)

∣∣∣∣ P(x) = ∑y P(x , y)P(X ,Y ) ∈ ext[K (X ,Y )]}

= CH{P(X |y)

∣∣∣∣∣ P(x |y) = P(x,y)∑y P(x,y)P(X ,Y ) ∈ ext[K (X ,Y )]}

= CHP(X ,Y )∣∣∣∣∣∣

P(x , y)=P(x |y)P(y)P(X |y) ∈ ext[K (X |y)]P(Y ) ∈ ext[K (Y )]

[EXE]Prove it!

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Basic operations with credal sets (vertices)

Joint

Marginalization

Conditioning

Combination

IMPRECISECredal sets

K (X ,Y )

K (X ) ={P(X)

∣∣∣∣ P(x) = ∑y P(x , y)P(X ,Y ) ∈ K (X ,Y )}

K (X |y) ={P(X |y)

∣∣∣∣∣ P(x |y) = P(x,y)∑y P(x,y)P(X ,Y ) ∈ K (X ,Y )}

K (X |Y ) ⊗ K (Y ) =P(X ,Y )∣∣∣∣∣∣

P(x , y)=P(x |y)P(y)P(X |y) ∈ K (X |y)P(Y ) ∈ K (Y )

IMPRECISEExtremes

= CH{

Pj (X ,Y )}nv

j=1

= CH{P(X)

∣∣∣∣ P(x) = ∑y P(x , y)P(X ,Y ) ∈ ext[K (X ,Y )]}

= CH{P(X |y)

∣∣∣∣∣ P(x |y) = P(x,y)∑y P(x,y)P(X ,Y ) ∈ ext[K (X ,Y )]}

= CHP(X ,Y )∣∣∣∣∣∣

P(x , y)=P(x |y)P(y)P(X |y) ∈ ext[K (X |y)]P(Y ) ∈ ext[K (Y )]

[EXE]Prove it!

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

Basic operations with credal sets (vertices)

Joint

Marginalization

Conditioning

Combination

IMPRECISECredal sets

K (X ,Y )

K (X ) ={P(X)

∣∣∣∣ P(x) = ∑y P(x , y)P(X ,Y ) ∈ K (X ,Y )}

K (X |y) ={P(X |y)

∣∣∣∣∣ P(x |y) = P(x,y)∑y P(x,y)P(X ,Y ) ∈ K (X ,Y )}

K (X |Y ) ⊗ K (Y ) =P(X ,Y )∣∣∣∣∣∣

P(x , y)=P(x |y)P(y)P(X |y) ∈ K (X |y)P(Y ) ∈ K (Y )

IMPRECISEExtremes

= CH{

Pj (X ,Y )}nv

j=1

= CH{P(X)

∣∣∣∣ P(x) = ∑y P(x , y)P(X ,Y ) ∈ ext[K (X ,Y )]}

= CH{P(X |y)

∣∣∣∣∣ P(x |y) = P(x,y)∑y P(x,y)P(X ,Y ) ∈ ext[K (X ,Y )]}

= CHP(X ,Y )∣∣∣∣∣∣

P(x , y)=P(x |y)P(y)P(X |y) ∈ ext[K (X |y)]P(Y ) ∈ ext[K (Y )]

[EXE]Prove it!

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

[Exe #2] An imprecise bivariate (graphical?) model

Two Boolean variables: Smoker,Lung Cancer

Eight “Bayesian” phisicians, eachone assessing Pj (S,C)

K (S,C) = CH{

Pj (S,C)}8

j=1

Compute:Marginal K (S)ConditioningK (C|S) := {K (C|s),K (C|s)}Combination (marg ext)K ′(C,S) := K (C|S)⊗ K (S)

Is this a (I)PGM?

Smoker Cancer

j Pj (s, c) Pj (s,¬c) Pj (¬s, c) Pj (¬s,¬c)

1 1/8 1/8 3/8 3/8

2 1/8 1/8 9/16 3/16

3 3/16 1/16 3/8 3/8

4 3/16 1/16 9/16 3/16

5 1/4 1/4 1/4 1/4

6 1/4 1/4 3/8 1/8

7 3/8 1/8 1/4 1/4

8 3/8 1/8 3/8 1/8

(0,1,0,0)

(0,0,1,0)

(1,0,0,0)

(0,0,0,1)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

[Exe #2] An imprecise bivariate (graphical?) model

Two Boolean variables: Smoker,Lung Cancer

Eight “Bayesian” phisicians, eachone assessing Pj (S,C)

K (S,C) = CH{

Pj (S,C)}8

j=1

Compute:Marginal K (S)ConditioningK (C|S) := {K (C|s),K (C|s)}Combination (marg ext)K ′(C,S) := K (C|S)⊗ K (S)

Is this a (I)PGM?

Smoker Cancer

j Pj (s, c) Pj (s,¬c) Pj (¬s, c) Pj (¬s,¬c)

1 1/8 1/8 3/8 3/8

2 1/8 1/8 9/16 3/16

3 3/16 1/16 3/8 3/8

4 3/16 1/16 9/16 3/16

5 1/4 1/4 1/4 1/4

6 1/4 1/4 3/8 1/8

7 3/8 1/8 1/4 1/4

8 3/8 1/8 3/8 1/8

(0,1,0,0)

(0,0,1,0)

(1,0,0,0)

(0,0,0,1)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

[Exe #2] An imprecise bivariate (graphical?) model

Two Boolean variables: Smoker,Lung Cancer

Eight “Bayesian” phisicians, eachone assessing Pj (S,C)

K (S,C) = CH{

Pj (S,C)}8

j=1

Compute:Marginal K (S)ConditioningK (C|S) := {K (C|s),K (C|s)}Combination (marg ext)K ′(C,S) := K (C|S)⊗ K (S)

Is this a (I)PGM?

Smoker Cancer

j Pj (s, c) Pj (s,¬c) Pj (¬s, c) Pj (¬s,¬c)

1 1/8 1/8 3/8 3/8

2 1/8 1/8 9/16 3/16

3 3/16 1/16 3/8 3/8

4 3/16 1/16 9/16 3/16

5 1/4 1/4 1/4 1/4

6 1/4 1/4 3/8 1/8

7 3/8 1/8 1/4 1/4

8 3/8 1/8 3/8 1/8

(0,1,0,0)

(0,0,1,0)

(1,0,0,0)

(0,0,0,1)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

[Exe #2] An imprecise bivariate (graphical?) model

Two Boolean variables: Smoker,Lung Cancer

Eight “Bayesian” phisicians, eachone assessing Pj (S,C)

K (S,C) = CH{

Pj (S,C)}8

j=1

Compute:Marginal K (S)ConditioningK (C|S) := {K (C|s),K (C|s)}Combination (marg ext)K ′(C,S) := K (C|S)⊗ K (S)

Is this a (I)PGM?

Smoker Cancer

j Pj (s, c) Pj (s,¬c) Pj (¬s, c) Pj (¬s,¬c)

1 1/8 1/8 3/8 3/8

2 1/8 1/8 9/16 3/16

3 3/16 1/16 3/8 3/8

4 3/16 1/16 9/16 3/16

5 1/4 1/4 1/4 1/4

6 1/4 1/4 3/8 1/8

7 3/8 1/8 1/4 1/4

8 3/8 1/8 3/8 1/8

(0,1,0,0)

(0,0,1,0)

(1,0,0,0)

(0,0,0,1)

Introduction Credal sets Credal networks Modelling observations Decision making Inference algorithms Other guys

[Exe #2] An imprecise bivariate (graphical?) model

Two Boolean variables: Smoker,Lung Cancer

Eight “Bayesian” phisicians, eachone assessing Pj (S,C)

K (S,C) = CH{

Pj (S,C)}8

j=1

Compute:Marginal K (S)ConditioningK (C|S) := {K (C|s),K (C|s)}Combination (marg ext)K ′(C,S) := K (C|S)⊗ K (S)

Is this a (I)PGM?

Smoker Cancer

j Pj (s, c) Pj (s,¬c) Pj (¬s, c) Pj (¬s,¬c)

1 1/8 1/8 3/8 3/8

2 1/8 1/8 9/16 3/16

3 3/16 1/16 3/8 3/8

4 3/16 1/16 9/16 3/16

5 1/4 1/4 1/4 1/4

6 1/4 1/4 3/8 1/8

7 3/8 1/8 1/4 1/4

8 3/8 1/8 3/8 1/8

(0,1,0,0)

(0,0,1,0)

(1,0,0,0)

(0,0,0,1)