imprecise probabilistic graphical models - sipta · probabilistic graphical models data mining and...
TRANSCRIPT
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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
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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)
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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)