building bayesian networksmarinav/teaching/bdminai/bdmaibuildbn.pdf · dynamic bayesian network for...
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Building Bayesian Networks
Lecture3: Building BN – p.1
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The focus today . . .
Problem solving by Bayesian networks
Designing Bayesian networksQualitative part (structure)Quantitative part (probability assessment)
Simplified Bayesian networksIn structure: Naïve Bayes, Tree-AugmentedNetworksIn probability assessment: Parent divorcing, CausalIndependence
Lecture3: Building BN – p.2
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Problem solving
Bayesian networks: a declarative (knowing what)knowledge-representation formalism, i.e.,:
mathematical basis
problem to be solved determined by (1) enteredevidence E (including potential decisions); (2) givenhypothesis H : P (H | E)
Examples:
Description of population (or prior information)
Classification and diagnosis: D = arg maxH P (H | E) i.e.D is the hypothesis with maximum P (H | E)
Prediction
Decision making based on what-if scenario’s
Lecture3: Building BN – p.3
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Prior information
Gives description of the population on which the assessedprobabilities are based, i.e., the original probabilities before newevidence is uncovered
Marginal probabilities P (V ) for every vertex V , e.g.,P (WILSON’S DISEASE = yes)
Lecture3: Building BN – p.4
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Diagnostic problem solving
Gives description of the subpopulation of the original population orindividual cases
Marginal probabilities P ∗(V ) = P (V | E) for every vertex V , e.g.,P (WILSON’S DISEASE = yes | E) for entered evidence E (red vertices,with probability for one value equal to 1)
Lecture3: Building BN – p.5
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Prediction of associated findings
Gives description of the findings associated with a given class orcategory, such as Wilson’s disease
Marginal probabilities P ∗(V ) = P (V | E) for every vertex V , e.g.,P (Kayser-Fleischer Rings = yes | E) with E evidence
Lecture3: Building BN – p.6
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Design of Bayesian network
Current design principle: start modelling qualitatively(different from traditional knowledge-based systems)
causalgraph
qualitativeprobabilistic network
quantitativenetwork
Bayesiannetwork
refinement
variables/relationships
domains ofvariables/qualitative
probabilisticinformation
numericalassessment
experts/dataset
evaluation
Lecture3: Building BN – p.7
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Terminology
NOYES
FLU
noyes
FEVER
noyes
SARS
noyes
VisitToChina
noyes
DYSPNOEA
<=37.5>37.5
TEMP
Parent SARS of Child FEVER
SARS is Ancestor of TEMP
DYSPNOEA is Descendant of VisitToChina
Query node, e.g., FEVER
Evidence, e.g., VisitToChina and TEMP
Markov blanket, e.g.,for SARS: {VisitToChina,DYSPNOEA,FEVER,FLU}
Lecture3: Building BN – p.8
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Causal graph: Topology (structure)
cause1
causen
... effect
Identify factors that are relevant
Determine how those factors are causally related toeach other
The arc cause→ effect does mean that cause is a factorinvolved in causing effect
Lecture3: Building BN – p.9
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Causal graph: Common effect
cause1
causen
... effect
An effect that has two or more ingoing arcs from othervertices is a common effect of those causes
Kinds of causal interactionSynergy: POLUTION −→ CANCER←− SMOKING
Prevention: VACCINE −→ DEATH←− SMALLPOX
XOR: ALKALI −→ DEATH←− ACID
Lecture3: Building BN – p.10
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Causal graph: Common cause
cause
effect1
effectn
...
A cause that has two or more outgoing arcs to othervertices is a common cause (factor) of those effects
The effects of a common cause are usually observables(e.g. manifestations of failure of a device or symptomsin a disease)
Lecture3: Building BN – p.11
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Causal graph: Example
FLU
PNEUMONIA
FEVER
MYALGIA
TEMP
FEVER and PNEUMONIA are two alternative causes offever (but may enhance each other)
FLU has two common effects: MYALGIA and FEVER
High body TEMPerature is an indirect effect of FLU andPNEUMONIA, caused by FEVER
Lecture3: Building BN – p.12
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Check independence relationship
FLU
PNEUMONIA
FEVER
MYALGIA
TEMP
Conditional independence: X ⊥⊥ Y | Z
{FEVER} ⊥⊥ {MYALGIA} | {FLU}
? | {FEVER}
{PNEUMONIA} ⊥⊥ {FLU} | ?{PNEUMONIA} 6⊥⊥ {FLU} | {FEVER}
Lecture3: Building BN – p.13
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Choose variables
Factors are mutually exclusive (cannot occur togetherwith absolute certainty): put as values in the samevariable, or
Factors may co-occur: multiple variables
DISEASEpneu/flu
FEVER
MYALGIA
(a) Single variable
FLUyes/no
FEVER
PNEUMONIAyes/no
MYALGIA
(b) Multiple variables
Lecture3: Building BN – p.14
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Choose values
Discrete valuesMutually exclusive and exhaustiveTypes:
binary, e.g., FLU = yes/no, true/false, 0/1ordinal, e.g., INCOME = low, medium, highnominal, e.g., COLOR = brown, green, redintegral, e.g., AGE = {1, . . . , 120}
Continuous values
Discretization (of continuous values)Example for TEMP:[−50,+5)→ cold[+5,+20)→ mild[+20,+50]→ hot
Lecture3: Building BN – p.15
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Probability assessment
qualitativeorders andequalities
indegreeof vertex
divorcing/causalindependence
quantitativeassessment
experts/dataset
probabilitydistribution
compareprediction with
literature
compare withtest dataset
checkconsistency
> 4
≤ 4
EVALUATION
8
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:
Lecture3: Building BN – p.16
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Expert judgements
Qualitative probabilities:Qualitative orders:
AGE P (General Health Status | AGE)
10-69 good > average > poor70-79 average > good > poor80-89 average > poor > good≥ 90 poor > average > good
Equalities:
P (CANCER = T1|AGE = 15− 29) =P (CANCER = T2|AGE = 15− 29)
Lecture3: Building BN – p.17
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Expert judgements (cont.)
Quantitative, subjective probabilities:
P (GHS | AGE)
AGE good average poor
10-69 0.99 0.008 0.00270-79 0.3 0.5 0.280-89 0.1 0.5 0.4≥ 90 0.1 0.3 0.6
Lecture3: Building BN – p.18
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A bottleneck in Bayesian networks
FEV ER
FLU
RABIES EARINF
BRONCH
The number of parameters for the effect given n causesgrows exponentially: ≥ 2n (for binary causes)
Unlikely evidence combination:P (fever|flu, rabies, ear_infection) =?
Problem: for many BNs too many probabilities have to beassessed
Lecture3: Building BN – p.19
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Special form Bayesian networks
Solution: use simpler probabilistic model, such that either
the structure becomes simpler, e.g.,
naive (independent) form BNT ree-Augmented Bayesian Network (TAN)
or,
the assessment of the conditional probabilities becomessimpler (even though the structure is still complex), e.g.,
parent divorcingcausal independence BN
Lecture3: Building BN – p.20
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Independent (Naive) form BN
C
E1
· · ·E2
Em
C is a class variable
Ei are evidence variables and E ⊆ {E1, . . . , Em}. Wehave Ei ⊥⊥ Ej | C, for i 6= j. Hence, using Bayes’ rule:
P (C | E) =P (E | C)P (C)
P (E)with:
P (E | C) =∏
E∈E
P (E | C) by cond. ind.
P (E) =∑
C
P (E | C)P (C) marg. & cond.
Lecture3: Building BN – p.21
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Example of Naive Bayes
Lecture3: Building BN – p.22
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Example of Naive Bayes (1)
Learning phase
Rain
Overcast
Sunny
NoYesOutlook
Rain
Overcast
Sunny
NoYesOutlook
Cool
Mild
Hot
NoYesTemperature
Cool
Mild
Hot
NoYesTemperature
Normal
High
oYesHumidity
Normal
High
oYesHumidity
Weak
Strong
NoYesWind
Weak
Strong
NoYesWind
P =Yes) = P =No) =
Lecture3: Building BN – p.23
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Example of Naive Bayes (2)
Testing phase (inference)
Play Tennis
Outlook Temp Humidity Wind
Play Tennis
Outlook Temp Humidity Wind
Play Tennis
Outlook Temp Humidity Wind
Lecture3: Building BN – p.24
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Example of Naive Bayes (3)
Testing phase (inference)
Play Tennis
Outlook Temp Humidity Wind
P Yes x P x Ye P Yes x ∝∝ Sunny es Cool Yes High es Strong Yes Yes
0.0053
No x ∝ Sunny o Cool o High No Strong No No 0.0206
P Yes x P No x x PlayTennis No
Note
P Yes x P Yes x P Yes x P No x
Play Tennis
Outlook Temp Humidity Wind
Play Tennis
Outlook Temp Humidity Wind
P Yes x P x Ye P Yes x ∝∝ Sunny es Cool Yes High es Strong Yes Yes
0.0053
No x ∝ Sunny o Cool o High No Strong No No 0.0206
P Yes x P No x x PlayTennis No
Note
P Yes x P Yes x P Yes x P No xLecture3: Building BN – p.25
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Tree-Augmented BN (TAN)
C
E1
E3E2
E5
E4
Extension of Naive Bayes: reduce the number ofindependent assumptions
Each node has at most two parents (one is the classnode)
Lecture3: Building BN – p.26
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Divorcing multiple parents
Surgery
DrugTreatment
GeneralHealth
Survival
(a) Original network
Surgery
DrugTreatment
GeneralHealth
Post-therapySurvival
Survival
(b) Divorced network
Reduction in number of probabilities to assess:
Identify a potential common effect of two or more parentvertices of a vertex
Introduce a new variable into the network, representingthe common effect
Lecture3: Building BN – p.27
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Causal Independence
C1 C2 . . . Cn
I1 I2 . . . In
Ef
with:
cause variables Cj, intermediate variables Ij , and theeffect variable E
P (E | I1, . . . , In) ∈ {0, 1}
interaction function f , defined such that
f(I1, . . . , In) =
{
e if P (e | I1, . . . , In) = 1
¬e if P (e | I1, . . . , In) = 0
Lecture3: Building BN – p.28
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Causal Independence: BN
C1 C2 . . . Cn
I1 I2 . . . In
Ef
P (e | C1, . . . , Cn) =X
I1,...,In
P (e | I1, . . . , In)P (I1, . . . , In | C1, . . . , Cn)
=X
f(I1,...,In)=e
P (e | I1, . . . , In)P (I1, . . . , In | C1, . . . , Cn)
Note that as Ii ⊥⊥ Ij | ∅, and Ii ⊥⊥ Cj | Ci, for i 6= j, it holds that:
P (I1, . . . , In | C1, . . . , Cn) =n
Y
k=1
P (Ik | Ck)
Conclusion: assessment of P (Ii |Ci) instead of P (E |C1, . . . , Cn), i.e., 2n vs. 2n probabilities
Lecture3: Building BN – p.29
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Causal independence: Noisy OR
C1 C2
I1 I2
EOR
Interactions among causes, as represented by thefunction f and P (E | I1, I2), is a logical OR
Meaning: presence of any one of the causes Ci withabsolute certainty will cause the effect e (i.e. E = true)
P (e|C1, C2) = ?
Lecture3: Building BN – p.30
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Causal independence: Noisy OR (cont.)
C1 C2
I1 I2
EOR
P (e|C1, C2) =∑
I1,I2
P (e|I1, I2, C1, C2)P (I1, I2|C1, C2)
= ?
Lecture3: Building BN – p.31
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Causal independence: Noisy OR (cont.)
C1 C2
I1 I2
EOR
P (e|C1, C2) =∑
I1,I2
P (e|I1, I2, C1, C2)P (I1, I2|C1, C2)
=∑
f(I1,I2)=e
P (e|I1, I2)∏
k=1,2
P (Ik|Ck)
Lecture3: Building BN – p.32
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Causal independence: Noisy OR (cont.)
C1 C2
I1 I2
EOR
I1 I2 P (e | I1, I2) f(I1, I2) = I1 ∨ I2
0 0 0 ⇒ ¬e
0 1 1 ⇒ e
1 0 1 ⇒ e
1 1 1 ⇒ e
P (e|C1, C2) =X
f(I1,I2)=e
P (e|I1, I2)Y
k=1,2
P (Ik|Ck)
= P (i1|C1)P (i2|C2) + P (¬i1|C1)P (i2|C2) + P (i1|C1)P (¬i2|C2)
Lecture3: Building BN – p.33
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Noisy OR: Real-world example
Dynamic Bayesian network for predicting the development ofhypertensive disorders during pregnancy
VascRisk (Vascular risk) has 11 causes and its original CPTrequires the estimation of 20736!!! entries. Practically impossible!
Solution: use noisy OR to simplify it
Lecture3: Building BN – p.34
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Causal independence: Noisy AND
C1 C2
I1 I2
EAND
Interactions among causes, as represented by thefunction f and P (E | I1, I2), is a logical AND
Meaning: presence of all causes Ci with absolutecertainty will cause the effect e (i.e. E = true);otherwise, ¬e
P (e|C1, C2) = ?
Lecture3: Building BN – p.35
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Are Bayesian networks always suitable?
“Essentially, all models are wrong but some are useful”George Box, Norman Draper (1987),
Empirical Model-Building and Response Surfaces, Wiley
Problem (modelling) objective, e.g., for functionapproximation or pure numeric prediction without aneed to explain the results a “black box” model such asneural networks can be sufficient
Sufficient knowledge about the problem (domainexperts, literature, data)
Complexity of the problem e.g., is it decomposable
Lecture3: Building BN – p.36
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Refining causal graphs
Model refinement is necessary.
How:ManualAutomatic
What:Probability adjustmentRemoving irrelevant factorsAdding previously hidden, unknown factorsCausal relationships adjustment, e.g., including,removing independence relations
Lecture3: Building BN – p.37