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Inference in Bayesian NetworksInference in Bayesian Networks

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Query TypesQuery Types

• Pr: – Evidence: Pr(e)– Posterior marginals: Pr(x|e) for every X

• MPE: Most probable instantiation:– Instantiation y such that Pr(y|e) is maximal (Y = E)

• MAP: Maximum a posteriori hypothesis:– Intantiation y such that Pr(y|e) is maximal (Y is subset of E)

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Pr: Posterior MarginalsPr: Posterior MarginalsBattery Age Alternator Fan Belt

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Diagnosis ScenarioDiagnosis ScenarioBattery Age Alternator Fan Belt

BatteryCharge Delivered

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ok on yes no

.001

ok off yes no

.090

A. Darwiche

Battery Age Alternator Fan Belt

BatteryCharge Delivered

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MPE: Most Probable ExplanationMPE: Most Probable Explanation

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BatteryCharge Delivered

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MPE: Most Probable ExplanationMPE: Most Probable Explanation

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MAP: Maximum a Posteriori MAP: Maximum a Posteriori HypothesisHypothesis

Battery Age Alternator Fan Belt

BatteryCharge Delivered

Battery Power

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MAP: Maximum a Posteriori MAP: Maximum a Posteriori HypothesisHypothesis

Battery Age Alternator Fan Belt

BatteryCharge Delivered

Battery Power

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Radio Lights Engine Turn Over

Gas Gauge

Gas

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A. Darwiche

Battery Age Alternator Fan Belt

BatteryCharge Delivered

Battery Power

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Gas

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Engine Start

MAP: Maximum a Posteriori MAP: Maximum a Posteriori HypothesisHypothesis

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Probability of EvidenceProbability of Evidence

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false

A B

true

false

A

.3

.7

ØABA

.1truetrue.9true false

ØB

.2falsefalse

.8false true

**

* *

λ~b λ~aλbλa

+

+ +

* * * *

.3 .1 .9 .8 .2 .7

Factoring

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NotationNotation• A binary variable X:

– is variable with two values (true, false)– x is short notation for X=true– ~x is short notation for X=false

• If X is a variable with parents Y and Z, then:

represents the probability Pr(X=x | Y=y, Z=y)

• If X is a binary variable with parents Y and Z (also binary), then:

represents the probability Pr(X=true | Y=false, Z=true)

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NotationNotation• An instantiation is a set of variables with their values:

– X=true,Y=false, Z=true is an instantiation– A=a, B=b, C=c is an instantiation

• x, ~y, z is short notation for the instantiationX=true, Y=false, Z=true

• a,b,c is short notation for the instantiation A=a, B=b, C=c

• Two instantiations are inconsistent iff they assign different values to the same variable:– x,~y,z and x,y,z are inconsistent– x,~y,z and a,b,c are consistent

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Pr(a) = .03 + .27 = .3

Joint Probability DistributionJoint Probability Distribution

false

false

B

.03

.27

A

.56

.14

truetrue

true

false

false

false

Pr

false

true

A. DarwichePr(~b) = .27 + .14 = .41

false

false

B

.03

.27

A

.56

.14

truetrue

true

false

false

false

Pr

false

true

Joint Probability DistributionJoint Probability Distribution

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F = .03λaλb + .27λaλ~b + .56λ~aλb + .14λ~a λ~b

false

false

B

.03

A

truetrue

true

false

false

false

Pr

false

true

.27

.14

.56

λaλb .03

λaλ~b .27

λ~aλb .56

λ~aλ~b .14

λa λb …are called evidence indicators

F is called the polynomial of the given probability distribution

Evidence IndicatorsEvidence Indicators

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Computing ProbabilitiesComputing Probabilities

• To compute the probability of instantiation e:Evaluate polynomial F while replacing each indicator

-by 1 if the instantiation is consistent with the indicator;-by 0 if the instantiation is inconsistent with the indicator

• Examples:– Indicator λa is consistent with instantiation a,~b,c

– Indicator λb is inconsistent with instantiation a,~b,c

– Indicator λd is consistent with instantiation a,~b,c

– Indicator λ~d is consistent with instantiation a,~b,c

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F = .03λaλb + .27λaλ~b + .56λ~aλb + .14λ~a λ~b

Computing ProbabilitiesComputing Probabilities

To compute the probability of instantiation a, ~b:

F(a,~b) = .03*1*0 + .27*1*1 + .56*0*0 + .14*0*1 = .27

To compute the probability of instantiation ~a:

F(~a) = .03*0*1 + .27*0*1 + .56*1*1 + .14*1*1 = .70

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A B

true

false

A

.3

.7

ØABA

.1truetrue.9true false

ØB

.2falsefalse

.8false true

false

false

BA

truetrue

true

false

false

false

Pr

false

true

.03=.3*.1

.27=.3*.9

56=.7*.8

.14=.7*.2

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A B

true

false

A ØABA

truetruetrue false

ØB

falsefalsefalse true

θaθ~a

θ b|a

θ~b|a

θ b|~a

θ ~b|~a

false

false

BA

truetrue

true

false

false

false

Pr

false

true

θa θ b|a

θa θ~b|a

θ~a θ b|~a

θ~a θ ~b|~a

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A B

true

false

A ØABA

truetruetrue false

ØB

falsefalsefalse true

θaθ~a

θ b|a

θ~b|a

θ b|~a

θ ~b|~a

false

false

BA

truetrue

true

false

false

false

Pr

false

true

λaλb θa θ b|a

λaλ~b θa θ~b|a

λ~aλb θ~a θ b|~a

λ~aλ~b θ~a θ ~b|~a

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A B

true

false

A ØABA

truetruetrue false

ØB

falsefalsefalse true

θaθ~a

θ b|a

θ~b|a

θ b|~a

θ ~b|~a

false

BA

truetrue

true

false

false

false

Pr

false

true

λaλb θa θ b|a

λaλ~b θa θ~b|a

λ~aλb θ~a θ b|~a

λ~aλ~b θ~a θ ~b|~a

λa λb θa θb|a+ λa λ~b θa θ~b|a+ λ~aλb θ~a θb|~a+ λ~a λ~b θ~a θ~b|~a

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A B

truefalse

A

θaθ~a

ØA

false

B

θ b|aθ~b|a

A

θ b|~aθ ~b|~a

truetruetrue

false

falsefalse

ØB

falsetrue

λa λb θa θb|a+ λa λ~b θa θ~b|a+ λ~aλb θ~a θb|~a+ λ~a λ~b θ~a θ~b|~a

The Polynomial of a Bayesian The Polynomial of a Bayesian NetworkNetwork

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F = λa λb λc λd θa θb|a θc|a θd|bc +

λa λb λc λ~d θa θb|a θc|a θ~d|bc +

….

A

B

C

D

The Polynomial of a Bayesian The Polynomial of a Bayesian NetworkNetwork

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Arithmetic CircuitArithmetic Circuit λa λb θa θb|a+ λa λ~b θa θ~b|a+ λ~aλb θ~a θb|~a+ λ~a λ~b θ~a θ~b|~a

+

**

* *

λ~b λ~aλbλa

+ +

* * * *

θa θab θa~b θ~ab θ~a~b θ~a

Factoring

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**

* *

λ~b λ~aλbλa

+

+ +

* * * *

θa θab θa~b θ~ab θ~a~b θ~a

1 1 1 0

.3

.3 .1 .9 .8 .2 0

.3 0

1 1

Arithmetic CircuitArithmetic Circuit

.3 .1 .9 .8 .2 .7

Pr(a)

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false

A B

true

false

A

.3

.7

ØABA

.1truetrue.9true false

ØB

.2falsefalse

.8false true

**

* *

λ~b λ~aλbλa

+

+ +

* * * *

θa θb|a θ~b|a θb|~a θ~b|~a θ~a

Factoring

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Factoring the Polynomial of a Factoring the Polynomial of a Bayesian NetworkBayesian Network

S1

T

S2 S3 Sn…

õtòtQ

i=1n (õsiòsijt+õøsiòø sijt)

+õø tòø tQ

i=1n (õsiòsijø t+õøsiòø sijø t)

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Primitive Platforms (embedded)

Embedding Probabilistic Embedding Probabilistic Reasoning SystemsReasoning Systems

Sophisticated Platform(desktop)

compiler

Eval Eval EvalA. Circuit A. Circuit A. Circuit

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TreeWidthTreeWidth(Measures connectivity of Networks)(Measures connectivity of Networks)

Higher treewidth

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TreeWidthTreeWidth(Measures connectivity of Networks)(Measures connectivity of Networks)

Singly-connected network(polytree)

Multiply-connected networks

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TreewidthTreewidth• The treewidth of a polytree is m, where m is

the maximum number of parents that any node

• If each node has at most one parent, the polytree is called a tree

• The treewidth of a tree is 1

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TreewidthTreewidth• Given a Bayesian network NN with:

– Number of nodes: n– Treewith: w

• We can generate an arithmetic circuit for NN:– In O(n 2w) time– In O(n 2w) space

• It is easy to do inference on polytrees