Graphical' Models' Local'Structure' Overview'spark-university.s3.amazonaws.com/stanford-pgm/slides/...CPDs.pdf · Daphne Koller General CPD • CPD P(X | Y 1, …, Y k) specifies

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Overview''

Probabilis.c'Graphical'Models' Local'Structure'

Representa.on'

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Tabular Representations

0.3 0.08 0.25

0.4 g2

0.02 0.9 i1,d0 0.7 0.05 i0,d1

0.5

0.3 g1 g3

0.2 i1,d1

0.3 i0,d0

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Pneu- monia Flu TB

Bron- chitis

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General CPD •  CPD P(X | Y1, …, Yk) specifies distribution

over X for each assignment y1, …, yk •  Can use any function to specify a factor φ(X, Y1, …, Yk) such that

∑x φ(x, y1, …, yk) = 1 for all y1, …, yk

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Many Models •  Deterministic CPDs •  Tree-structured CPDs •  Logistic CPDs & generalizations •  Noisy OR / AND •  Linear Gaussians & generalizations

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Context-Specific Independence

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Tree$Structured$CPDs$

Probabilis3c$Graphical$Models$ Local$Structure$

Representa3on$

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A

S

L

(0.8,0.2)

(0.9,0.1) (0.4,0.6)

(0.1,0.9)

s1

a0 a1

s0

l1 l0 Letter SAT

Job

Apply

Tree CPD

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Letter1 Letter2

Job

Choice

Tree CPD C c1 c2

L2

(0.8,0.2) (0.1,0.9)

l1 l0 L1

(0.9,0.1) (0.3,0.7)

l1 l0

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C c1 c2

L

(0.8,0.2) (0.1,0.9)

l1 l0 L

(0.9,0.1) (0.3,0.7)

l1 l0

Letter1 Letter2

Job

Choice

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Multiplexer CPD Z1 Zk Z2 . . .

Y A

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$$

Microsoft Troubleshooters

#$of$parameters:$145$to$55$

Thanks to: Eric Horvitz, Microsoft Research

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Summary •  Compact CPD representation that

captures context-specific dependencies •  Relevant in multiple applications: – Hardware configuration variables – Medical settings – Dependence on agent’s action – Perceptual ambiguity

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Independence'of'Causal'Influence'

Probabilis4c'Graphical'Models' Local'Structure'

Representa4on'

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. . . Cough

Pneu- monia Flu TB

Bron- chitis

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Noisy OR CPD X1

Y

Z1

Xk

Zk

X2

Z2

. . . Z0

P(Zi=1|Xi) = Xi=0

Xi=1

P(Z0=1) =λ0

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Independence of Causal Influence

Y

X1

Z

Z1

Xk

Zk

X2

Z2

. . . Z0

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Sigmoid CPD

Y

X1

Z

Z1

Xk

Zk

X2

Z2

. . .

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Behavior of Sigmoid CPD

w0 = -5 multiply w and w0 by 10

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CPCS

# of parameters: 133,931,430 to 8254

M. Pradhan G. Provan B. Middleton M. Henrion UAI 1994

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Con$nuous'Variables'

Probabilis$c'Graphical'Models' Local'Structure'

Representa$on'

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Continuous Variables

Temperature Sensor

Temperature’ Door

Outside Temp if D=0

if D=1

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Y

X1 Xk X2 . . . Linear Gaussian

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Y

X1 Xk X2 . . . Conditional Linear Gaussian A

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Robot Localization

Fox, Burgard, Thrun

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Nonlinear Gaussians

Fox, Burgard, Thrun

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Robot Motion Model

Fox, Burgard, Thrun