Daphne Koller

Overview''

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

Representa.on'

Daphne Koller

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

Cough

Pneu- monia Flu TB

Bron- chitis

Daphne Koller

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

Daphne Koller

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

Daphne Koller

Context-Specific Independence

Daphne Koller

Tree$Structured$CPDs$

Probabilis3c$Graphical$Models$ Local$Structure$

Representa3on$

Daphne Koller

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

Daphne Koller

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

Daphne Koller

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

Daphne Koller

Multiplexer CPD Z1 Zk Z2 . . .

Y A

Daphne Koller

$$

Microsoft Troubleshooters

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

Thanks to: Eric Horvitz, Microsoft Research

Daphne Koller

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

Daphne Koller

Independence'of'Causal'Influence'

Probabilis4c'Graphical'Models' Local'Structure'

Representa4on'

Daphne Koller

. . . Cough

Pneu- monia Flu TB

Bron- chitis

Daphne Koller

Noisy OR CPD X1

Y

Z1

Xk

Zk

X2

Z2

. . . Z0

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

Xi=1

P(Z0=1) =λ0

Daphne Koller

Independence of Causal Influence

Y

X1

Z

Z1

Xk

Zk

X2

Z2

. . . Z0

Daphne Koller

Sigmoid CPD

Y

X1

Z

Z1

Xk

Zk

X2

Z2

. . .

Daphne Koller

Behavior of Sigmoid CPD

w0 = -5 multiply w and w0 by 10

Daphne Koller

CPCS

# of parameters: 133,931,430 to 8254

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

Daphne Koller

Con$nuous'Variables'

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

Representa$on'

Daphne Koller

Continuous Variables

Temperature Sensor

Temperature’ Door

Outside Temp if D=0

if D=1

Daphne Koller

Y

X1 Xk X2 . . . Linear Gaussian

Daphne Koller

Y

X1 Xk X2 . . . Conditional Linear Gaussian A

Daphne Koller

Robot Localization

Fox, Burgard, Thrun

Daphne Koller

Nonlinear Gaussians

Fox, Burgard, Thrun

Daphne Koller

Robot Motion Model

Fox, Burgard, Thrun