05 probabilistic graphical models

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Bayesian Networks Unit 5 Probabilistic Graphical Models (PGM) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright Wang, Yuan-Kai, 王元凱 [email protected] http://www.ykwang.tw Department of Electrical Engineering, Fu Jen Univ. 輔仁大學電機工程系 2006~2011 Reference this document as: Wang, Yuan-Kai, “Probabilistic Graphical Models," Lecture Notes of Wang, Yuan-Kai, Fu Jen University, Taiwan, 2011.

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Page 1: 05 probabilistic graphical models

Bayesian Networks

Unit 5 Probabilistic Graphical Models (PGM)

Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright

Wang, Yuan-Kai, 王元凱[email protected]

http://www.ykwang.tw

Department of Electrical Engineering, Fu Jen Univ.輔仁大學電機工程系

2006~2011

Reference this document as: Wang, Yuan-Kai, “Probabilistic Graphical Models,"

Lecture Notes of Wang, Yuan-Kai, Fu Jen University, Taiwan, 2011.

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Fu Jen University Department of Electrical Engineering Yuan-Kai Wang Copyright

Bayesian Networks Unit : Probabilistic Graphical Models p. 2

Goal of This Unit• Learn how to

– Build graphical model (network model) by graph theory

– Inference under uncertainty according to probability theory

• Theory of Bayesian networks– Conditional independence– D-Separation– Basic algorithm:

• Variable Elimination• Introduce some BN models

– MRF, HMM, DBN, Naïve Bayes, …

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Bayesian Networks Unit : Probabilistic Graphical Models p.

Related Units• Background

– Statistical inference– Graph theory

• Next units– Exact inference algorithms– Approximate inference algorithms

3

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Bayesian Networks Unit : Probabilistic Graphical Models p. 4

References for Self-Study• Chapter 14, Artificial Intelligence-a modern

approach, 2nd, by S. Russel & P. Norvig, Prentice Hall, 2003

• E. Charniak, Bayesian networks without tears, AI Magazine

• T. A. Stephenson, An introduction to Bayesian network theory and usage, IDIAP research report, IDIAP-RR-00-03, 2000

• B. D’Ambrosio, Inference in Bayesian networks, AI Magazine, 1999

• M. I. Jordan & Y. Weiss, Probabilistic Inference in graphical models,

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Contents

1. Representing Uncertain Knowledge .............. 182. Various PGM Models ..................................... 523. Conditional Independence …………………. 664. Inference .......................................................... 885. Applications on Computer Vision ................. 1366. Summary ……………………………………. 1467. References …………………………………… 152

Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright

5

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Example – Car Diagnosis

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Examples on Computer Vision

Anthropological Measurements

A

LeftHand

Hl

JointsJ

ComponentsC

ObservationsO

ObservationsOij

LeftForearm

Fl

LeftUpper Arm

Ul

HeadH

TorsoT

RightUpper Arm

Ur

RightForearm

Fr

RightHand

Hl

HandSizeSh

HeadSizeShd

TorsoSizeSt

UpperArm Size

Sa

ForearmSize Sf

LeftWrist

Wl

LeftElbow

El

LeftShoulder

Sl

NeckN

RightShoulder

Sr

RightElbow

Er

RightWrist

Wr

HandSizeSh

UpperArm Size

Sa

ForearmSize Sf

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Where do PGMs come from ?• Common problems in real life :

– Complex, Uncertain

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Graph + Probability• Graph has

– Node + Edge• Two kinds of graph

– Directed graph– Undirected graph

• Probability has– Random variable Node– Probability Edge

• Directed graph : conditional probability• Undirected graph: joint probability

X YP(X,Y)

X YP(X|Y)

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Probabilistic Modeling of Problems (1/2)

• Usually node has two semantics– Cause– Effect

• Causal relationships between nodes– Probabilistic– Conditional probability P(Y|X): P(Effect|Cause)– X and Y are not independent– Directed graph

Burglary Earthquake

Alarm

John Calls Mary Calls

P(A|B,E)

P(J|A) P(M|A)

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Probabilistic Modeling of Problems (2/2)

• If node has no causal semantics• But happens together

(influence each other)– Probabilistic– Joint probability P(X,Y)– X and Y are not independent– Undirected graph

Student X

Student Y

P(X,Y)

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Cause/Effect Class/Feature (1/2)• In pattern recognition

/computer vision– Cause class– Effect feature

FaceExpression

EyebrowMotion

MouthMotion

Facial expression image Base image (neutral expression)

P(f2|class)P(f1|class)

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Cause/Effect Class/Feature (2/2)• Face detection:

2-class classificationFace

object

Skin Color

Eyepattern

P(f2|class)P(f1|class)

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Cause/Effect State/Observation• In video analysis

(Tracking)– Cause State– Effect Observation

( | )t tP z x

Detected position : zt

Real position : xt Predicted positionx-

t+1

Reallocation

Observedlocation

Reallocation

Observedlocation

Reallocationxt

xt+1

zt

xt-1

zt-1

P(xt|xt-1)

P(zt-1|xt-1)

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What Are PGMs Good For?

Classification: P(class|feature) Prediction: P(Effect|Cause)=? Diagnosis: P(Cause|Effect)=?

Medicine Bio-informatics

Computer troubleshooting

Stock market

Text Classification

SpeechrecognitionComputer

Vision

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Three Problems in PGM• Representation

– Given a problem– Build its graphical model

(Construction of Bayesian network)• Inference

– Given a set of evidences nodes– Get probabilities of node(s)

• Learning– Learn the CPT of a BN– Learn the graphical structure

of a BN

Reallocation

Observedlocation

Reallocation

Observedlocation

Reallocation

Reallocation

Observedlocation

Reallocation

Observedlocation

Reallocation

xt xt+1

zt

xt-1

zt-1

x z1 32 63 9

P(xt|xt-1)P(zt-1|xt-1)

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Structure of Related Lecture Notes

PGM Representation

Inference

Problem

Learning

Data

Unit 5 : Probabilistic GraphicalUnit 9 : Hybrid BNUnits 10~15: Naïve Bayes, MRF,

HMM, DBN,Kalman filter

Unit 6: Exact inferenceUnit 7: Approximate inferenceUnit 8: Temporal inference

Units 16~ : MLE, EM

StructureLearning

ParameterLearning

B E

A

J M

P(A|B,E)P(J|A)P(M|A)

P(B)P(E)

Query

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1. Representing Uncertain Knowledge

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Review (1/3)

)()()|()|(

ePhPhePehP

Posterior

Prior

Probability of Evidence

Likelihood

• Probability of an hypothesis, h, can be updated when evidence, e, has been obtained

• It is usually not necessary to calculate P(e) directly •As it can be obtained by normalizing the posterior probabilities, P(hi | e)

Bayes’ Theorem

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Review (2/3)Marginalization

Hh

hXPXP ),()(

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Review (3/3)• Full joint probability distribution

– Can answer any question P(X|E=e)P(X|E=e) = hP(X, e, h)

– But become intractably large as the number of variables grows

• Independence and conditional independence among random variables – CPTs = FJD– But can greatly reduce the number of

probabilities that need to specified

FJD

CPT

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A Simple Bayesian Network• 1 FJD = 2 CPTs

– P(Cavity, Toothache)= P(Toothache|Cavity)

* P(Cavity)– P(X,Y)=P(X|Y)P(Y)

=P(Y|X)P(X)• Graphical model

can represent– Causal relationship– Joint relationship

Cavity

Toothache

P(C)0.002

T P(T|C)T 0.70F 0.01

CausalRelationship

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A Burglary Network

Burglary Earthquake

Alarm

John Calls Mary Calls

P(B)0.001

P(E)0.002

B E P(A|B,E)T T 0.95T F 0.95F T 0.29F F 0.001

A P(J|A)T 0.90F 0.05

A P(M|A)T 0.70F 0.01

The graph is directedand acyclic

A conditional probability distribution quantifies the effects of the parents on node

(random) variables

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Compact Representation• If all n nodes have k parents• O(2k n) vs. O(2n) parameters

Burglary Earthquake

Alarm

John Calls Mary Calls

P(B)0.001

P(E)0.002

B E P(A|B,E)T T 0.95T F 0.95F T 0.29F F 0.001

A P(J|A)T 0.90F 0.05

A P(M|A)T 0.70F 0.01

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Formal Definition of a BN• Directed Acyclic Graph (DAG)

–Nodes : Random variables–Edges : Direct influence between 2 variables

• CPTs : Quantifies the dependency of two variables P(X|Parent(X))

–Ex : P(C|A,B), P(D|A)• A priori distribution :

for each node with no parents–Ex : P(A) and P(B)

A

E

D C

B

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Conditional Independence in the Directed Acyclic Graph

• Topology of network encodes dependency/independence

• Weather is independentof the other variables

• Cavity has direct influence on Tooth and Catch

• Toothache and Catchare conditionally independent given Cavity

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Conditional Probability Table (CPT)

P(W)0.001

P(C)0.02

C P(T|C)T 0.90F 0.05

C P(Catch|C)T 0.70F 0.01

P(Xi|Parent(Xi)) or P(Xi|Pa(Xi))

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Causality and Bayesian Networks• Not every BN describes causal relationships

between the variables • Consider the dependence between Lung

Cancer, L, and the X-ray test, X. • A BN with causality

• Another BN represents the same distribution and independencies without causality

L XP(l)=0.001 P(x|l)=0.6P(x|l)=0.02

L X P(x1)=0.02058P(l1|x1)=0.02915P(l1|x2)=0.00041

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Example - Construction of BN (1/3)• I have a burglar alarm installed at

home• I am at work• Neighbor John calls to say my

alarm is ringing • But neighbor Mary doesn't call• Sometimes it's set off by minor

earthquakes • Is there a burglar?

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Example - Construction of BN (2/3)• Step 1: Find Random variables

– Burglar, Earthquake, Alarm, JohnCalls, MaryCalls

• Step 2: Represent the causal relationshipsamong random variables

– A burglar can set the alarm off– An earthquake can set the alarm off– The alarm can cause Mary to call– The alarm can cause John to call

• Step 3: Use network topology withprobability

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Example - Construction of BN (3/3)

Burglary Earthquake

Alarm

John Calls Mary Calls

P(B)0.001

P(E)0.002

B E P(A|B,E)T T 0.95T F 0.95F T 0.29F F 0.001

A P(J|A)T 0.90F 0.05

A P(M|A)T 0.70F 0.01

• 5 Boolean random variables + 5 CPTs

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Marginalization in Bayesian Network

mmMHh

MjaebPhjaebPjaebP,

),,,,(),,,,(),,,(

mmM aaA jjJHh

MJAebPhebPebP, , ,

),,,,(),,(),(

Burglary Earthquake

Alarm

John Calls Mary Calls

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Markov Chain, Conditional Probability, Independence, and Directed Edge

• Markov chain

– L and X are dependent, not independent• Markov chain Has conditional prob. Not independent Has directed edge

L XP(X|L)

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Common CausesSmoking

Bronchitis Lung Cancer

• Markov condition: I(B, L | S), i.e. P(b | l, s) = P(b | s)

• If SB and SL, and “Joe is a smoker”• “Joe has Bronchitis” v.s. “Joe has Lung Cancer” ?• “Joe has Bronchitis” will not give us any more

information about the probability of “Joe has Lung Cancer”

It is a DAG

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Common Effects

Alarm

Burglary Earthquake

• Markov condition: I(B, E), i.e. P(b | e) = P(b)• Burglary and Earthquake are independent of

each other• However they are conditionally dependent given

Alarm• If the alarm has gone off, news that there had

been an earthquake would ‘explain away’ the idea that a burglary had taken place

It is a DAG

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Markov Assumption• Markov chain v.s.

independence• Random variable X

is independent of its non-descendents, given its parents Pa(X)– Formally,

I (X, NonDesc(X) | Pa(X))Non-descendent

Descendent

Ancestor

Parent

X

Y1 Y2

Non-descendent

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Markov Assumption Example

• In this example:– I ( E, B )– I ( B, {E, R} )– I ( R, {A, B, C} | E )– I ( A, R | B,E )– I ( C, {B, E, R} | A)

Earthquake

Radio

Burglary

Alarm

Call

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Joint Probability Distribution

)|(),|()|()|()(),,,,( acPbeaPerPebPePcarbeP

),,,|(),,|(),|()|()(),,,,( arbecPrbeaPberPebPePcarbeP

• Note that our joint distribution with 5 variables can be represented as:

But due to the Markov condition we have, for example,)|(),,,|( acParbecP

The joint probability distribution can be expressed as

• Ex: the probability that someone has a smoking history, lung cancer but not bronchitis, suffers from fatigue and tests positive in an X-ray test is

000135.06.05.0003.075.02.0),,,,( xflbsP

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Representing the Joint Distribution

n

iiin xpaxPxxxP

121 ))(|(),...,,(

• For a BN with nodes X1, X2, …, Xn

• An enormous saving can be made regarding the number of values required for the joint distribution

• For n binary variables•2n – 1 values are required for FJD

• For a BN with n binary variables and •Each node has at most k parents •Less than 2kn values are required for CPTs

FJD n CPTs

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Exercise (1/2)S

G

D

U

HE

),,,,,( ChAeugdsP

)(),|(),|()|()()( u|ChPugAePdsuPsgPdPsP

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Exercise (2/2)• P(a, b, c, d, e)

= P(e | a, b, c, d) P(a, b, c, d)by the product rule

= P(e | c) P(a, b, c, d)by cond. indep. assumption

= P(e | c) P(d | a, b, c) P(a, b, c) = P(e | c) P(d | b, c) P(c | a, b) P(a, b)= P(e | c) P(d | b, c) P(c | a) P(b | a) P(a)

a

b c

d e

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Exercises• Facial Expression Recognition• Face Detection• Face Tracking• Body Segmentation

Using GeNIehttp://genie.sis.pitt.edu/

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Another Example : Water-Sprinkler

WetGrass

Cloudy

Sprinkler Rain

P(C)0.5

S R P(W|S,R)T T 0.99T F 0.9F T 0.9F F 0.0

C P(S|C)T 0.1F 0.5

C P(R|C)T 0.8F 0.2

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Inference in Water-Sprinkler (1/2)• If the grass is wet (WetGrass=True)

– Two possible explanations : rain or sprinkler– Which is the more likely?

)Pr(),Pr()|Pr(

TWTWTSTWTS

)Pr(),Pr()|Pr(

TWTWTRTWTR

The grass is more likely to be wet because of the rain

Sprinkler

Rain

430.06471.02781.0

)Pr(

),,,Pr(,

TW

TWTSRCrc

708.06471.04581.0

)Pr(

),,,Pr(,

TW

TWTRSCsc

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Inference in Water-Sprinkler (2/2)

),,|Pr(),|Pr()|Pr()Pr(),,,Pr( SCRWCRSCRCWSRC 2 x 4 x 8 x 16 = 1024

2 x 4 x 4 x 8 = 256

Time needed for calculations

Using conditional independency properties :

Using Bayes chain rule :

),|Pr()|Pr()|Pr()Pr(),,,Pr( SRWCSCRCWSRC

WetGrass

Cloudy

Sprinkler Rain

P(C)0.5

S R P(W|S,R)T T 0.99T F 0.9F T 0.9F F 0.0

C P(S|C)T 0.1F 0.5

C P(R|C)T 0.8F 0.2

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Inference (1/5)

00.10.20.30.40.50.60.70.80.9

1

00.10.20.30.40.50.60.70.80.9

1

Earthquake Burglary

AlarmRadio

Call

P(E=t|C=t)=0.1 P(B=t|C=t) = 0.7

C=t

E B P(A|E,B)e b 0.9 0.1e b 0.2 0.8e b 0.9 0.1e b 0.01 0.99

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Inference (2/5)

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1

00.10.20.30.40.50.60.70.80.9

1

Earthquake Burglary

AlarmRadio

Call

P(E=t|C=t)=0.1 P(B=t|C=t) = 0.7

C=t

R=t

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Inference (3/5)

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1

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1

00.10.20.30.40.50.60.70.80.9

1

Earthquake Burglary

AlarmRadio

Call

P(E=t|C=t)=0.1 P(B=t|C=t) = 0.7

C=t

R=t

00.10.20.30.40.50.60.70.80.9

1

P(E=t|C=t,R=t)=0.97 P(B=t|C=t,R=t) = 0.1

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Inference (4/5)

00.10.20.30.40.50.60.70.80.9

1

00.10.20.30.40.50.60.70.80.9

1

00.10.20.30.40.50.60.70.80.9

1

Earthquake Burglary

AlarmRadio

Call

P(E=t|C=t)=0.1 P(B=t|C=t) = 0.7

C=t

R=t

00.10.20.30.40.50.60.70.80.9

1

P(E=t|C=t,R=t)=0.97 P(B=t|C=t,R=t) = 0.1

Explaining away effect

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Inference (5/5)

00.10.20.30.40.50.60.70.80.9

1

00.10.20.30.40.50.60.70.80.9

1

00.10.20.30.40.50.60.70.80.9

1

Earthquake Burglary

AlarmRadio

Call

P(E=t|C=t)=0.1 P(B=t|C=t) = 0.7

C=t

R=t

00.10.20.30.40.50.60.70.80.9

1

P(E=t|C=t,R=t)=0.97 P(B=t|C=t,R=t) = 0.1

“Probability theory is nothing but common sense reduced to calculation”– Pierre Simon Laplace

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2. Various PGM Models

Factor Graph

NaïveBayes

Taxonomy

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Directional v.s. Undirectional

Directed ( Bayesian networks)

Undirected( Markov networks)

x1 x2

y1 y2

x1 x2

y1 y2

j

jpaji

ipai ppp )|()|()( )()( xyxxyx, a

aZp )(1)( yx,yx,

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Naive Bayes Model• Strong (Naive) assumption of problems

– A single cause directly influences a number of effects

– All effects are conditionally independent, given the cause

i

i

niCauseEffectPCauseP

EffectEffectCauseP)|()(

),,(

2n+1 probabilities O(n)More details on another unit

n

iiin xpaxPxxxP

121 ))(|(),...,,(

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Naïve Bayesian Classifier (NBC)• Use Naïve Bayes for classification

n

ii

n

n

ClassFeaturePClassP

ClassFeatureFeaturePFeatureFeatureClassP

1

1

1

)|()(

),,(),|(

Class

Feature 1 Feature n

FaceExpression

EyebrowMotion

MouthMotion

Faceobject

Skin Color

Eyepattern

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Temporal Causality Represented by Bayesian Networks

• Temporal Causality– In many systems, data arrives sequentially– Dealing causality with time

• Dynamic Bayes nets (DBNs) can be used to model such time-series (sequence) data

• Special cases of DBNs include– State-space models (Kalman filter)– Hidden Markov models (HMMs)

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State Space Models (SSM)

n

iiin xpaxPxxxP

121 ))(|(),...,,(

t = 1 2 3

n

iiiii

TTTT

TTTT

XPXXPXYPXXP

XYPXXPXYPXXPXYPXPYXPYYXXP

11011

12212111

:1:111

)()|( where,)|()|(

)|()|()|()|()|()(),(),,,,...,(

Y1 Y3

X1 X2 X3

Y2 YT

XT• Hidden Markov Model• Kalman Filter

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DBN (1/2)

Slice 1(DAG)

Slice 2(DAG)

+

Repeat

t = 1 2 3 4 5

More complex temporal models than HMM & Kalman

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DBN (2/2)t = 1 2 3 4 5

n

iiin xpaxPxxxP

121 ))(|(),...,,(

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Bayesian SSM

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Factorial SSM• Multiple hidden sequences• Avoid exponentially large hidden space

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Example: Markov Random Field• Typical application: image region

labelling

yiyi

xixi

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Example: Conditional Random Field

y

y

y

y

xixi

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Markov Random Fields (1/2)Undirected graph

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MRF (2/2)

Local evidence (compatibility with image)Compatibility with neighbors

y

x

Parametertying

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• A Bayesian network/probabilistic graphical model G, represents a set of Markov Independencies P

• There is a factorization theorem

• This section inspects deeper meanings of conditional independence for– The factorization theorem– Inference algorithms in later units

3. Conditional Independencies

i

iin PaXPXXP )|(),...,( 1

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Conditional Independence• Dependencies

– Two connected nodes influence each other

• Independent– Example: I(B;E)

• Conditional Independent– Example

• I(J;M|A)?• I(B;E|A)?

– d-seperation

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D-Separation• It is a rule describing the influences

between nodes

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Serial (Intermediate Cause)

• Indirect causal effect, no evidence

• Clearly burglary will effect Marry call

• Same situation for indirect evidence effect, because independence is symmetric

• If I(E;M|A) then I(M;E|A)

B

A

M

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Diverging (Common Cause)

A

J M

• Influence can flow from John call to Mary call if we don‘t know whether or not there is alarm.

• But I(J;M|A)

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Converging (Common Effect)

E B

A

• Influence can‘t flow from Earthquake to burglary if we don‘t know whether or not there is alarm

• So I(E;B)• Special structure which

cause independence. • V-Structure

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Independence of Two Events

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D-Separation for 3 Nodes

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• Three cases:–Common cause

– Intermediate cause

–Common Effect

Blocked Unblocked

E

R A

E

R A

Path Blockage (1/3)

Blocked Active

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Blocked UnblockedE

C

A

E

C

A

Path Blockage (2/3)

Blocked Active• Three cases:

–Common cause

– Intermediate cause

–Common Effect

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Blocked Unblocked

E B

A

C

E B

A

CE B

A

C

Path Blockage (3/3)

Three cases:– Common cause

– Intermediate cause

– Common Effect

Blocked Active

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General Case

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D-Separation in General• X is d-separated from Y, given Z,

– If all paths from a node in X to a node in Yare blocked, given Z

• Checking d-separation can be done efficiently(linear time in number of edges)– Bottom-up phase:

Mark all nodes whose descendents are in Z– X to Y phase:

Traverse (BFS) all edges on paths from Xto Y and check if they are blocked

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Paths (1/2)• Intuition: dependency must “flow” along

paths in the graph• A path is a sequence of neighboring

variables

Examples:• R E A B• C A E R

Earthquake

Radio

Burglary

Alarm

Call

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Paths (2/2)• For a path between two end nodes X, Y• The path is a

– Active path• If we can find dependency between X & Y

– Blocked path• If we cannot find dependency between X & Y• X & Y are conditional independent• X & Y are D-Separated

• We want to classify situations in which paths are active

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–d-sep(R,B)?

D-Separation Example 1 (1/3)

A

E B

C

R

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D-Separation Example 1 (2/3)

E B

A

C

R

–d-sep(R,B) = yes–d-sep(R,B|A)?

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–d-sep(R,B) = yes–d-sep(R,B|A) = no–d-sep(R,B|E,A)?

D-Separation Example 1 (3/3)

E B

A

C

R

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D-Separation Example 2

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D-Separation Example 3

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d-separation: Car Start Problem• 1. ‘Start’ and ‘Fuel’ are dependent on each other.• 2. ‘Start’ and ‘Clean Spark Plugs’ are dependent on each other.• 3. ‘Fuel’ and ‘Fuel Meter Standing’ are dependent on each other.• 4. ‘Fuel’ and ‘Clean Spark Plugs’ are conditionally dependent on

each other given the value of ‘Start’.• 5. ‘Fuel Meter Standing’ and ‘Start’ are conditionally

independent given the value of ‘Fuel’.

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Exercises

Reallocation

Observedlocation

Reallocation

Observedlocation

Reallocationxt

xt+1

zt

xt-1

zt-1

P(xt|xt-1)

P(zt-1|xt-1)

FaceExpression

EyebrowMotion

MouthMotion

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4. Inference• 4.1 What Is Inference• 4.2 How Inference• 4.3 Inference Methods

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4.1 What Is Inference

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Exercises (1/2)• Face detection Facial Expression Recog.

FaceExpression

EyebrowMotion

MouthMotion

Faceobject

Skin Color

Eyepattern

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Exercises (2/2)• Face tracking

( | )t tP z x

Detected position : zt

Real position : xt Predicted positionx-

t+1

Reallocation

Observedlocation

Reallocation

Observedlocation

Reallocationxt

xt+1

zt

xt-1

zt-1

P(xt|xt-1)

P(zt-1|xt-1)

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3 Kinds of Variables in Inference• Remember the general inference

procedure in previous unit (uncertainty inference unit)

• Let P(X|E=e) be the query–X be the query variable–E be the set of evidence variables

• e be the observed values of E–H be the remaining

unobserved variables(Hidden variables)

V S

LT

A B

X D

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The Burglary Example

Burglary Earthquake

Alarm

John Calls Mary Calls

Query : P(Burglary|John Calls=true)

Query variables: XBurglary

Evidence variables: E=eJohn Calls = true

Hidden variables: HEarthquake, Alarm,Marry Calls

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The Asia Example• Query P(L|v,s,d)

–Query variables: L–Evidence variables:

V=true, S=true, D=true–Hidden variables:

T, X, A, B

V S

LT

A B

X D

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arg max P(X|e)

• arg maxx P(X=x|e) will get a decision

P(X=true | e) = 0.8P(X=false | e) = 0.2

• For P(X | e), if X is a Boolean variable • P(X | e) will compute 2 probabilities

P(X=true | e) = 0.8P(X=false | e) = 0.2

Max X = True

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Five Types of Queries in Inference• For a probabilistic graphical model G• Given a set of evidence E=e• Query the PGM with

– P(e) : Likelihood query– arg max P(e) :

Maximum likelihood query– P(X|e) : Posterior belief query– arg maxx P(X=x|e) : (Single query variable)

Maximum a posterior (MAP) query– arg maxx1…xt

P(X1=x1, …, Xt=xt|e) :Most probable explanation (MPE) queryAlso called Viterbi decoding

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Likelihood Query P(e) (1/2)

X1

E1

X2 Xt

E2 Et

e1

e2

et

e1:t

Input video

P (E1:t=e1:t)

An HMM for Surprise

Probability of Evidence

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Likelihood Query P(e) (2/2)

tXX

ttt XXeEP

1

),,,( 1:1:1

Hh

hHeEP ),(• Marginalization of all hidden variables

X1

E1

X2 Xt

E2 Et

)()|( where,)|()|( 1011

11

XPXXPXEPXXPtXX

n

iiiii

1 2

),,,( 1:1:1X X X

tttt

XXeEP

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Maximum Likelihood Query arg max P(e)X1

E1

X2 Xt

E2 Et

e1

e2

et

e1:t

Input video

P Surprise(e1:t)

An HMM for SurprisePS(Xt|Xt-1),PS(Ei|Xi)

Cry HMM PC(Xt|Xt-1),PC(Ei|Xi)

P Cry(e1:t)Max

X1

E1

X2 Xt

E2 Et

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Maximum Likelihood Query arg max P(e)

• Likelihood query P(E=e))( eEP

Hh

hHeEP ),(

Step 1:Step 2:

Bayes theorem

Marginalizationof all hidden variables

• Query arg max P(E=e)

Hh

hHeEP ),(maxarg

Step 1: Step 2:

Bayes theorem

Marginalizationof all hidden variables

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Posteriori Belief Query P(X|e)• Usually applied on tracking

–Use temporal models of PGM• 4 query types

–Filtering: P(Xt | E1=e1,…, Et=et)=P(Xt |e1:t)–Prediction: P(Xt+1 | e1:t)–Smoothing: P(Xt-k | e1:t)

(Fixed-lag smoothing)X1

E1

X2 Xt

E2 Et

Xt+1

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P(X|e) – Filtering (1/2)• P(Xt | e1:t) X1

E1

X2 Xt

E2 Et

( | )t tP z x

Detected position: ei

Real position: xi Filtered position: x’t

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P(X|e) – Filtering (2/2)• Inference of the query P(Xt|e1:t) is

),()(

),()|(

:1

:1

:1:1

tt

t

tttt

eXPeP

eXPeXP

11

),,( 11:1tXX

ttt XXeXP

)|()|(11 ~1

1 iiXX ti

ii XePXXPt

Step 1:

Step 3:

Step 2:

Bayes theorem

Marginalizationof all hidden variables

Chaining by conditional independence

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P(X|e) – Prediction (1/2)• P(Xt+k | e1:t) for k > 0

X1

E1

X2 Xt

E2 Et

Xt+1For k=1

Detected position : ei

Real position : xi Predicted positionx’

t+1

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P(X|e) – Prediction (2/2)• Inference of the query P(Xt+1|e1:t) is

),()(

),()|(

:11

:1

:11:11

tt

t

tttt

eXPeP

eXPeXP

tXX

ttt XXeXP

1

),,( 1:11

)|()|()|(1 ~1

11 iiXX ti

iitt XePXXPXXPt

Step 1:

Step 3:

Step 2:

Bayes theorem

Marginalizationof all hidden variables

Chaining by conditional independence

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P(X|e) – Smoothing (1/3)• P(Xk | e1:t) for 1 k < t

X1

E1

X2 Xt

E2 Et

Xk

Ek

Detected position: zt

Real position: xt Smoothed position: xt

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P(X|e) – Smoothing (2/3)• Inference of the query P(Xk|e1:t) is

),()(

),()|(

:1

:1

:1:1

tk

t

tktk

eXPeP

eXPeXP

tKk XXXX

tt XXeP,,,

1:1111

),(

)|()|(,,, ~1

1111

iiXXXX ti

ii XePXXPtKk

Step 1:

Step 3:

Step 2:

Bayes theorem

Marginalizationof all hidden variables

Chaining by conditional independence

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P(X|e) – Smoothing (3/3)• Fixed-lag smoothing

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MAP Query (1/2)• arg maxx P(Xi=x|e)• Usually applied on Classification

– Find most likely class X=x, given the evidence e (feature)

FacialExpression

EyebrowMotion

MouthMotion

X={Surprise, Smile, …}

P(X=Surprise|e)

P(X=Smile|e)

If P(X=Smile|e) is the max probabilitySmile = arg maxx P(Xi=x|e)

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MAP Query (2/2)• MAP query arg maxx P(X=x|E=e)

Hhx

hHexXP ),,(maxarg

Step 1:

Step 2:

Bayes theorem

Marginalizationof all hidden variables

),(maxarg)(

),(maxarg

)|(maxarg

exXPeP

exXP

exXP

x

x

x

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MPE Query• Also called Viterbi decoding• arg maxx P(X1=x1,…, Xt=xt|e1:t) • = arg maxx1:t

P(X1:t|e1:t)• = Smoothing for X1:t-1 + Filtering

X1

E1

X2 Xt

E2 Et

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Exercises• Face Detection• Facial Expression Recognition• Face Tracking• Body Segmentation

FacialExpression

EyebrowMotion

MouthMotion

X={Surprise, Smile, …}

Reallocation

Observedlocation

Reallocation

Observedlocation

Reallocationxt

xt+1

zt

xt-1

zt-1

P(xt|xt-1)

P(zt-1|xt-1)

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4.2 How Inference• Inference of the query P(X|E=e) is

),()(

),()|(

eEXPeEP

eEXPeEXP

Hh

hHeEXP ),,(

Hh ni

ii XPaXP~1

))(|(

Step 1:

Step 3:

Step 2:

Bayes theorem

Marginalizationof all hidden variables

Chaining by conditional independence

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The 4th Step of Inference

• Step 4: Compute the sum product?– Need an efficient algorithm– First, we will explain the computation of

the sum-product by an enumeration algorithm• Easy but not efficient

– Then, more efficient methods will be explained in next two units

Hh ni

ii XPaXPeEXP~1

))(|()|( Steps 1 - 3

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The Burglary Example (1/3)• A posterior query on the burglary

network–P(B|j, m)–= P(B, j, m) / P(j, m)–= P(B, j, m)–= e a P(B, e, a, j, m)

This will use the full joint distribution table

E and A are hidden variables

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The Burglary Example (2/3)• Rewrite the full joint entries using

product of CPT entries–P(B|j,m)–= E A P(B, E, A, j, m)–= E A P(j, m, A, B , E)–= E A P(j|m,A,B,E)P(m|A,B,E)

P(A|B,E)P(B|E)P(E) (Chain rule)–= eaP(B)P(e)P(a|B,e)P(j|a)P(m|a)

(Conditional Independence)(All probabilities are CPT entries)

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The Burglary Example (3/3)• P(B|j,m) = < P(b|j,m), P(b|j,m)>

–= EAP(B)P(E)P(A|B,E)P(j|A)P(m|A)– =< EAP(b)P(E)P(A|b,E)P(j|A)P(m|A),EAP(b)P(E)P(A|b,E)P(j|A)P(m|A)>

–= <0.00059224, 0.0014919>– <0.284, 0.716>

• The chance of a burglary, given calls from both neighbors, is about 28%

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Inference by Enumeration• A query P(X|e)

–= h Xi P(Xi | Pa(Xi))• Enumerate all P(Xi | Pa(Xi))• Multiply all P(Xi | Pa(Xi))• Sum all produts

Please refer the Unit - Uncertainty Inference

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Expression Tree of Sum Product

• a + bc a1b1+a2b2

2

1iiiba

+

* acb

+

*

b1a1

*

b2a2

?1

n

iiiba

Hh ni

ii XPaXPeEXP~1

))(|()|(

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Problem of Sum Product :Repeated Computation

• a1b + a2b • a1b1+a1b2+a1b3+a2b1+a2b2+a2b3

2

1

3

1i jjiba

2

1iiba

+

*

ba1

*

ba2

*

+

a2a1

b

2

1

3

1i jji ba

+ +

*

a2a1

b1 b2

b3+

= (a1+a2) *(b1+b2+b3)

b3

+

*

b1a1

*

b2a2

*

b3a3

+*

b1a1

*

b2a2

*

a3

++ +

2

1iiab

= (a1+a2)b

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The Burglary Example (1/3)• Rewrite the full joint entries using

product of CPT entries–P(B|j,m)–= EAP(B)P(E)P(A|B,E)P(j|A)P(m|A)–= P(B) EP(E) AP(A|B,e)P(j|A)P(m|A)

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The Burglary Example (2/3)• P(b|j,m)= EAP(b)P(E)P(A|b,E)P(j|A)P(m|A)

+

**

P(j|a)P(m|a)

*

P(e)P(a|b,e)

P(b)

*A=a

**

P(j|a)P(m|a)

*

P(e)P(a|b,e)

P(b)

*A= a

E=e+

P(b)

**

P(j|a)P(m|a)

*

P(e)P(a|b,e)

*A=a

**

P(j|a)P(m|a)

*

P(e)P(a|b,e)

P(b)

*A= a

E= e+

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The Burglary Example (3/3)• P(b|j,m)= P(b)EP(E)AP(A|b,E)P(j|A)P(m|A)

**

P(j|a)P(m|a)

P(a|b,e)

E=eA=a

P(b)

**

P(j|a)P(m|a)

P(a|b,e)

E=eA= a *

*

P(j|a)P(m|a)

P(a|b,e)

E=eA=a *

*

P(j|a)P(m|a)

P(a|b,e)

E=eA= a

P(e)+

*

P(e)+

*

+*

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Enumeration as Depth-First Search• Recursive depth-first enumeration

–Suppose n Boolean variables–O(n) space–O(2n) time

• Enumeration is inefficient–Repeated computation–e.g., computes P(j|a)P(m|a) for each

value of e

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Enumeration Algorithm (1/2)

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Enumeration Algorithm (2/2)

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Exercise: Enumeration

smart study

prepared fair

pass

p(smart)=.8 p(study)=.6

p(fair)=.9

p(prep|…) smart smartstudy .9 .7study .5 .1

p(pass|…)smart smart

prep prep prep prepfair .9 .7 .7 .2fair .1 .1 .1 .1

Query: What is the probability that a student studied, given that they pass the exam?

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Complexity of InferenceTheorem:

Computing queries in a Bayesian network is NP-hard

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Hardness• Hardness does not mean we cannot

solve inference– It implies that we cannot find a general

procedure that works efficiently for all networks

– For particular families of networks, we can have provably efficient procedures

– We will characterize such families in the next two classes

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4.3 Inference Methods• Steps of inference• Step 1: Bayesian theorem• Step 2: Marginalization• Step 3: Conditional independence• Step 4: Sum product computation

–Enumeration–Exact inference–Approximate inference

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Exact v.s. Approximate Inference• Exact inference

– Get exact probability of the query• Approximate inference

– Get approximate probability of the query– Avoid exponential complexity of exact

inference in discrete loopy graphs– Because cannot compute messages in

closed form (even for trees) in the non-linear/non-Gaussian case

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Exact Inference• Enumeration• Variable Elimination• Belief Propagation

– Message Passing• Junction Tree

– Clustering, Join tree

Unit –Exact Inference

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Approximate Inference• Stochastic Simulation

– Monte Carlo method– Sampling method– Include: direct sampling (logic sampling),

likelihood weighting sampling• Markov Chain Monte Carlo Sampling

(MCMC)• Loopy Belief Propagation

Unit – Approximate Inference

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Software Implemented Inference Methods

PNL GeNIe/SmileEnumeration v (Naïve)Variable Elimination

Junction Tree v v (Clustering)

Belief Propagation v (Pearl) v (Polytree)Direct Sampling v (Logic)

Likelihood Sampling v(LWSampling) v(Likelihood sampling)MCMC Sampling v(Gibbswithanneal) (Other 5 sampleings)

Loopy Belief Propagation

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Exercise• Use GeNIe (http://genie.sis.pitt.edu/)

to – Generate a PGM– Inference the PGM

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5. Applications on CV• Face Recognition• Human Body Tracking• Super-resolution

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Face Modeling and Recognition Using Bayesian NetworksGang Song*, Tao Wang, Yimin Zhang, Wei Hu, Guangyou Xu*, Gary Bradski

Face feature finder (separate)

Learn Gabor filter “jet” at each point

System:

Add Pose switching variable

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Face Modeling and Recognition Using Bayesian NetworksGang Song*, Tao Wang, Yimin Zhang, Wei Hu, Guangyou Xu*, Gary Bradski

Results:

Results:

BNPFR – Bayesnet with PoseBNFR – Bayesnet w/o PoseEHMM – Embedded HMMEGM – Gabor jets

Pose

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Looking for all possible joint configuration J is computationally impractical. Therefore, segmentation takes place in two stages. First, we segment the head and torso, and determine the position of the neck. Then, we jointly segment the upper arms, forearms and hands, and determine the position of the remaining joints.

The Segmentation Problem

AHT ji

Aijijji

HTijijF qPqPPQQQJ,QJ,

JOJOQJO,,

),|(),|(maxarg),|(maxarg

HTA QQ , state assignments for the arm and head&torso regionsHTA JJ , joints for the arms and head&torso components.

Step I Step II

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Upper Body Model

LeftHand

Hl

JointsJ

ComponentsC

ObservationsOObservations

Oij

F A J CF

F AAJAJAJOji

ijijq

ijijijji

ij uPPqPqOPuOPPij,,

})()|(),|(),,|({})({)(

Anthropological Measurements

A

LeftForearm

Fl

LeftUpper Arm

Ul

HeadH

TorsoT

RightUpper Arm

Ur

RightForearm

Fr

RightHand

Hl

HandSizeSh

HeadSizeShd

TorsoSizeSt

UpperArm Size

Sa

ForearmSize Sf

LeftWrist

Wl

LeftElbow

El

LeftShoulder

Sl

NeckN

RightShoulder

Sr

RightElbow

Er

RightWrist

Wr

HandSizeSh

UpperArm Size

Sa

ForearmSize Sf

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Body Tracking Results

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MRF (1/3)Undirected graph

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MRF (2/3)

Local evidence (compatibility with image)Compatibility with neighbors

y

x

Parametertying

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MRF (3/3)Image patches

(xi, yi)

(xi, xj)

image

scene

Scenepatches

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MRFs for Super-Resolution

ActualCubic SplineInput Bayesian Net

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6. Summary• What we know the representation

problem–What is a Bayesian network–A little about “how to inference”

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Probabilistic Graphical Models• Given a PGM = given a joint

probability function• We can immediately write down

– Its joint probability, and– Its equivalent conditional probability

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Directed Graph

),,( 1 ClassFeatureFeatureP n Class

Feature 1 Feature n

n

ii ClassFeaturePClassP

1

)|()(

=

X1

E1

X2 Xt

E2 Et

),,,,( 11 tt EEXXP

t

itttt XEPXXPXEPXP

21111 )|()|()|()(

=

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Undirected Graph

),,,,(),(

11 NN YYXXPYXP

Ni

iiiNeighborj

ji YXXXP~1)(

),(),(

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All Queries have the same Form

n

ii

n

n

ClassFeaturePClassP

ClassFeatureFeaturePFeatureFeatureClassP

1

1

1

)|()(

),,(),|(

Class

Feature 1 Feature n

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We Still Need to Know• Inference

–Details of inference algorithms• Learning

–How to learn CPTs–How to build or automatically learn

the structure of a Bayesian network by given a set of learning data

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7. References• A Brief Introduction to Graphical Models and

Bayesian Networks (Kevin Murph, 1998)– http://www.cs.ubc.ca/~murphyk/Bayes/bnintro.html

• Artificial Intelligence I (Dr. Dennis Bahler)– http://www.csc.ncsu.edu/faculty/bahler/courses/csc520f02/bayes1.html

• Nir Friedman– http://www.cs.huji.ac.il/~nir/

• Judea Pearl, Causality (on-line book)– http://bayes.cs.ucla.edu/BOOK-2K/index.html

• Introduction to Bayesian Networks– A tutorial for the 66th MORS symposium– Dennis M. Buede, Joseph A. Tatmam, Terry A.

Bresnick

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References• An introduction to Bayesian network theory

and usage, T. A. Stephenson, IDIAP Research report IDIAP-RR 00-03, Feb. 2000. [Available: http://www.rpi.edu/~liaow/file/Intro_BN.pdf]

• Bayesian network without tears, E. Charniak, AI Magazine, 1991. [Available: http://www.rpi.edu/~liaow/file/BNtears.pdf]

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References• Inference in belief Networks : A procedural

guide (Cecile Huang)• Tutorial on graphical models and BNT

– presented to the Mathworks, May 2003 • Java Applets : Prof R.D. Boyle

([email protected])• HMMs – Summary (L R Rabiner and B H Juang )

– http://www.comp.leeds.ac.uk/roger/HiddenMarkovModels/html_dev/summary/s1_pg2.html