Selected Topics in Graphical Models

Petr Šimeček

Independence Unconditional Independence:

Discrete r.v. Continuous r.v.

Conditional Independence: Discrete r.v.

Continuous r.v.

YX yxyxyx ,)p()p(),p( ..P)f()f(),f( sayxyx

ZYX |

zyx

zyzxzzyx

,,

),p(),p()p(),,p(

..P

),f(),f()f(),,f(

sa

zyzxzzyx

List of Independence RelationshipsN random variables X1, X2, …, XN and their

distribution P

List of all conditional and unconditional independence relations between them

)},(|),(),(

,},...,1{,,);|,{(

CiXBiXAiX

disjunctNCBACBA

iii

Representation by Graph

X6

X5 X4

X3

X2

X1

X1

X3

X2

X4

Example – Sprinkler Network

Rain

WetGrass

Sprinkler

Cloudy

Example – Sprinkler Network

Rain

WetGrass

Sprinkler

CloudyCLOUDY

T F

0.5 0.5

Example – Sprinkler Network

Rain

WetGrass

Sprinkler

Cloudy

SPRINK T F

C=T 0.1 0.9

C=F 0.5 0.5

CLOUDY

T F

0.5 0.5

RAIN T F

C=T 0.8 0.2

C=F 0.2 0.8

Example – Sprinkler Network

Rain

WetGrass

Sprinkler

Cloudy

SPRINK T F

C=T 0.1 0.9

C=F 0.5 0.5

CLOUDY

T F

0.5 0.5

WET GRASS T F

R=T S=T 0.99 0.01

R=T S=F 0.9 0.1

R=F S=T 0.9 0.1

R=F S=F 0 1

RAIN T F

C=T 0.8 0.2

C=F 0.2 0.8

Example – Sprinkler Network

R

W

S

C

),|P().|P().|P().P(),,,P( SRWCSCRCWSRC

The number of parameters needn’t grow

exponentially with the number of

variables!

It depends on the number of parents

of nodes.

Purpose 1– Propagation of Evidence

Rain

WetGrass

Sprinkler

Cloudy

What is the probability that it is raining if we know that grass is wet?

Propagation of EvidenceIn general: I have observed some

variable(s). What is the probability of other variable(s)? What is the most probable value(s)?

Why don’t transfer BN to contingency table? Marginalization does not work for N large: needs 2N memory, much time, has low precision…

Propagation of EvidenceIn general: I have observed some

variable(s). What is the probability of other variable(s)? What is the most probable value(s)?

Why don’t transfer BN to contingency table? Marginalization does not work for N large: needs 2N memory, much time, has low precision…

Purpose 2 – Parameter Learning

Rain

WetGrass

Sprinkler

Cloudy

SPRINK T F

C=T ? ?

C=F ? ?

CLOUDY

T F

? ?

WET GRASS T F

R=T S=T ? ?

R=T S=F ? ?

R=F S=T ? ?

R=F S=F ? ?

RAIN T F

C=T ? ?

C=F ? ?

Parameter LearningWe know: graph (CI structure) sample (observations) of BN

We don’t know: conditional probabilistic distributions

(could be estimated by MLE, Bayesian stat.)

Purpose 3 – Structure Learning

CLOUDY SPRINKLER RAIN WET GRASS

TRUE FALSE FALSE FALSE

FALSE TRUE FALSE FALSE

TRUE FALSE TRUE FALSE

FALSE FALSE FALSE FALSE

FALSE TRUE FALSE FALSE

FALSE FALSE TRUE TRUE

TRUE FALSE TRUE TRUE

TRUE FALSE TRUE FALSE

… … … …

Structure LearningWe know: independent observations (data) of BN sometimes, the casual ordering of vars

We don’t know: graph (CI structure) conditional probabilistic distributions

Solution: CI tests maximization of some criterion – huge s. space

(AIC, BIC, Bayesian approach)

Example – Entry Examination

Markov EquivalenceSome of arcs can be changed without

changing CI relationships.

The best one can hope to do is to identify the model up to Markov equivalence.

RainWet

GrassRain

WetGrass

Structure Learning Theory

algorithms proved to be asymptotically right Janžura, Nielsen (2003)

1 000 000 observations for 10 binary variables

Practice in medicine – usually 50-1500 obs. BNs are often used in spite of that

Structure Learning - Simulation 3 variables, take m from 100 to 1000 for each m do 100 times

generate of Bayesian network generate m samples use K2 structure learning algorithm

count the probability of successful selection for each m

This should give an answer to the question:“Is it a chance to find the true model?”

200 400 600 800 1000

45

50

55

60

65

70

75

Number of Observations

Pro

ba

bili

ty o

f Tru

e M

od

el S

ele

ctio

n (%

)

To Do List:

software: free, open source, easy to use, fast, separated API

more simulation: theory x practice

popularization of structural learning

Czech literature: maybe my PhD. thesis

References: Castillo E. et al. (1997): Expert Systems

and Probabilistic Network Models, Springer Verlag.

Neapolitan R. E. (2003): Learning Bayesian Networks, Prentice Hall.

Janžura N., Nielsen J. (2003): A numerical method for learning Bayesian Networks from Statistical Data, WUPES.