Probabilistic Relational Models: A Tutorial Lise Getoor University of Maryland, College Park May 4, 2005

Probabilistic Relational Models: A Tutorial

Lise GetoorUniversity of Maryland, College

ParkMay 4, 2005

PRMs

• Developed by Daphne Koller’s group at Stanford– representation: Avi Pfeffer

• builds on work in KBMC (knowledge-based model construction) by Haddawy, Poole, Wellman and others…

• Object Oriented Bayesian Networks • Relational Probability Models

– learning: myself, Nir Friedman, Avi, Ben• Attribute Uncertainty• Structural Uncertainty• Class Uncertainty• Identity Uncertainty

– undirected models: Ben Taskar, Eran Segal

Motivation: Discovering Patterns in Structured Data

Patient

Treatment

Strain Contact

Learning Statistical Models

Traditional approaches– work well with flat representations– fixed length attribute-value vectors – assume independent (IID) sample

Patient

flatten

Problems:– introduces statistical skew– loses relational structure

• incapable of detecting link-based patterns

– must fix attributes in advance

Contact

Roadmap

• Background: » Bayesian Networks (BNs) [Pearl, 1988]– Probabilistic Relational Models (PRMs)

• Learning PRMs w/ Attribute Uncertainty

• PRMs w/ Structural Uncertainty

• PRMs w/ Class Hierarchies

Bayesian Networks

nodes = random variablesedges = direct probabilistic

influence

Network structure encodes independence assumptions: XRay conditionally independent of Pneumonia given Infiltrates

XRay

Lung Infiltrates

Sputum Smear

TuberculosisPneumonia

Bayesian Networks

XRay

Lung Infiltrates

Sputum Smear

TuberculosisPneumonia

• Associated with each node Xi there is a conditional probability distribution P(Xi|Pai:) — distribution over Xi for each assignment to parents

– If variables are discrete, P is usually multinomial– P can be linear Gaussian, mixture of Gaussians, …

0.8 0.2

p

t

p

0.6 0.4

0.010.99

0.2 0.8

tp

t

t

p

TP P(I |P, T )

BN Semantics

• Compact & natural representation:– nodes have k parents 2k n vs. 2n params

conditionalindependenciesin BN structure

+local

probabilitymodels

full jointdistribution

over domain=

t)|sP(i)|P(xt),p|P(iP(t))pP()sx,i,t,,pP(

X

I

S

TP

Roadmap

• Background: • Bayesian Networks (BNs)» Probabilistic Relational Models (PRMs)

• Learning PRMs w/ Attribute Uncertainty



Probabilistic Relational Models

• Combine advantages of relational logic & Bayesian networks: – natural domain modeling: objects, properties,

relations;– generalization over a variety of situations;– compact, natural probability models.

• Integrate uncertainty with relational model:– properties of domain entities can depend on

properties of related entities;– uncertainty over relational structure of domain.

Relational Schema

Strain

Unique

Infectivity

Infected with

Interacted with

• Describes the types of objects and relations in the database

ClassesClasses

RelationshipsRelationshipsContact

Close-Contact

Skin-Test

Age

Patient

Homeless

HIV-Result

Ethnicity

Disease-Site AttributesAttributes

Contact-Type

Probabilistic Relational Model

Close-Contact

Transmitted

Contact-Type

Disease Site

Strain

Unique

Infectivity

Patient

Homeless

HIV-Result

POB

Contact Age

Cont.Contactor.HIVCont.Close-Contact

Cont.Transmitted |

P

4.06.0

3.07.0

2.08.0

1.09.0

,

,

,

,,

tt

ft

tf

ff

P(T | H, C)CH

Relational Skeleton

Fixed relational skeleton – set of objects in each class– relations between them

Uncertainty over assignment of values to attributes

PRM defines distribution over instantiations of attributes

Strains1

Patientp2

Patientp1

Contactc3

Contactc2

Contactc1

Strains2

Patientp3

A Portion of the BN

P1.Disease Site

P1.Homeless

P1.HIV-Result

P1.POB

C1.Close-Contact

C1.Transmitted

C1.Contact-Type

C1.Age

C2.Close-Contact

C2.Transmitted

C2.Contact-Type

truefalse

true

4.06.0

3.07.0

2.08.0

1.09.0

,

,

,

,,

tt

ft

tf

ff

P(T | H, C)CH

4.06.0

3.07.0

2.08.0

1.09.0

,

,

,

,,

tt

ft

tf

ff

P(T | H, C)CH

C2.Age

PRM: Aggregate Dependencies

sum, min, max, avg, mode, count

Disease Site

Patient

Homeless

HIV-Result

POB

Age

Close-Contact

Transmitted

Contact-Type

Contact

Age

.

.

PatientJane Doe

POB US

Homeless no

HIV-Result negative

Age ???

Disease Site pulmonary

A

.

Contact#5077

Contact-Typecoworker

Close-Contact no

Agemiddle-aged

Transmitted false

Contact#5076

Contact-Typespouse

Close-Contact yes

Agemiddle-aged

Transmitted true

Contact#5075

Contact-Typefriend

Close-Contact no

Agemiddle-aged

Transmitted false

mode

6.03.01.0

2.06.02.0

2.04.04.0

o

m

yomym

PRM with AU Semantics

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

AxparentsAxPP Sx Ax

I

AttributesObjects

probability distribution over completions I:

PRM relational skeleton + =

Strain

Patient

Contact

Strain s1

Patient p1

Patient p2

Contactc3

Contactc2

Contactc1

Strain s2

Patient p3

Learning PRMs w/ AU

Database Patient

Strain

Contact

Relational

Schema

PatientContact

Strain

• Parameter estimation• Structure selection

Parameter Estimation in PRMs• Assume known dependency structure S• Goal: estimate PRM parameters

– entries in local probability models,

• is good if it is likely to generate the observed data, instance I .

• MLE Principle: Choose so as to maximize l

),|(log),:( SPSl II

).(|. AxparentsAx

As in Bayesian network learning,

crucial property: decomposition

separate terms for different X.A

ML Parameter Estimation

ContactCloseContact

Transmitted

PatientHIV

DiseaseSite

Count

Query for counts:

Patienttable

Contacttable

ctCloseContaC

HIVP

dTransmitteC

.

.

.

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

tCCfHPNtCCfHPfTCN

P

??

??

??

??

,

,

,

,,

tt

ft

tf

ff

P(T | H, C)CH

Cont.Contactor.HIVCont.Close-Contact

Cont.Transmitted |

P

Structure Selection

• Idea: – define scoring function – do local search over legal structures

• Key Components:– legal models – scoring models– searching model space

Structure Selection


• Key Components:» legal models– scoring models– searching model space

Legal Models

author-of

• PRM defines a coherent probability model over a skeleton if the dependencies between object attributes is acyclic

How do we guarantee that a PRM is acyclic for every skeleton?

ResearcherProf. Gump

Reputationhigh

PaperP1

Accepted yes Paper

P2Accepted

yes

sum

Attribute Stratification

PRMdependency structure S

dependencygraph

Paper.Accecpted

Researcher.Reputation

if Researcher.Reputation depends directly on Paper.Accepted

dependency graph acyclic acyclic for any Attribute stratification:

Algorithm more flexible; allows certain cycles along guaranteed acyclic relations

Structure Selection


• Key Components:– legal models» scoring models– searching model space

Scoring Models

• Bayesian approach:

])()|(log[)|(log):(

priorlikelihoodmarginal

SPSPSPSScore

III

• Standard approach to scoring models; used in Bayesian network learning

Structure Selection


• Key Components:– legal models – scoring models» searching model space

Searching Model Space

Contact

Strain Patient

score

Delete C.CC.T Contact

Strain Patient

scoreAdd S.IS.U

Strain Contact

Patient

Phase 0: consider only dependencies within a class

Contact

Strain Patient scoreAdd S.IP.D

score

Add P.HC.TContact

Strain Patient

Contact

PatientStrain

Phase 1: consider dependencies from “neighboring” classes, via schema relations

Phased Structure Search

Phased Structure Search

scoreAdd S.IC.T

score

Add C.PS.I

Phase 2: consider dependencies from “further” classes, via relation chains

Contact

Strain Patient

Contact

Strain Patient

Contact

Strain Patient

Experimental Evaluation

Synthetic Data

• Simple ‘genetic’ domain• Construct training set of various sizes• Compare the log-likelihood of test set of

size 100,000– ‘gold’ standard model– Learn parameters (model structure given)– Learn model (learn both structure and

parameters)

Blood Type

M-chromosome

P-chromosome Person

Result

Contaminated

Blood Test

Blood Type

M-chromosome

P-chromosome

Person Blood Type

M-chromosome

P-chromosome

Person

(Father)

(Mother)

Error on Test Set

-3

-2.5

-2

-1.5

-1

-0.5

0

0 1000 2000 3000 4000

Dataset Size

Av

g L

og

-Lik

elih

oo

d

Gold

Learned Parameters

Learned Models

Error Variance

0

0.5

1

1.5

2

2.5

0 1000 2000 3000 4000

Dataset Size

Av

g E

rro

rLearned Parameters

Learned Models

Errors in Learned Structure

0

2

4

6

8

10

12

500 1300 1800 2500 3000 3800 4300

Dataset Size

Nu

mb

er

of

Le

arn

ed M

od

els

too simple

correct

too complex

TB Cases in SF

Patient (2300)Ethnicity

Homeless

Age @ diagnosis

HIV result

Disease-site

X-ray

Contact (20000)Contact-type

Age

Care

Infected

Strain (1000)

Unique

Drug-Resistance

hivres

# contacts

result

transmitted

infectivity

smrpos

care

closecont

ageatdx

closecont

hh_oohh

ethnic

# infected

% infected

hh_oohh

contype

homeless

gender

contype

disease site

contage

xray

pob

ContactStrain

Subcase

Patient

TB PRM

total assets

# roles

rtn earn assets

age

rtn assets

fired

# employees

top_roletop_role

total_assets

retired retired

salary salary

Company

Role

Prev-Role

Person

SEC PRM

20,000

120,000

40,000

Roadmap

• Motivation and Background

• PRMs w/ Attribute Uncertainty

» PRMs w/ Structural Uncertainty


An Example

Topic

Theory AI

Agent

Theory papers

Cornell

Scientific Paper

Topic

Theory AI

•Attributes of object•Attributes of linked objects

•Attributes of heterogeneous linked objects•Collective Classification

Structural Uncertainty

• Motivation: relational structure provides useful information for density estimation and prediction

• Construct probabilistic models of relational structure that capture structural uncertainty

• Two new mechanisms:– Reference uncertainty– Existence uncertainty

PRMs w/ AU: another example

Vote

Rank

Movie

Income

Gender

Person

AgeGenre

PRM consists of:

Relational Schema

Dependency Structure

Vote.Person.Gender,Vote.Person.Age

Vote.Movie.Genre,Vote.Rank |

P

Local Probability Models

Fixed relational skeleton :– set of objects in each class– relations between them

Movie m1

Vote v1 Movie: m1 Person: p1

Person p2

Person p1

Movie m2

Uncertainty over assignment of values to attributes

PRM w/ Attribute Uncertainty



Primary Keys

Foreign Keys

PRM w/ AU Semantics

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

AxparentsAxPP Sx Ax

I

AttributesObjects

Ground BN defining distribution over complete instantiations of attributes I:

PRM relational skeleton + =

Patient p2

Vote

Movie Person Movie

Vote

Vote

Person

Person

Movie

Vote

Issue

• PRM w/ AU applicable only in domains where we have full knowledge of the relational structure

Next we introduce PRMs which allow uncertainty over relational structure…

PRMs w/ Structural Uncertainty

Advantages:– Applicable in cases where we do not have full

knowledge of relational structure– Incorporating uncertainty over relational structure

into probabilistic model can improve predictive accuracy

Two approaches:– Reference uncertainty– Existence uncertainty

• Different probabilistic models; varying amount of background knowledge required for each

Citation Relational Schema

Wrote

PaperTopic

Word1

WordN

…Word2

PaperTopic

Word1

WordN

…Word2Cites

CountCiting Paper

Cited Paper

AuthorInstitution

Research Area

Attribute Uncertainty

Paper

Word1

Topic

WordN

Wrote

Author

...

Research Area

P( WordN | Topic)

P( Topic | Paper.Author.Research Area

Institution P( Institution | Research Area)

Reference Uncertainty

Bibliography

Scientific Paper

`1. -----2. -----3. -----

???

Document Collection

PRM w/ Reference Uncertainty

CitesCitingCited

Dependency model for foreign keys

PaperTopicWords

PaperTopicWords

Naïve Approach: multinomial over primary key• noncompact• limits ability to generalize

Reference Uncertainty Example

PaperP5

Topic AI

PaperP4

Topic AI

PaperP3

Topic AI

PaperM2

Topic AI

Paper P1Topic Theory

CitesCitingCited

Paper P5Topic AI

PaperP3

Topic AI




Paper.Topic = AIPaper.Topic = Theory

P1

P2

PaperTopicWords P1 P2

3.0 7.0

P1 P2

1.0 9.0

Topic

99.0 01.0 Theory

AI

PRMs w/ RU Semantics

PRM-RU + entity skeleton

probability distribution over full instantiations I

Cites

Cited

Citing

PaperTopic

Words

PaperTopic

Words

PRM RU

Paper P5Topic AI



Paper P3Topic AI

Paper P1Topic ???

Paper P5Topic AI



Paper P3Topic AI

Paper P1Topic ???

RegReg

RegRegCites

entity skeleton

Structure Search: New Operators

CitesCitingCited

PaperTopicWords

PaperTopicWords

Cited

Papers

1.0

Paper Paper

Paper Paper

Paper Paper

Paper Paper

Paper Paper

Paper

Topic = AI

ΔscoreRefine on Topic

Paper Paper

Paper Paper

Paper

Paper Paper

Paper Paper

Paper

Paper Paper

Paper Paper

Paper

Paper

Paper Paper

Δscore

Refine on Author.Instition

AuthorInstitution

Institution = MIT

PRMs w/ RU Summary

• Define semantics for uncertainty over foreign-key values

• Search now includes operators Refine and Abstract for constructing foreign-key dependency model

• Provides one simple mechanism for link uncertainty

Existence Uncertainty

Document CollectionDocument Collection

? ??

PRM w/ Exists Uncertainty

Cites

Dependency model for existence of relationship

PaperTopicWords

PaperTopicWords

Exists

Exists Uncertainty Example

Cites

PaperTopicWords

PaperTopicWords

Exists

Citer.Topic Cited.Topic

0.995 0005 Theory Theory

False True

AI Theory 0.999 0001

AI AI 0.993 0008

AI Theory 0.997 0003

PRMs w/ EU Semantics

PRM-EU + object skeleton

probability distribution over full instantiations I

Paper P5Topic AI



Paper P3Topic AI

Paper P1Topic ???

Paper P5Topic AI



Paper P3Topic AI

Paper P1Topic ???

object skeleton

???

PRM EU

Cites

Exists

PaperTopic

Words

PaperTopic

Words

Learning PRMs w/ EU

• Idea: just like in PRMs w/ AU– define scoring function – do greedy local structure search

• Issues:– efficiency

•Computation of sufficient statistics for exists attribute

•Do not explicitly consider relations that do not exist

Experiment I: EachMovie+

thriller

action

horror

gender

theater_status gendervideo_status

age

animationart_foreign

classic

personal_income

comedy

drama

rankhousehold_income

family

romance Movie

Person

Movie

Actor

MOVIE

ROLE

VOTEPERSO

N

ACTOR

education

* © 1999 -2000 Internet Movie Database Limited† http://www.research.digital.com/SRC/EachMovie

Size: 1600

Size: 35,000Size: 50,000

Size: 25,000Size: 300,000

*

†

EachMovie+ PRM-RU

thriller

actionhorror

gender

theater_status

gender

video_status

ageanimation

art_foreign

classic

personal_income

comedy

dramarank

household_incomefamily

romanceMovie

Person

Movie

Actor

MOVIE

ROLE

VOTE PERSON

ACTOR

education

M F

8.0 2.0

Action

7.0 3.0true

false

Typical Voter: male, young adult, college w/o degree, middle income

EachMovie+ PRM-EU

agecomedy

drama rank

gender

family

personal_income

horror

romance

exists

household_income

thriller

exists

gendertheater_status

video_status

action

education

animation

art_foreign

classic

MOVIE

ROLE

VOTEPERSO

N

ACTOR

+

-

Men much more likely to vote on action movies

Experiment II: Prediction

Paper P506

Paper P516Topic Reinforcement LearningWords

…Paper P1309Topic Probabilistic ReasoningWords

…Paper P289Topic Reinforcement LearningWords

…

Cited Papers

Paper P134Topic Reinforcement LearningWords

…Paper P1067Topic Reinforcement LearningWords

…

Citing Papers

Topic ??

w1 wN. . .

Domains

Cites

Exists

PaperTopic

w1 wN. . .

PaperTopic

w1 wN. . .

cited paper citing paper

Cora Dataset, McCallum, et. al

Link

Exists

Web PageCategory

w1 wN. . .

Category

w1 wN. . .

From Page To Page

Web Page

WebKB, Craven, et. al

Prediction Accuracy

Naïve-bayesRU Citing RU Cited ExistsCora 0.75 0.81 0.79 0.85WebKB 0.74 0.78 0.77 0.82

0.65

0.7

0.75

0.8

0.85

0.9

Cora WebKB

Acc

ura

cy

Naive-Bayes

RU Citing

RU Cited

Exists

Experiment III: Collective Classification

Paper#2 Topic Paper#3Topic

WordN

Paper#1Word1

Topic... ... ...

Author#1

Area Inst

#1-#2

Author#2

Area Inst

Exists

#2-#3

Exists

#2-#1

Exists

#3-#1

Exists

#1-#3

Exists

WordN

Word1WordN

Word1

Exists

WordNWord1

WordN

Word1WordN

Word1

ExistsExists Exists ExistsExists Exists

Inst Inst

TopicTopicTopic

Area Area

TopicTopicTopic

Area Area

Topic TopicTopic

Area Area

#3-#2

Inference in Unrolled BN

• Prediction requires inference in “unrolled” network– Infeasible for large networks– Use approximate inference for E-step

• Loopy belief propagation (Pearl, 88; McEliece, 98)– Scales linearly with size of network– Guaranteed to converge only for polytrees– Empirically, often converges in general nets

(Murphy,99)

• Local message passing– Belief messages transferred between related instances– Induces a natural “influence” propagation behavior

• Instances give information about related instances

...

From-Page Category

Word1 WordN

Exists

From

To

Link

Hub

To-Page

Word

AnchorHas

...

Category

Word1 WordN

Hub

Web Domain

WebKB Results*

0.54

0.56

0.58

0.6

0.62

0.64

0.66

0.68

0.7

cornell texas wisconsin washington

School

Acc

ura

cy

Naive-Bayes

Exists

Ex+Hubs+Anchors

* from “Probabilistic Models of Text and Link Structure for Hypertext Classification”, Getoor, Segal, Taskar and Koller in IJCAI 01 Workshop Text Learning: Beyond Classification

Roadmap




» PRMs w/ Class Hierarchies

From Instances to Classes in Probabilistic Relational Models

• Compare two approaches – Probabilistic Relational Models (PRMs)– Bayesian Network (BNs)

• PRMs with Class Hierarchies (PRM-CH)– bridge gap between BNs and PRMs

• Learning PRM-CHs– hierarchy supplied– discovering hierarchy

VoteProgram

Voter

Ranking

PRM for Collaborative Filtering

VoteProgram

Voter

Ranking IncomeIncomeIncome

1.06.03.0

5.04.01.0

4.05.01.0

1.04.05.0

bs

hssitcom

bsdoc

hsdoc

hmlEG

sitcom

+ Dependency Model

TV-ProgramGenre

Budget

Time-slot

Network

TV-ProgramGenre

Budget

Time-slot

Network

Relational Schema

PersonAge

Gender

Education

BN for Collaborative filtering

Law & Order

Frasier

NBC MondayNight Movies

Mad about you

Beverly Hills 90210

Seinfeld Friends

Melrose Place

Models Inc.

Breese, et al. UAI-98

Limitations of PRMs

• In PRM, all instances of the same classmust use the same dependency mode,it cannot distinguish:– documentaries and sitcoms – “60 Minutes” and Seinfeld

• PRM cannot have dependencies that are“cyclic”– ranking for Frasier depends on ranking for

Friends

Limitations of BNs

• In BN, each instance has its own dependency model, cannot generalize over instances– If John tends to like sitcoms, he will probably like

next season’s offerings– whether a person enjoys sitcom reruns depends

on whether they watch primetime sitcoms

• BN can only model relationships between atmost one class of instances at a time– In previous model, cannot model relationships

between people – if my roommate watches Seinfeld I am more

likely to join in

Desired Model

Allows both class and instance dependencies

WWWF

Person

Age

Gender

Education

Income

Soap Genre

Budget

Time-slot

Network

Genre

Budget

Time-slot

Network

Documentary

Sitcom-VoteProgram

Voter

Ranking

Doc-Vote

Program

Voter

Ranking

Vote

Program

Voter

Ranking

TV-Program Genre

Budget

Time-slot

Network

PRMs w/ Class Hierarchies

Allows us to:• Refine a “heterogenous” class into

more coherent subclasses• Refine probabilistic model along class

hierarchy– Can specialize/inherit CPDs– Construct new dependencies that were

originally “acyclic”Provides bridge from class-based model

to instance-based model

PRM-CH

PersonAge

TV-ProgramGenreBudgetTime-slotNetwork

GenderEducationIncome

VoteProgramVoterRanking

Relational Schema

Class Hierarchy

SoapOpera

TV-Program

SitCom DocumentaryDrama

Legal-Drama Medical-Drama

Dependency Model

BudgetSoapOpera

BudgetTV -Program

BudgetSitCom BudgetDocumentaryBudgetDrama

Budget Legal-Drama

BudgetMedical-Drama

Learning PRM-CHs

Relational

Schema

Database:

TVProgram Person

Vote

Person

Vote

TVProgram

Instance I

• Class hierarchy provided

• Learn class hierarchy

Structure Selection

• Idea: – define scoring function – do phased local search over legal

structures

• Key Components:– scoring models

– searching model space

PRM w/ CHs

new operators

unchanged

• Scenario 1: Class hierarchy is provided

• New Operators– Specialize/Inherit

Learning PRM-CH

BudgetSoapOpera

BudgetTV -Program

BudgetSitCom BudgetDrama

Budget Legal-Drama

BudgetMedical-Drama

BudgetDocumentaryBudgetDocumentar

y

Learning Class Hierarchy • Issue: partially observable data set• Construct decision tree for class defined over

attributes observed in training set

TV-Program.Genre

sitcomdrama

class1 class3

documentary

class2

class4

English

TV-.Network.Nationality

class5

French

class6

American

• New operator

– Split on class attribute– Related class attribute

MOVI

E

Animation

Family

Drama

Comedy

Romance

Action

Horror

Thriller

Theater Status

Video Status

Art/Foreign

Classic

VOTE

Rating PERSON

Gender

Age

Personal Income

Household Income

Education

EachMovie+ PRM

1400 Movies5000 People

240,000 Votes

http://www.research.digital.com/SRC/EachMovie

Theater Status

Video Status

Art/Foreign

ClassicDrama

ROMANCE-MOVIE

Animation

Family

Horror

Thriller

Gender

Age

Personal Income

Household Income

Education

PERSON

ROMANCE-VOTERating

OTHER-VOTERating

COMEDY-VOTERating

ACTION-VOTE

Rating

Theater Status

Video Status

Art/Foreign

ClassicDrama

ACTION-MOVIE

Animation

Family

Horror

Thriller

Theater Status

Video Status

Art/Foreign

ClassicDrama

COMEDY-MOVIE

Animation

Family

Horror

Thriller

PRM-CH

Theater Status

Video Status

Art/Foreign

Classic

OTHER-MOVIE

ThrillerDrama

Horror

Animation

Family

Comparison

• 5 Test Sets: 1000 votes, ~100 movies, ~115 people– PRM Mean LL: -12,079, std 475.68– PRM-CH Mean LL: -10558, std 433.10

• Using standard t-test, PRM-CH model outperforms PRM model with over 99% confidence

PRM-CH Summary

• PRMs with class hierarchies are a natural extension of PRMs:– Specialization/Inheritance of CPDs – Allows new dependency structures

• Provide bridge from class-based to instance-based models

• Learning techniques proposed– Need efficient heuristics – Empirical validation on real-world domains

Roadmap





Conclusions

• PRMs can represent distribution over attributes from multiple tables

• PRMs can capture link uncertainty• PRMs allow inferences about individuals

while taking into account relational structure (they do not make inappropriate independence assumptions)

Selected Publications• “Learning Probabilistic Models of Link Structure”, L. Getoor, N.

Friedman, D. Koller and B. Taskar, JMLR 2002.• “Probabilistic Models of Text and Link Structure for Hypertext

Classification”, L. Getoor, E. Segal, B. Taskar and D. Koller, IJCAI WS ‘Text Learning: Beyond Classification’, 2001.

• “Selectivity Estimation using Probabilistic Models”, L. Getoor, B. Taskar and D. Koller, SIGMOD-01.

• “Learning Probabilistic Relational Models”, L. Getoor, N. Friedman, D. Koller, and A. Pfeffer, chapter in Relation Data Mining, eds. S. Dzeroski and N. Lavrac, 2001.– see also N. Friedman, L. Getoor, D. Koller, and A. Pfeffer, IJCAI-99.

• “Learning Probabilistic Models of Relational Structure”, L. Getoor, N. Friedman, D. Koller, and B. Taskar, ICML-01.

• “From Instances to Classes in Probabilistic Relational Models”, L. Getoor, D. Koller and N. Friedman, ICML Workshop on Attribute-Value and Relational Learning: Crossing the Boundaries, 2000.

• Notes from AAAI Workshop on Learning Statistical Models from Relational Data, eds. L. Getoor and D. Jensen, 2000.

• Notes from IJCAI Workshop on Learning Statistical Models from Relational Data, eds. L. Getoor and D. Jensen, 2003.

• Notes from ICML Workshop on Statistical Relational Learning and its Connections to Other Fields, T. Dietterich, L. Getoor and K. Murphy, 2004

See http://www.cs.umd.edu/~getoor

Documents

Probabilistic Relational Models: A Tutorial Lise Getoor University of Maryland, College Park May 4, 2005