UIUC CS 497: Section EA Lecture #8 Reasoning in Artificial Intelligence Professor: Eyal Amir Spring...

UIUC CS 497: Section EALecture #8

Reasoning in Artificial Intelligence

Professor: Eyal Amir

Spring Semester 2004

(Based on slides by Lise Getoor (UMD))

Last Time

• Approximate Inference with Probabilistic Graphical Models

• Monte Carlo techniques

• Markov Chain Monte Carlo

Today

• Probabilistic Relational Models (PRMs)

• PRMs w/ Attribute Uncertainty

• PRMs w/ Link Uncertainty

Patterns in Structured Data

Patient

Treatment

Strain Contact

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


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

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

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

Probabilistic Relational Models

• Combine advantages of FOL & Bayes Nets:

– natural domain modeling

– 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 SchemaStrain

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 distr. 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 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

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 StratificationPRM

dependency 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

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)

Outline


» PRMs w/ Attribute Uncertainty

• PRMs w/ Link Uncertainty

Attribute UncertaintyTopic

Theory AI

Agent

Theory papers

Cornell

Scientific Paper

Topic

Theory AI

•Attributes of object•Attributes of linked objects

•Attributes of heterogeneous linked objects

PRMs w/ AU: 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 with Attribute Uncertainty 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…

Outline


• PRMs w/ Attribute Uncertainty

» PRMs w/ Link Uncertainty

Approach

• Construct probabilistic models of relational structure that capture link uncertainty

• Two new mechanisms:– Reference uncertainty– Existence uncertainty

• Advantage:– Applicable with partial knowledge of relational

structure

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

CitesCitedCiting

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

CitesCitedCiting

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

Existence Uncertainty

Document CollectionDocument Collection

? ??

PRM w/ Existence 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

Inference in Unrolled BN• Exact Inference in “unrolled” BN

– Infeasible for large networks– Structural (Attr/Reference/Exists) Uncertainty creates

very large cliques– Use caching (Pfeffer ’00)– FOL-Resolution-style techniques

• 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)

• Use approx. inference: MCMC (Pasula etal. ’01)

MCMC with PRMs

Prof1.$$

Prof2.$$

Prof3.$$

Prof1.fame

Prof2.fame

Prof3.fame

Student1.advisor

Student1.success

MCMC with PRMs

Prof1.$$

Prof2.$$

Prof3.$$

Prof2.fame

Student1.advisor

Student1.success

=Prof2

Networkstructurechanged

Gibbs Sampling with PRMs

• For each complex attribute A: reference attribute Ref[A], w/finite domain Val[Ref[A]]

• Reference uncertainty modifies chain of attributes


• For each complex attribute A: reference attribute Ref[A], w/finite domain Val[Ref[A]]

• Reference uncertainty modifies chain of attributes

• Gibbs for simple attributes: Use MB

• Gibbs for complex attributes (RU):– Add reference variables

M-H Sampling with PRMs

Prof1.$$

Prof2.$$

Prof3.$$

Prof2.fame

Student1.advisor

Student1.success

=Prof2

P(s1.a=P3,...X…) q(s1.a=P2,...X…| s1.a=P3,...X…) --------------------------------------------------------------------- =P(s1.a=P2,...X…) q(s1.a=P3,...X…| s1.a=P2,...X…)

Prof3.fame

Changing a ref.variable

P(s1.a=P3,...X…) P(s1.a=P3 | P1.$$,…,Pn.$$) P(s1.s|P3.f,------------------------ = -----------------------------------------P(s1.a=P2,...X…) P(s1.a=P3,...X…)


Prof1.$$

Prof2.$$

Prof3.$$

Prof2.fame

Student1.advisor

Student1.success

=Prof2

Prof3.fame


P(s1.a=P3,...X…) ------------------------ = P(s1.a=P2,...X…)

P(s1.a=P3 | P1.$$,…,Pn.$$) P(s1.s | P3.f,S1.a=P3)-------------------------------------------------------------------P(s1.a=P2 | P1.$$,…,Pn.$$) P(s1.s | P2.f,S1.a=P2)

P(s1.a=P3 | P1.$$,…,Pn.$$) P(s1.s | P3.f,S1.a=P3)-------------------------------------------------------------------- =P(s1.a=P2 | P1.$$,…,Pn.$$) P(s1.s | P2.f,S1.a=P2)


Prof1.$$

Prof2.$$

Prof3.$$

Prof2.fame

Student1.advisor

Student1.success

=Prof2

Prof3.fame


P(s1.a=P3 | P3.$$) P(s1.s | P3.f,S1.a=P3)--------------------------------------------------------P(s1.a=P2 | P2.$$) P(s1.s | P2.f,S1.a=P2)

Whenaggregationfunction(e.g.,max, softmax)

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 inapproriate independence assuptions)

Next Time

• Dynamic Bayesian Networks

THE END

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.

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

QueriesFull joint distribution specifies answer to any query: P(variable | evidence about others)

XRay

Lung Infiltrates

Sputum Smear


XRay Sputum Smear

UIUC CS 497: Section EA Lecture #8 Reasoning in Artificial Intelligence Professor: Eyal Amir Spring...

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