Learning Probabilistic Relational Models

Daphne KollerStanford University

koller@cs.stanford.edu

Nir FriedmanHebrew Universitynir@cs.huji.ac.il

Lise GetoorStanford University

getoor@cs.stanford.edu

Avi PfefferStanford University

avi@cs.stanford.edu

• Data sources– relational and object-oriented databases– frame-based knowledge bases – World Wide Web

Learning from Relational Data

• Problem:– must fix attributes in advance

can represent only some limited set of structures– IID assumption may not hold

• Traditional approaches– work well with flat representations– fixed length attribute-value vectors – assume IID samples

Our Approach• Probabilistic Relational Models (PRMs)

– rich representation language models• relational dependencies• probabilistic dependencies

• Learning PRMs – parameter estimation– model selection

from data stored in relational databases

Outline• Motivation• Probabilistic relational models

– Probabilistic Logic Programming[Poole, 1993]; [Ngo & Haddawy 1994]

– Probabilistic object-oriented knowledge[Koller & Pfeffer 1997; 1998]; [Koller, Levy & Pfeffer; 1997]

• Learning PRMs• Experimental results• Conclusions

Probabilistic Relational Models

• Combine advantages of predicate logic & BNs: – 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 SchemaStudentIntelligencePerformance

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• Describes the types of objects and relations in the database

ClassesClasses

RelationshipsRelationships

AttributesAttributes

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What’s Uncertain?

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Attribute Values

ObjectsStudent

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Uncertainty– over assignments of values to attributes

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PRM: Dependencies

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PRM: Summary• A PRM specifies

– a probabilistic dependency structure S• a set of parents for each attribute X.A

– a set of local probability models

• Given a skeleton structure , a PRM specifies a probability distribution over instances I:– over attribute values of all objects in

Classes Objects

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

. axparentsX Xx AX

axPSP III

Value of attribute A in object xAttributes

Learning PRMs

Relational

Schema

Database:

• Parameter estimation

• Structure selection

Course Student

Reg

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Instance I

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

– entries in local probability models,

• A parameterization 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

crucial property: decompositionseparate terms for different X.A

ML parameter estimation

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DB technology well-suited to the computation of suff statistics:

Coursetable

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...

Count

sufficient statistics

Model Selection• Idea:

– define scoring function – do local search over legal structures

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

Scoring Models

• Bayesian approach:

• closed form solution

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

priorlikelihoodmarginal

SPSPSPSScore

III

Legal Models

• Dependency ordering over attributes:

x.a

y.b

axby .. if X.A depends on Y.B

PaperAccepted

ResearcherReputation author-of

• PRM defines a coherent probability model over skeleton if is acyclic

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

PRMdependency structure S

dependencygraph

Y.B

X.A

if X.A depends directly on Y.B

dependency graph acyclic acyclic for any Attribute stratification:

Limitation of stratificationPersonM-chromosome

P-chromosome

Blood-type

PersonM-chromosome

P-chromosome

Blood-type

PersonM-chromosome

P-chromosome

Blood-type

Father Mother

Person.M-chrom Person.P-chrom

Person.B-type ???

Guaranteed acyclic relations

PersonM-chromosome

P-chromosome

Blood-type

PersonM-chromosome

P-chromosome

Blood-type

PersonM-chromosome

P-chromosome

Blood-type

Father Mother

• Prior knowledge: the Father-of relation is acyclic– dependence of Person.A on Person.Father.B cannot induce cycles

Guaranteeing acyclicity• With guaranteed acyclic relations, some cycles in

the dependency graph are guaranteed to be safe.• We color the edges in the dependency graph

A cycle is safe if– it has a green edge– it has no red edge

yellow: withinsingle object

X.B

X.Agreen: viag.a. relation

Y.B

X.Ared: viaother relations

Y.B

X.A

Person.M-chrom Person.P-chrom

Person.B-type

Searching Model Space

Student

Course Reg scoreAdd C.AC.B

score

Delete S.IS.P Student

Course Reg

Student

RegCourse

Phase 0: consider only dependencies within a class

Phased structure search

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Course Reg scoreAdd C.AR.B

score

Add S.IR.CStudent

Course Reg

Student

RegCourse

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

Phased structure search

scoreAdd C.AS.P

score

Add S.IC.B

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

Student

Course Reg

Student

Course Reg

Student

Course Reg

Experimental Results:Movie Domain (real data)

11,000 movies, 7,000 actors

ActorGender

AppearsRole-type

MovieProcess

Decade

Genre

source: http://www-db.stanford.edu/movies/doc.html

Genetics domain (synthetic data)PersonM-chromosome

P-chromosome

Blood-type

PersonM-chromosome

P-chromosome

Blood-type

PersonM-chromosome

P-chromosome

Blood-type

Father Mother

Blood-TestContaminated

Result

Experimental Results

-32000

-30000

-28000

-26000

-24000

-22000

-20000

-18000

200 300 400 500 600 700 800

Sco

re

Dataset Size

Median LikelihoodGold Standard

Future directions• Learning in complex real-world domains

– drug treatment regimes– collaborative filtering

• Missing data• Learning with structural uncertainty• Discovery

– hidden variables– causal structure– class hierarchy

Conclusions• PRMs natural extension of BNs:

– well-founded (probabilistic) semantics– compact representation of complex models

• Powerful learning techniques– builds on BN learning techniques– can learn directly from relational data

• Parameter estimation– efficient, effective exploitation of DB technology

• Structure identification– builds on well understood theory– major issues:

• guaranteeing coherence• search heuristics