probabilistic relational models for link prediction problem

Probabilistic Relational Models for Link Prediction Problem

By: Sina SajadmaneshAdvisor: Dr. Hamid Reza Rabiee

2 DMLLink Prediction DMLDMLLink Prediction2

Outline

Introduction Probabilistic Relational Models Directed vs. Undirected Networks

Relational Bayesian NetworksRelational Markov Networks


Introduction

Probabilistic Relational Models Combines probabilistic graphical models

with entity-relationship models Defines a joint probability distribution over

attributes of entities and relations A method for describing probabilistic

relationships among attributes of entities and related entities

4 DMLLink Prediction DML4

Introduction

Probabilistic Relational Network Directed Graphical Model

• Relational Bayesian Network (RBN)• Focus on causal interactions

Undirected Graphical Model• Relational Markov Network (RMN)• Focus on symmetric interactions


Outline

IntroductionRelational Bayesian Networks

Relational Schema Probabilistic Model Relational Skeleton Attribute Uncertainty Semantics Link Prediction

Relational Markov Networks


Relational Bayesian Networks



Relational Schema

AuthorGood Writer

Author ofHas Review

Review

PaperQualityAccepted

Mood

LengthSmart



Probabilistic Model

Length

MoodAuthor

Good Writer

Paper

Quality

Accepted

ReviewSmart



Relational Skeleton

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

Author A1

Paper P1 Author: A1 Review: R1

Review R2

Review R1

Author A2



Review R2

10

DMLLink Prediction DMLDMLLink Prediction10


Attribute Uncertainty

RBN defines distribution over instantiations of attributes

11



Aggregate DependenciesReview R1

Length

Mood

Review R2

Length

Mood

Review R3

Length

Mood

Paper P1

Accepted

Quality

12



Aggregate Dependencies

sum, min, max, avg, mode, count

Review R1

Length

Mood

Review R2

Length

Mood

Review R3

Length

Mood

Paper P1

Accepted

Quality

mode

3.07.04.06.08.02.09.01.0

,,,,

,

ttfttfff

P(A | Q, M) MQ

13



Semantics

Probability distribution over instantiation I

Author

Paper

Review

Author A1

Paper P2

Paper P1

ReviewR3

ReviewR2

ReviewR1

Author A2

Paper P3

𝑃 ( 𝐼|𝜎 ,𝑆 ,𝜃 )=∏𝑥∈𝜎

∏𝑥 . 𝐴

𝑃 (𝑥 .𝐴|𝑝𝑎𝑟𝑒𝑛𝑡𝑠(𝑥 .𝐴)¿ ¿

14



Link Prediction

? ??

15



Link Prediction

Cites

PaperTopicWords

PaperTopicWords

Exists

Citer.Topic Cited.Topic

0.995 0.005 Theory Theory

False True

AI Theory 0.999 0.001

AI AI 0.993 0.008 AI Theory 0.997 0.003

16



Link Prediction

Paper#2 Topic Paper#3Topic

WordN

Paper#1

Word1

Topic

... ... ...

Author #1Area Inst

#1-#2

Author #2Area Inst

Exists

#2-#3Exists

#2-#1Exists

#3-#1Exists

#1-#3Exists

WordN

Word1WordN

Word1

Exists

#3-#2

17


Outline

IntroductionRelational Bayesian NetworksRelational Markov Networks

Advantages of Undirected Models Markov instead of Bayesian network Relational Clique Templates Formal Definition Link Prediction

18



Advantages of Undirected Models Cycles are not a problem Easy to learn discriminatively Symmetric, non-causal interactions Handles patterns involving multiple entities

• Triangle patterns• Transitive patterns

Devised for collective classification

19



Markov instead of Bayesian network

Author2 Paper2TopicArea

Venue

Paper1Topic

Author1

SubArea

Area

1.8AI

THTH

0.3

1.50.2

AIAITHAI

TH

T2T1 (T1,T2)

Template potential

20



Relational Clique Templates Specify tuples of variables in the network

instantiation using relational query language Components:

• F a set of entity variables (FROM)• W condition about the attributes (WHERE)• S subset of attributes (SELECT)

Query results to cliques

SELECT p1.topic, p2.topicFROM Paper p1, Paper p2, Cite cWHERE c.citer=p1.key AND c.cited=p2.key

21



Formal Definition Set of clique templates C Set of potential functions Ф

Defines a conditional distribution over labels of an instantiation

𝑃 ( 𝐼|Ф ,𝐶 )= 1𝑍 ∏

𝐶∏

𝑐∈𝐶(𝐼 )Φ𝑐 ( 𝐼 .𝑣𝑐 )

22



Link Prediction Factors affecting the relations of different

entities

Entity’s attributes:• Properties, Labels

Entity’s structural properties:• Similarity, Transitivity

More complex patterns can be captured using cliques that represents dependencies and correlations

23


References[1] Getoor, Lise. Introduction to statistical relational learning. MIT press, 2007.

[2] Pfeffer, Avrom J., and Daphne Koller. Probabilistic reasoning for complex systems. Stanford: Stanford University, 2000.

[3] Koller, Daphne, and Avi Pfeffer. "Probabilistic frame-based systems."AAAI/IAAI. 1998.

[4] Taskar, Ben, Pieter Abbeel, and Daphne Koller. "Discriminative probabilistic models for relational data." Proceedings of the Eighteenth conference on Uncertainty in artificial intelligence. Morgan Kaufmann Publishers Inc., 2002.

[5] Taskar, Ben, et al. "Link prediction in relational data." Advances in neural information processing systems. 2003.

probabilistic relational models for link prediction problem

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