Statistical Relational Learning: A Quick Intro

1
Statistical Relational Learning A Quick Intro

• Lise Getoor
• University of Maryland, College Park

2
acknowledgements

• Synthesis of ideas of many individuals who have
  participated in various SRL workshops
• Hendrik Blockeel, Mark Craven, James Cussens,
  Bruce DAmbrosio, Luc De Raedt, Tom Dietterich,
  Pedro Domingos, Saso Dzeroski, Peter Flach, Rob
  Holte, Manfred Jaeger, David Jensen, Kristian
  Kersting, Daphne Koller, Heikki Mannila, Tom
  Mitchell, Ray Mooney, Stephen Muggleton, Kevin
  Murphy, Jen Neville, David Page, Avi Pfeffer,
  Claudia Perlich, David Poole, Foster Provost, Dan
  Roth, Stuart Russell, Taisuke Sato, Jude
  Shavlik, Ben Taskar, Lyle Ungar and many others
• and students
• Indrajit Bhattacharya, Mustafa Bilgic, Rezarta
  Islamaj, Louis Licamele, Qing Lu, Galileo Namata,
  Vivek Sehgal, Prithviraj Sen

3
Why SRL?

• Traditional statistical machine learning
  approaches assume
• A random sample of homogeneous objects from
  single relation
• Traditional ILP/relational learning approaches
  assume
• No noise or uncertainty in data
• Real world data sets
• Multi-relational, heterogeneous and
  semi-structured
• Noisy and uncertain
• Statistical Relational Learning
• newly emerging research area at the intersection
  of research in social network and link analysis,
  hypertext and web mining, graph mining,
  relational learning and inductive logic
  programming
• Sample Domains
• web data, bibliographic data, epidemiological
  data, communication data, customer networks,
  collaborative filtering, trust networks,
  biological data, sensor networks, natural
  language, vision

4
SRL Approaches

• Directed Approaches
• Bayesian Network Tutorial
• Rule-based Directed Models
• Frame-based Directed Models
• Undirected Approaches
• Markov Network Tutorial
• Frame-based Undirected Models
• Rule-based Undirected Models

5
Probabilistic Relational Models

• PRMs w/ Attribute Uncertainty
• Inference in PRMs
• Learning in PRMs
• PRMs w/ Structural Uncertainty
• PRMs w/ Class Hierarchies

Representation Inference Koller Pfeffer
98, Pfeffer, Koller, Milch Takusagawa 99,
Pfeffer 00 Learning, Structural Uncertainty
Class Hierarchies Friedman et al. 99, Getoor,
Friedman, Koller Taskar 01 02, Getoor 01
6
Relational Schema
Author
Review
Good Writer
Mood
Smart
Length
Paper
Quality
Accepted
Has Review
Author of

• Describes the types of objects and relations in
  the database

7
Probabilistic Relational Model
Review
Author
Smart
Mood
Good Writer
Length
Paper
Quality
Accepted
8
Probabilistic Relational Model
Review
Author
Smart
Mood
Good Writer
Length
Paper
Quality
Accepted
9
Probabilistic Relational Model
Review
Author
Smart
Mood
Good Writer
Length
Paper
P(A Q, M)
,
M
Q
9
.
0
1
.
0
,
f
f
Quality
8
.
0
2
.
0
,
t
f
Accepted
4
.
0
6
.
0
,
f
t
3
.
0
7
.
0
,
t
t
10
Relational Skeleton
Paper P1 Author A1 Review R1
Author A1
Review R1
Paper P2 Author A1 Review R2
Review R2
Author A2
Review R2
Paper P3 Author A2 Review R2

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

11
PRM w/ Attribute Uncertainty
Paper P1 Author A1 Review R1
Author A1
Review R1
Paper P2 Author A1 Review R2
Author A2
Review R2
Paper P3 Author A2 Review R2
Review R3
PRM defines distribution over instantiations of
attributes
12
A Portion of the BN
P2.Accepted
P3.Accepted
13
A Portion of the BN
P(A Q, M)
,
M
Q
9
.
0
1
.
0
,
f
f
8
.
0
2
.
0
,
t
f
P2.Accepted
4
.
0
6
.
0
,
f
t
3
.
0
7
.
0
,
t
t
P3.Accepted
14
PRM Aggregate Dependencies
Paper
Review
Mood
Quality
Length
Accepted
15
PRM Aggregate Dependencies
Paper
Review
Mood
Quality
Length
Accepted
P(A Q, M)
,
M
Q
9
.
0
1
.
0
,
f
f
8
.
0
2
.
0
,
t
f
4
.
0
6
.
0
,
f
t
3
.
0
7
.
0
,
t
t
mode
sum, min, max, avg, mode, count
16
PRM with AU Semantics
Author
Review R1
Author A1
Paper
Paper P1
Review R2
Author A2
Review
Paper P2
Review R3
Paper P3
PRM
relational skeleton ?


probability distribution over completions I
17
Probabilistic Relational Models

• PRMs w/ Attribute Uncertainty
• Inference in PRMs
• Learning in PRMs
• PRMs w/ Structural Uncertainty
• PRMs w/ Class Hierarchies

18
PRM Inference

• Simple idea enumerate all attributes of all
  objects
• Construct a Bayesian network over all the
  attributes

19
Inference Example
Review R1
Skeleton
Paper P1
Review R2
Author A1
Review R3
Paper P2
Review R4
Query is P(A1.good-writer) Evidence is
P1.accepted T, P2.accepted T
20
PRM Inference Constructed BN
A1.Smart
A1.Good Writer
21
PRM Inference

• Problems with this approach
• constructed BN may be very large
• doesnt exploit object structure
• Better approach
• reason about objects themselves
• reason about whole classes of objects
• In particular, exploit
• reuse of inference
• encapsulation of objects

22
PRM Inference Interfaces
Variables pertaining to R2 inputs and internal
attributes
A1.Smart
A1.Good Writer
P1.Quality
P1.Accepted
23
PRM Inference Interfaces
Interface imported and exported attributes
A1.Smart
A1.Good Writer
R2.Mood
P1.Quality
R2.Length
P1.Accepted
24
PRM Inference Encapsulation
R1 and R2 are encapsulated inside P1
A1.Smart
A1.Good Writer
25
PRM Inference Reuse
A1.Smart
A1.Good Writer
26
Structured Variable Elimination
Author 1
A1.Smart
A1.Good Writer
Paper-1
Paper-2
27
Structured Variable Elimination
Author 1
A1.Smart
A1.Good Writer
Paper-1
Paper-2
28
Structured Variable Elimination
Paper 1
A1.Smart
A1.Good Writer
Review-2
Review-1
P1.Quality
P1.Accepted
29
Structured Variable Elimination
Paper 1
A1.Smart
A1.Good Writer
Review-2
Review-1
P1.Quality
P1.Accepted
30
Structured Variable Elimination
Review 2
A1.Good Writer
R2.Mood
R2.Length
31
Structured Variable Elimination
Review 2
A1.Good Writer
R2.Mood
32
Structured Variable Elimination
Paper 1
A1.Smart
A1.Good Writer
Review-2
Review-1
P1.Quality
P1.Accepted
33
Structured Variable Elimination
Paper 1
A1.Smart
A1.Good Writer
Review-1
R2.Mood
P1.Quality
P1.Accepted
34
Structured Variable Elimination
Paper 1
A1.Smart
A1.Good Writer
Review-1
R2.Mood
P1.Quality
P1.Accepted
35
Structured Variable Elimination
Paper 1
A1.Smart
A1.Good Writer
R2.Mood
R1.Mood
P1.Quality
P1.Accepted
36
Structured Variable Elimination
Paper 1
A1.Smart
A1.Good Writer
R2.Mood
R1.Mood
P1.Quality
True
P1.Accepted
37
Structured Variable Elimination
Paper 1
A1.Smart
A1.Good Writer
38
Structured Variable Elimination
Author 1
A1.Smart
A1.Good Writer
Paper-1
Paper-2
39
Structured Variable Elimination
Author 1
A1.Smart
A1.Good Writer
Paper-2
40
Structured Variable Elimination
Author 1
A1.Smart
A1.Good Writer
Paper-2
41
Structured Variable Elimination
Author 1
A1.Smart
A1.Good Writer
42
Structured Variable Elimination
Author 1
A1.Good Writer
43
Benefits of SVE

• Structured inference leads to good elimination
  orderings for VE
• interfaces are separators
• finding good separators for large BNs is very
  hard
• therefore cheaper BN inference
• Reuses computation wherever possible

44
Limitations of SVE

• Does not work when encapsulation breaks down
• But when we dont have specific information about
  the connections between objects, we can assume
  that encapsulation holds
• i.e., if we know P1 has two reviewers R1 and R2
  but they are not named instances, we assume R1
  and R2 are encapsulated
• Cannot reuse computation when different objects
  have different evidence

R3 is not encapsulated inside P2
45
Probabilistic Relational Models

• PRMs w/ Attribute Uncertainty
• Inference in PRMs
• Learning in PRMs
• PRMs w/ Structural Uncertainty
• PRMs w/ Class Hierarchies

46
Four SRL Approaches

• Directed Approaches
• BN Tutorial
• Rule-based Directed Models
• Frame-based Directed Models
• PRMs w/ Attribute Uncertainty
• Inference in PRMs
• Learning in PRMs
• PRMs w/ Structural Uncertainty
• PRMs w/ Class Hierarchies
• Undirected Approaches
• Markov Network Tutorial
• Frame-based Undirected Models
• Rule-based Undirected Models

47
Learning PRMs w/ AU
Author
Database
Paper
Review
PRM
Author
Paper
Review

• Parameter estimation

• Structure selection

Relational Schema
48
ML Parameter Estimation
Review
Mood
Paper
Length
Quality
Accepted
49
ML Parameter Estimation
Review
Mood
Paper
Length
Quality
Accepted
q
50
Structure Selection

• Idea
• define scoring function
• do local search over legal structures
• Key Components
• legal models
• scoring models
• searching model space

51
Structure Selection

• Idea
• define scoring function
• do local search over legal structures
• Key Components
• legal models
• scoring models
• searching model space

52
Legal Models

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

Paper P1 Accepted yes
author-of
Researcher Prof. Gump Reputation high
Paper P2 Accepted yes
sum
How do we guarantee that a PRM is acyclic for
every skeleton?
53
Attribute Stratification
PRM dependency structure S
dependency graph
Paper.Accepted
if Researcher.Reputation depends directly on
Paper.Accepted
Researcher.Reputation
Algorithm more flexible allows certain cycles
along guaranteed acyclic relations
54
Structure Selection

• Idea
• define scoring function
• do local search over legal structures
• Key Components
• legal models
• scoring models same as BN
• searching model space

55
Structure Selection

• Idea
• define scoring function
• do local search over legal structures
• Key Components
• legal models
• scoring models
• searching model space

56
Searching Model Space
Phase 0 consider only dependencies within a class
Author
Review
Paper
57
Phased Structure Search
Phase 1 consider dependencies from neighboring
classes, via schema relations
Author
Review
Paper
Author
Review
Paper
Add P.A?R.M
? score
Author
Review
Paper
58
Phased Structure Search
Phase 2 consider dependencies from further
classes, via relation chains
Author
Review
Paper
Author
Review
Paper
Add R.M?A.W
Author
Review
Paper
? score
59
Probabilistic Relational Models

• PRMs w/ Attribute Uncertainty
• Inference in PRMs
• Learning in PRMs
• PRMs w/ Structural Uncertainty
• PRMs w/ Class Hierarchies

60
Kinds of structural uncertainty

• How many objects does an object relate to?
• how many Authors does Paper1 have?
• Which object is an object related to?
• does Paper1 cite Paper2 or Paper3?
• Which class does an object belong to?
• is Paper1 a JournalArticle or a ConferencePaper?
• Does an object actually exist?
• Are two objects identical?

61
Structural Uncertainty

• Motivation PRM with AU only well-defined when
  the skeleton structure is known
• May be uncertain about relational structure
  itself
• Construct probabilistic models of relational
  structure that capture structural uncertainty
• Mechanisms
• Reference uncertainty
• Existence uncertainty
• Number uncertainty
• Type uncertainty
• Identity uncertainty

62
Citation Relational Schema
Author
Institution
Research Area
Wrote
Paper
Paper
Topic
Topic
Word1
Word1
Word2
Cites

Word2

Citing Paper
WordN
Cited Paper
WordN
63
Attribute Uncertainty
Author
Institution
P( Institution Research Area)
Research Area
Wrote
P( Topic Paper.Author.Research Area
Paper
Topic
P( WordN Topic)
...
Word1
WordN
64
Reference Uncertainty

Bibliography
1. ----- 2. ----- 3. -----
Scientific Paper
Document Collection
65
PRM w/ Reference Uncertainty
Paper
Paper
Topic
Topic
Cites
Words
Words
Citing
Cited
Dependency model for foreign keys

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

66
Reference Uncertainty Example
Paper P5 Topic AI
Paper P4 Topic AI
Paper P3 Topic AI
Paper M2 Topic AI
Paper P5 Topic AI
C1
Paper P4 Topic Theory
Paper P1 Topic Theory
Paper P2 Topic Theory
Paper P1 Topic Theory
Paper P3 Topic AI
C2
Paper.Topic AI
Paper.Topic Theory
Cites
Citing
Cited
67
Reference Uncertainty Example
Paper P5 Topic AI
Paper P4 Topic AI
Paper P3 Topic AI
Paper M2 Topic AI
Paper P5 Topic AI
C1
Paper P4 Topic Theory
Paper P1 Topic Theory
Paper P2 Topic Theory
Paper P6 Topic Theory
Paper P3 Topic AI
C2
Paper.Topic AI
Paper.Topic Theory
C1
C2
Topic
Cites
Theory
Citing
AI
Cited
68
Introduce Selector RVs
P2.Topic
Cites1.Selector
P3.Topic
Cites1.Cited
P1.Topic
P4.Topic
Cites2.Selector
P5.Topic
Cites2.Cited
P6.Topic
Introduce Selector RV, whose domain is
C1,C2 The distribution over Cited depends on
all of the topics, and the selector
69
PRMs w/ RU Semantics
Paper
Paper
Topic
Topic
Cites
Words
Words
Cited
Citing
PRM RU
70
Learning
PRMs w/ RU

• Idea
• define scoring function
• do phased local search over legal structures
• Key Components
• legal models
• scoring models
• searching model space

model new dependencies
unchanged
new operators
71
Legal Models
Review
Mood
Paper
Paper
Important
Important
Accepted
Cites
Accepted
Citing
Cited
72
Legal Models
Cites1.Selector
Cites1.Cited
P2.Important
R1.Mood
P3.Important
P1.Accepted
P4.Important
When a nodes parent is defined using an
uncertain relation, the reference RV must be a
parent of the node as well.
73
Structure Search
Cites
Author
Citing
Institution
Cited
Cited
74
Structure Search New Operators
Cites
Author
Citing
Institution
Cited
Refine on Topic
Cited
?score
Paper
Paper
Paper
Paper
Paper
75
Structure Search New Operators
Cites
Author
Citing
Institution
Cited
Refine on Topic
Cited
Paper
Paper
Paper
Paper
Paper
Refine on Author.Instition
?score
Paper
Paper
Paper
76
PRMs w/ RU Summary

• Define semantics for uncertainty over which
  entities are related to each other
• Search now includes operators Refine and Abstract
  for constructing foreign-key dependency model
• Provides one simple mechanism for link
  uncertainty

77
Existence Uncertainty
Document Collection
Document Collection
78
PRM w/ Exists Uncertainty
Paper
Paper
Topic
Topic
Cites
Words
Words
Exists
Dependency model for existence of relationship
79
Exists Uncertainty Example
Paper
Paper
Topic
Topic
Cites
Words
Words
Exists
False
True
Cited.Topic
Citer.Topic
80
Introduce Exists RVs
Author 2
Author 1
Inst
Area
Inst
Area
Paper3
Paper1
Paper2
Topic
Topic
Topic
WordN
WordN
Word1
Word1
Word1
...
...
...
WordN
Exists
Exists
Exists
Exists
Exists
Exists
1-2
2-3
2-1
3-1
1-3
3-2
81
Introduce Exists RVs
Author 2
Author 1
Inst
Area
Inst
Area
Paper3
Paper1
Paper2
Topic
Topic
Topic
WordN
WordN
Word1
Word1
Word1
...
...
...
WordN
Exists
Exists
Exists
Exists
Exists
Exists
1-2
2-3
2-1
3-1
1-3
3-2
82
PRMs w/ EU Semantics
Paper
Paper
Topic
Topic
Cites
Words
Words
Exists
PRM EU
83
Learning
PRMs w/ EU

• Idea
• define scoring function
• do phased local search over legal structures
• Key Components
• legal models
• scoring models
• searching model space

model new dependencies
unchanged
unchanged
84
Four SRL Approaches

• Directed Approaches
• BN Tutorial
• Rule-based Directed Models
• Frame-based Directed Models
• PRMs w/ Attribute Uncertainty
• Inference in PRMs
• Learning in PRMs
• PRMs w/ Structural Uncertainty
• PRMs w/ Class Hierarchies
• Undirected Approaches
• Markov Network Tutorial
• Frame-based Undirected Models
• Rule-based Undirected Models

85
PRMs with classes

• Relations organized in a class hierarchy
• Subclasses inherit their probability model from
  superclasses
• Instances are a special case of subclasses of
  size 1
• As you descend through the class hierarchy, you
  can have richer dependency models
• e.g. cannot say Accepted(P1) lt- Accepted(P2)
  (cyclic)
• but can say Accepted(JournalP1) lt-
  Accepted(ConfP2)

Venue
Journal
Conference
86
Type Uncertainty

• Is 1st-Venue a Journal or Conference ?
• Create 1st-Journal and 1st-Conference objects
• Introduce Type(1st-Venue) variable with possible
  values Journal and Conference
• Make 1st-Venue equal to 1st-Journal or
  1st-Conference according to value of
  Type(1st-Venue)

87
Learning PRM-CHs
Vote
Database
Person
Instance I
PRM-CH
Vote

• Class hierarchy provided

TVProgram
Person
Relational Schema
88
Learning
PRMs w/ CH

• Idea
• define scoring function
• do phased local search over legal structures
• Key Components
• legal models
• scoring models
• searching model space

model new dependencies
unchanged
new operators
89
Guaranteeing Acyclicity w/ Subclasses
90
Learning PRM-CH

• Scenario 1 Class hierarchy is provided
• New Operators
• Specialize/Inherit

AcceptedPaper
AcceptedJournal
AcceptedConference
AcceptedWorkshop
91
Learning Class Hierarchy

• Issue partially observable data set
• Construct decision tree for class defined over
  attributes observed in training set

• New operator

• Split on class attribute

• Related class attribute

92
PRMs w/ Class Hierarchies

• Allow 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
93
Summary Directed Frame-based Approaches

• Focus on objects and relationships
• what types of objects are there, and how are they
  related to each other?
• how does a property of an object depend on other
  properties (of the same or other objects)?
• Representation support
• Attribute uncertainty
• Structural uncertainty
• Class Hierarchies
• Efficient Inference and Learning Algorithms

94
Butwhat about Probabilistic DBs?

• Similarities
• Representation, e.g.
• PRMs can model attribute correlations compactly
  (or)
• PRMs can model tuple uncertainty by introducing
  exists random variable for each uncertain tuple
  (maybe)
• PRMs can model join dependencies compactly
• Differences
• ML emphasis on generalization and compact
  modeling
• DB emphasis on loss-less data storage
• Commonality
• Need for efficient query processing

95
Conclusion

• Statistical Relational Learning
• Supports multi-relational, heterogeneous domains
• Supports noisy, uncertain, non-IID data
• aka, real-world data!
• Different approaches
• rule-based vs. frame-based
• directed vs. undirected
• Many common issues
• Need for collective classification and
  consolidation
• Need for aggregation and combining rules
• Need to handle labeled and unlabeled data
• Need to handle structural uncertainty
• etc.
• Great opportunity for combining machine learning
  for hierarchical statistical models with
  probabilistic databases which can efficiently
  store, query, update models