Statistical Relational Learning

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Statistical Relational Learning

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Acknowledgements

• Lise Getoor, Nir Friedman, Daphne Koller, Ben
  Taskar, Avi Pfeffer, David Jensen, Pedro
  Domingos, Indrajit Bhattacharya and many others

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Why SRL?

• Probabilistic graphical models (particularly
  Bayesian network) have been shown to be a useful
  way of representing statistical patterns in real
  world domain
• Probabilistic relational models (PRMs) are a
  recent development the extend the standard
  attribute-based Bayesian network representation
  to incorporate a much richer relational structure
  
• Allow the specification of a probability model
  for classes of objects rather than simple
  attributes.
• Allow properties of an entity to depend
  probabilistically on properties of other related
  entities.

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An example

• A simple model of the performance of a student in
  a course
• Random variables
• Course difficulty high, medium, low
• Student intelligence high, low
• Understands material yes, no
• Good test taker yes, no
• Homework grade A, B, C, D, E
• Exam grade A, B, C, D, E

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Complete joint probability distribution

• Must specify a probability for each of the
  exponentially many different instantiations of
  the set
• P(I,D,G,U,E,H) will consider all possible
  assignment of the value of these variables
• 222355600
• The naïve representationof the joint
  distribution isinfeasible

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Bayesian networks

• Key insight each variable is directly influenced
  by only a few others
• Probabilistic conditional independence each node
  is conditionally independent of its
  non-descendants given values for its parents
• Associate with each node a conditional
  probability distribution (CPD), which specifies
  for each node X the probability distribution over
  the values of X given each combination of values
  for its parents, denoted as Pa(X)
• The joint distribution can be factorized into a
  product of CPDs of all the variables

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Bayesian networks

• Bayesian networks provide a compact
  representation of complex joint distributions

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However

• Bayesian networks are often inadequate to
  properly model aspects of complex relational
  domains.
• A Bayesian network for a given domain involves a
  pre-specified set of random variables, whose
  relationship to each other is fixed in advance.
• They cannot deal with domains where we may
  encounterseveral entities in a varietyof
  configurations, becausethey lack the concept
  ofan object (or domain entity)

If we treat the circles as random variables,then
how to handle the AVG dependence
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Introduction to PRMs

• PRMs extend Bayesian networks with the concepts
  of individuals, their properties and relations
  between them
• The relational framework is motivated primarily
  by the concepts of relational database
• Relational database vs. PRM schema instance

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PRMs definition Relational schema

• A schema for a relational model describe a set of
  classes, ?X1, X2,Xn.
• The domain entities of a class is called objects
• Each class is associated with a set of
  descriptive attributes and a set of reference
  slots
• Each class correspond to a single table
• Descriptive attributes correspond to standard
  attributes in the table
• Reference slots correspond to attributes that are
  foreign keys

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A more complex school example

• The rectangles are classes
• The underlined attributes are reference slots and
  the others are descriptive attributes

Each single table is a class
Foreign keys are reference slots
Standard attributes are descriptive attributes
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Another movie example

• Movies and actors

Each single table is a class
Foreign keys are reference slots
Standard attributes are descriptive attributes
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Descriptive attributes

• The set of descriptive attribute of a class X is
  denoted A(X), attributes of class X is denoted
  X.A, and its domain value is V(X.A)
• Examples
• A(Student) Intelligence, Ranking
• V(Student.Intellegence) high, low
• V(Actor.Gender) male, female

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Reference slots

• The set of reference slots of a class X is
  denoted R(X). we use X.? to denote the reference
  slot ? of X
• The domain type Dom? X
• The range type Range? Y, Y is some class in ?
• Examples
• R(Registration) Course, Student
• RangeCourse.Instructor Professor
• R(Role) Actor, Movie
• RangeRole.Actor Actor

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Inverse slot slot chain

• For each reference slot ?, we can define an
  inverse slot ?-1, interpreted as the inverse
  function of ?
• We define a slot chain t ?1,,?k to be a
  sequence of slots such that for all i. Range?i
  Dom?i1. The slot chain allows us to compose
  slots, defining functions from objects to other
  indirectly related objects
• Examples
• The inverse slot for the Student slot of
  Registration is called Registered-In
• Student.Registered-In.Course.Instructor can be
  used to denote a students set of instructors

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Schema Instance

• An instance I of a schema specifies
• A set of objects x, partitioned into classes
• A value for each descriptive attribute x.A
• A value for each reference slot x.?, which is an
  object of the appropriate range type
• A complete instantiation I is a set of object
  with no missing values and no dangling references
• School example
• One Professor
• Two Classes
• Three Registration
• Two Students

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PRMs definition relational skeleton

• A relational skeleton s of a relational schema is
  a partial specification of an instance of the
  schema.
• It specifies the set of objects for each class
  and the relations(dependency structure) that hold
  between objects (similar to Bayesian network
  structure)
• It leaves the value of the attributes
  unspecified
• A PRM will specify a probability distributions
  over all complete instantiations that extend
  the skeleton

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PRMs definition relational skeleton

• The dependency structure of a relational skeleton
  is defined by associating with each attributes
  X.A a set of formal parents Pa(X.A)
• X.A can depend on another probabilistic attribute
  B of X
• X.A can also depend on attributes of related
  object X.t.C, where t is a slot chain
• The class-level dependencies are instantiated
  according to the relational skeleton, to define
  object-level dependency
• We use s(X) to refer the set of objects of class
  X
• Let x be some object in s(X), the actual parent
  of x.A is x.B
• The formal parent of x.A is y.C, where y belongs
  to x.t

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Differences to Bayesian networks

• The PRM defines the dependency model at the class
  level, allowing it to be used for any object in
  the class. The class dependency model is
  universally quantified and instantiated for every
  object in the class domain
• The PRM explicitly uses the relational structure
  of the model, in that it allows the probabilistic
  model of an attribute of an object to depend also
  on attribute of related objects.The specific set
  of related objects can vary with the relational
  skeleton s

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Definition of PRM

• A probabilistic relational model (PRM) ? for a
  relational schema S define for each class X ??and
  each descriptive attribute A ? A(X), a set of
  formal parents Pa(X.A) ,and a conditional
  probability distribution (CPD) that represents
  P(X.APa(X.A))
• A PRM consists of two components the qualitative
  dependency structure S, and the set of parameters
  ?S associated with it. For the basic PRM for
  attribute uncertainty, we assume that the
  relational skeleton sr is given

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Qualitative structure

• The qualitative structure of the network is
  defined via an instance dependency graph Gs,
  whose nodes correspond to descriptive attributes
  x.A of objects in the skeleton, and the edges
  correspond to the direct attributes dependence
  and the slot chain dependence
• Note that the slot chain x.t might be
  multi-valued, we must specify the probabilistic
  dependence of x.A on the multi-set y.B y ?
  x.t.
• It is impractical to provide a dependency model
  for each of the unboundedly many possible
  multi-set size.
• We use an aggregate function and define a
  dependence on the computed aggregate value

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Aggregate function

• The dependence of x.A on x.t.B is interpreted as
  a probabilistic dependence of x.A on some
  aggregate property of this multi-set.
• Many natural and useful notions of
  aggregate?Mean, median, maximum, cardinality
  etc.
• We allow X.A to have a parent ?(X.t.B ).The
  semantic is that for any x ? X, x.A will depend
  on the value of ?(x.t.B ). We define V(?(X.t.B
  )) to be the set of possible values of this
  aggregate

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Parameters

• A PRM associates a CPD for each attribute of each
  class. As for dependencies, we assume that the
  parameters are shared by each object in the
  class.
• The school example
• P(GD,I)
• P(Ravg(G))An aggregate is used here

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PRM semantics

• Given a skeleton sr, we have a set of random
  variables of interest. The set of random
  variables for sr is the set of attributes of the
  form x.A, where x ?s(Xi) ,and A ? A(Xi) fpr some
  class Xi
• The PRM specifies a probability distribution over
  the possible joint assignments if values to these
  random variables. It basically defines a ground
  Bayesian network.
• This ground Bayesian network leads to the
  following chain rule which defines a distribution
  over the instantiations compatible with the
  skeleton sr

Each attribute
Each object
Each class
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Differences to Bayesian networks

• Our random variables are the attributes of a set
  of objects.
• The set of parents of a random variable can vary
  according to the relational context of the object
  the set of object to which it is related
• The school example

The attribute Grade of the class Registration
depends on the attributes Intelligence of the
class Student.
For the Registration object 5639, it references
Jane-Doe.Intelligence.
But for some other Registration objects, they
might reference Bob.Intelligence or
Tony.Intelligence
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Coherent probability distribution

• We have to ensure that the resulting function
  from instances to numbers does indeed define a
  coherent probability, where the sum of the
  probability of all instances is 1
• In Bayesian network, the requirement is satisfied
  if the dependency graph is acyclic a variable is
  not an ancestor of itself
• We need to check whether a dependency structure S
  is acyclic relative to a fixed skeleton s

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Dependency graph

• A stronger guarantee acyclic class dependency
  graph
• The class dependency graph has an edge from Y.B
  to X.A if either XY and X.B is a parent of X.A,
  or ?(X.t.B ) is a parent of X.A and
  RangeX.tY
• It is clear that if the classdependency graph is
  acyclic, we can neverhave that x.A depends on
  itself. The school example

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HoweverAnother example

• Blood test
• A cycle in the class dependency graph does not
  imply that all skeleton induce cyclic instance
  dependencies
• Although the model appears to be cyclic at the
  class level, we know that the cyclicity is always
  resolved at the level of individual objects

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Learning PRMs

• Input
• A relational schema which specifies the basic
  vocabulary in the domain the set of classes,
  the attributes associated with different classes,
  and the possible types of relations between
  objects in the different classes
• The training data consists of a fully specified
  instance of that schema
• Learning task
• Parameter estimation
• Structure learning

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Parameter estimation

• We assume that the qualitative dependency
  structure S is known
• The key ingredient in parameter estimation is the
  likelihood function, the probability of the data
  given the model
• Then performing Maximum likelihood estimation
  (MLE), to find the parameter setting ?S that
  maximize the likelihood L(?SI,s,S) for a given
  I,s, and S. The maximum likelihood model is the
  model that best predicts the training data
• We can also take a Bayesian approach to parameter
  estimation by incorporating parameter priors

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Structure learning

• Hypothesis space
• Specify which structures are legal candidate
  hypothesisAcyclic dependency graph
• Scoring structures
• Bayesian scoreThis score is composed of the
  prior probability of the structure and the
  posterior probability of the structure given the
  data
• Structure search
• Hill-climbing search
• A heuristic search algorithm

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A heuristic search algorithm
Phase 0 consider only dependencies within a class
Author
Review
Paper
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A heuristic search algorithm
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
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A heuristic search algorithm
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
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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
  likenext seasons offerings
• whether a person enjoys sitcom reruns dependson
  whether they watch primetime sitcoms
• BN can only model relationships between atmost
  one class of instances at a time
• In previous model, cannot model
  relationshipsbetween people
• if my roommate watches Seinfeld I am morelikely
  to join in

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PRM Summary

• PRMs inherit key advantages of probabilistic
  graphical models
• Coherent probabilistic semantics
• Exploit structure of local interactions
• Relational models inherently more expressive
• Web of influence use multiple sources of
  information to reach conclusions
• Exploit both relational information and power of
  probabilistic reasoning

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Probabilistic Relational Models (PRMs)

• Developed by Daphne Kollers 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 Lise Getoor, Nir Friedman, Avi
• Attribute Uncertainty
• Structural Uncertainty
• Class Uncertainty
• Identity Uncertainty
• undirected models Ben Taskar
• Reference
• Learning Probabilistic Models of Link Structure.
   Lise Getoor, Nir Friedman, Daphne Koller,
  Benjamin Taskar. Journal of Machine Learning
  Research, Volume 3, page 679- -707 - 2002

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Families of SRL Approaches

• Frame-based Probabilistic Models
• Probabilistic Relational Models (PRMs),
• Probabilistic Entity Relation Models (PERs),
• Object Oriented Bayesian Networks (OOBNs)
• First Order Probabilistic Logic (FOPL)
• BLOGs
• Relational Markov Logic (RML)
• Stochastic Functional Programs
• PRISM
• Stochastic Logic Programs (SLPs)
• IBAL

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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

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Recent SRL Activities

• Dagstuhl Workshop on Probabilistic, Logical and
  Relational Learning - Towards a
  Synthesishttp//www.dagstuhl.de/05051/
• ICML 2004 workshop on Statistical Relational
  Learning and its Connections to Other
  Fieldshttp//www.cs.umd.edu/projects/srl2004/
• IJCAI 2003 workshop on Statistical Relational
  Learninghttp//kdl.cs.umass.edu/srl2003/
• AAAI 2000 workshop on Statistical Relational
  Learninghttp//robotics.stanford.edu/srl
• Several related workshops
• KDD MRDM workshops
• http//www-ai.ijs.si/SasoDzeroski/MRDM2004/
• http//www-ai.ijs.si/SasoDzeroski/MRDM2003/
• http//www-ai.ijs.si/SasoDzeroski/MRDM2002/