Docstoc

Satisfaction

Document Sample
Satisfaction Powered By Docstoc
					  Learning Probabilistic
    Relational Models
  Nir Friedman              Lise Getoor
  Hebrew University        Stanford University
   nir@cs.huji.ac.il     getoor@cs.stanford.edu

 Daphne Koller               Avi Pfeffer
  Stanford University      Stanford University
koller@cs.stanford.edu    avi@cs.stanford.edu
 Learning from Relational Data
• Data sources
  – relational and object-oriented databases
  – frame-based knowledge bases
  – World Wide Web
• Traditional approaches
  – work well with flat representations
  – fixed length attribute-value vectors
  – assume IID samples

• Problem:
  – must fix attributes in advance 
      can represent only some limited set of structures
  – IID assumption may not hold
             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 Schema
                         Classes
     Professor                                  Student
     Popularity                                 Intelligence

     Teaching-Ability                           Performance

     Stress-Level
                                                    Relationships
                        Teach    Take
Attributes
                            In
          Course                        Registration
          Difficulty                    Grade
          Rating                        Satisfaction


• Describes the types of objects and relations in the
  database
          Example instance I
Professor
    Prof. Gump                           Student
Popularity                                    John
                                          Student Doe
    high                                 Intelligence
                                                Jane Doe
Teaching Ability                              high
                                          Intelligence
    medium                               Performance
                                                high
Stress-Level                                  average
                                          Performance
    low                                       average

                        Reg
                         Reg#5639
       Course
                        Grade#5639
            Phil142
        Course            Reg
                            A
                         Grade
       Difficulty
              Phil101          #5639
                        Satisfaction
                             A
            low
        Difficulty        Grade
                            3
                         Satisfaction
       Rating low              A
            high
        Rating               3
                          Satisfaction
            high              3
           What’s Uncertain?
Professor
    Prof. Gump                             Student
Popularity
                         Objects                John
                                            Student Doe
    high                                   Intelligence
                                                  Jane Doe
                                                Student
Teaching Ability                                high
                                            Intelligence Dunn
                                                     Judy
    medium                                 Performance
                                                  high
                                                Intelligence
Stress-Level                                    average
                                            Performance
                                                     high
    low                                           average
                    Relations                   Performance
                                                     high
                          Reg
                           Reg#5639
       Course
                          Grade#5639
            Phil142
        Course              Reg
                              A
                           Grade
       Difficulty
              Phil101
            low
                                 #5639
                          Satisfaction
                               A             Attribute
        Difficulty          Grade
                              3
       Rating low          Satisfaction
                                 A           Values
            high
        Rating                 3
                            Satisfaction
            high                3
               Attribute Uncertainty
    Professor
         Prof. Gump                          Student
    Popularity                                      John
                                               Student Deer
         ???                                 Intelligence Doe
                                                      Jane
    Teaching Ability                                ???
                                               Intelligence
         ???                                 Performance
                                                      ???
    Stress-Level                                    ???
                                               Performance
         ???                                          ???
                            Reg
           Course            Reg #5639
                 Phil142    Grade #5639
            Course            Reg
           Difficulty             A
                             Grade #5639
                  Phil101
                 ???        Satisfaction
                                   A
            Difficulty        Grade
           Rating ???             3
                             Satisfaction
                                     ???
                 ???
            Rating                 3
                              Satisfaction
                  ???                ???


Fixed skeleton 
   – set of objects in each class
   – relations between them
Uncertainty
   – over assignments of values to attributes
             PRM: Dependencies
                 Professor
                                            Student
                 Popularity
                                            Intelligence
                 Teaching-Ability
                                            Performance
Course           Stress-Level
Difficulty

Rating                           D, I     A     B      C
                                    Reg.Grade
                                  h, h 0.5 |0.4 0.1 
                Reg
                                P  h, l Reg.In.Dif ficulty, 
                                          0 .1 0 .5 0 . 4
                 Grade              
                                     l , h Reg.Taker. Intell 
                                             0 .8 0 .1 0. 1 
                                                             
                 Satisfaction        l, l    0 . 3 0 .6 0. 1
      PRM: Dependencies (cont.)
Professor
    Prof. Gump                           Student
Popularity                                    John Deer
                                          Student Doe
    high                                 Intelligence
                                                Jane Doe
Teaching Ability                              high
                                              low
                                          Intelligence
    medium                               Performance
                                                high
Stress-Level                                  average
                                          Performance
    low                                         average

                        Reg
                        Reg
                         Reg#5639            D, I      A    B    C
       Course
                        Grade#5639
                             #5639
            Phil142
        Course            Reg                 h, h     0 .5 0 .4 0. 1
                            A
                        Grade
                         Grade
       Difficulty
              Phil101          #5639
                        Satisfaction
                             ?
            low
        Difficulty          3
                             A
                          Grade                h, l    0 .1 0 .5 0 . 4
       Rating           Satisfaction
                         Satisfaction
              low              ?
            high
        Rating
                             3
                             3
                          Satisfaction
                                               l, h    0 .8 0 .1 0. 1
            high              3                l, l    0 . 3 0 .6 0. 1
 PRM: aggregate dependencies
                Professor l
                      avg       m    h
                         A 0.1 0.2 0.7      Student
                 Popularity 0.2 0.4 0.4
                         B                    Student
                                            Intelligence
                         C 0.6 0.3 0.1            Jane Doe
                 Teaching-Ability            Intelligence
                                            Performance
                                                  high
Course       Reg
                 Stress-Level av             Performance
                              g                   average
Difficulty       #5077
             Grade
                 C Reg
Rating
             Satisfaction #5054                 Problem!!!
                 2 Grade
              Reg         C Reg
                                              Need CPTs of
                     Satisfaction#5639
                Grade 1 Grade                 varying sizes
                                 A
                Satisfaction Satisfaction
                                 3
 PRM: aggregate dependencies
                     Professor
                                           Student
                     Popularity
                                            Intelligence
                     Teaching-Ability
                                            Performance
Course               Stress-Level
Difficulty
                   count
Rating                         avg

                    Reg
                     Grade

             avg     Satisfaction       sum, min, max,
                                        avg, mode, count
               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 q

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

P(I | , S, q)             P(I          x.a   | I parents ( x.a ) )
                      X x( X ) X . A
                                            Value of attribute A
            Classes      Objects Attributes     in object x
                  Learning PRMs
                                           Reg




                                  Course
                                                 Student
    Database:
    Instance I

          Reg




Course
                      Student   • Parameter estimation
         Relational
          Schema                • Structure selection
   Parameter estimation in PRMs
• Assume known dependency structure S
• Goal: estimate PRM parameters q
  – entries in local probability models, q x. A| parents ( x. A )

• A parameterization q is good if it is likely to
  generate the observed data, instance I .

          l (q : I, S )  log P(I | S, q)
• MLE Principle: Choose q* so as to maximize l
       crucial property: decomposition
          separate terms for different X.A
           ML parameter estimation
    Course                                               Student
     Difficulty
                                Reg                      Intelligence
     Rating                       Grade                  Performance
                                  Satisfaction


   qR.G  A|C .Dl,S.I h 
       *
                                                   Reg.Grade |             
                                                                           
                                                 P     Reg.In.Dif ficulty, 
    N ( R.G  A,C . D  l , S . I  h )                Reg.Taker. Intell 
                                                                           
         N ( C . D l ,S . I h)
                            sufficient statistics
 DB technology well-suited to the computation of suff statistics:

Count  C .Diff
                          Course                  Reg                Student
                           table                 table                table
                R.Grade
                 S .Int
            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:
                                           marginal
                                           likelihood
                                            prior
                                           
  Score ( S : I )  log P( S | I )  log[ P(I | S )P( S )]

• closed form solution
                  Legal Models
Researcher                             Paper
 Reputation                author-of     Accepted


• Dependency ordering over attributes:
    y.b   x.a         if X.A depends on Y.B

• PRM defines a coherent probability model over
  skeleton  if  is acyclic
                              y.b

                              x.a
       Guaranteeing Acyclicity
How do we guarantee that a PRM is acyclic
for every skeleton?
           PRM
                                    dependency
       dependency
                                       graph
        structure S
                      Y.B
                            if X.A depends directly on Y.B

                      X.A


Attribute stratification:
 dependency graph acyclic   acyclic for any 
      Limitation of stratification
                    Father            Mother
Person                                           Person
M-chromosome                                     M-chromosome
P-chromosome        Person                       P-chromosome
Blood-type           M-chromosome                Blood-type
                     P-chromosome

                     Blood-type


         Person.M-chrom                   Person.P-chrom


                          Person.B-type             ???
    Guaranteed acyclic relations
                      Father            Mother
Person                                           Person
M-chromosome                                      M-chromosome
P-chromosome           Person                     P-chromosome

Blood-type              M-chromosome              Blood-type
                        P-chromosome

                        Blood-type


• 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
                 X.A                        X.A                     X.A
yellow: within              green: via            red: via
 single object              g.a. relation         other relations
                 X.B                        Y.B                     Y.B


   Person.M-chrom                Person.P-chrom
                                                       A cycle is safe if
                       Person.B-type                    – it has a green edge
                                                        – it has no red edge
         Searching Model Space
Phase 0: consider only dependencies within a class
                                  Course             Student
                                           Reg




Course         Student
         Reg




                         Course            Student
                                  Reg
         Phased structure search
Phase 1: consider dependencies from “neighboring”
         classes, via schema relations
                                  Course             Student
                                           Reg




Course         Student
         Reg




                         Course            Student
                                  Reg
         Phased structure search
Phase 2: consider dependencies from “further”
         classes, via relation chains
                                  Course              Student
                                           Reg




Course         Student
         Reg




                         Course             Student
                                  Reg
     Experimental Results:
    Movie Domain (real data)
 11,000 movies, 7,000 actors

Movie                                     Actor
Process                                    Gender

Decade
                     Appears
Genre
                      Role-type


        source: http://www-db.stanford.edu/movies/doc.html
Genetics domain (synthetic data)
               Father    Mother
Person                            Person
M-chromosome                      M-chromosome
P-chromosome   Person             P-chromosome
Blood-type      M-chromosome      Blood-type
                P-chromosome

                Blood-type


               Blood-Test
                Contaminated
                Result
                 Experimental Results
        -18000


        -20000


        -22000


        -24000
Score




                                               Median Likelihood
        -26000                                   Gold Standard


        -28000


        -30000


        -32000
                  200   300   400       500        600    700      800
                                    Dataset Size
             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

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:4
posted:11/14/2012
language:English
pages:32