1. Probabilistic Models of Object-Relational Domains
Daphne Koller
Stanford University
Joint work with:
Lise Getoor
Ben Taskar
Drago Anguelov
Nir Friedman
Pieter Abbeel
Rahul Biswas
Avi Pfeffer
Ming-Fai Wong
Evan Parker
Eran Segal

2. Bayesian Networks: Problem
Bayesian nets use propositional representation
Real world has objects, related to each other
Intelligence
These “instances” are not independent
Difficulty
Intell_Jane
Diffic_CS101
Grade_Jane_CS101
Intell_George
Diffic_CS101
Grade_George_CS101 A C
Grade
Intell_George
Diffic_Geo101
Grade_George_Geo101
One of the key benefits of the propositional representation was our ability to represent our knowledge without explicit enumeration of the worlds. In turns out that we can do the same in the probabilistic framework. The key idea, introduced by Pearl in the Bayesian network framework, is to use locality of interaction. This is an assumption which seems to be a fairly good approximation of the world in many cases.
However, this representation suffers from the same problem as other propositional representations. We have to create separate representational units (propositions) for the different entities in our domain. And the problem is that these instances are not independent. For example, the difficulty of CS101 in one network is the same difficulty as in another, so evidence in one network should influence our beliefs in another.

3. Probabilistic Relational Models
Combine advantages of relational 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

4. St. Nordaf University Geo101 CS101 Teaching-ability Prof. Jones
Prof. Smith
Teaches
Teaches
Grade
In-course
Registered
Intelligence
Satisfac
Welcome to
Geo101
George
Grade
Registered
Difficulty
Welcome to
CS101
In-course
Satisfac
Grade
Let us consider an imaginary university called St. Nordaf. St. Nordaf has two faculty, two students, two courses, and three registrations of students in courses, each of which is associated with a registration record. These objects are linked to each other: professors teach classes, students register in classes, etc. Each of the objects in the domain has properties that we care about.
Registered
Satisfac
Jane
In-course

5. Relational Schema Classes Attributes Relations
Specifies types of objects in domain, attributes of each type of object & types of relations between objects
Classes
Professor
Student
Teaching-Ability
Intelligence
Teach
Take
Attributes
Relations
Registration
Grade
Satisfaction
Course In
Difficulty
The university can be described in an organized form using a relational database. The schema of a database tells us what types of objects we have, what are the attributes of these objects that are of interest, and how the objects can relate to each other.

6. Representing the Distribution
Very large probability space for a given context
All possible assignments of all attributes of all objects
Infinitely many potential contexts
Each associated with a very different set of worlds
Need to represent
infinite set of complex distributions
Unfortunately, this is a very large number of worlds, and to specify a probabilistic model we have to specify a probability for each one.
Furthermore, the resulting distribution would be good only for a limited time. If St. Nordaf hired a new faculty member, or got a new student, or even if the two students registered for different classes next year, it would no longer apply, and St. Nordaf would have to pay Acme consulting all over again. Thus, we want a model that holds for the infinitely many potential universities that hold over this very simple schema. Thus, we are stuck with what seems to be an impossible problem. How do we represent an infinite set of possible distributions, each of which is by itself very complex.

7. Probabilistic Relational Models
Universals: Probabilistic patterns hold for all objects in class
Locality: Represent direct probabilistic dependencies
Links define potential interactions
Professor
Teaching-Ability
Student
Intelligence
Course
Difficulty A B C
Reg
Grade
Satisfaction
The two key ideas that come to our rescue derive from the two approaches that we are trying to combine. From relational logic, we have the notion of universal patterns, which hold for all objects in a class. From Bayesian networks, we have the notion of locality of interaction, which in the relational case has a particular twist: Links give us precisely a notion of “interaction”, and thereby provide a roadmap for which objects can interact with each other.
In this example, we have a template, like a universal quantifier for a probabilistic statement. It tells us: “For any registration record in my database, the grade of the student in the course depends on the intelligence of that student and the difficulty of that course.” This dependency will be instantiated for every object (of the right type) in our domain. It is also associated with a conditional probability distribution that specifies the nature of that dependence. We can also have dependencies over several links, e.g., the satisfaction of a student on the teaching ability of the professor who teaches the course.
[K. & Pfeffer; Poole; Ngo & Haddawy]

8. PRM Semantics Instantiated PRM BN
variables: attributes of all objects
dependencies: determined by
links & PRM
Teaching-ability
Prof. Jones
Prof. Smith
Grade
Satisfac
Intelligence
Welcome to
Geo101
George
Difficulty
Welcome to
CS101
The semantics of this model is as something that generates an appropriate probabilistic model for any domain that we might encounter. The instantiation is the embodiment of universals on the one hand, and locality of interaction, as specified by the links, on the other.
Jane

9. The Web of Influence CS101 C A low high Geo101 easy / hard low / high
Welcome to
CS101 C A
low
high
Welcome to
Geo101
This web of influence has interesting ramifications from the perspective of the types of reasoning patterns that it supports. Consider Forrest Gump. A priori, we believe that he is pretty likely to be smart. Evidence about two classes that he took changes our probabilities only very slightly. However, we see that most people who took CS101 got A’s. In fact, even people who did fairly poorly in other classes got an A in CS101. Therefore, we believe that CS101 is probably an easy class. To get a C in an easy class is unlikely for a smart student, so our probability that Forrest Gump is smart goes down substantially.
easy / hard
low / high

10. Reasoning with a PRM Generic approach:
Instantiate PRM to produce ground BN
Use standard BN inference
In most cases, resulting BN is too densely connected to allow exact inference
Use approximate inference: belief propagation
Improvement: Use domain structure — objects & relations — to guide computation
Kikuchi approximation where clusters = objects

11. Data  Model  Objects Database Learner Probabilistic Model
Course
Student
Reg
Learner
Probabilistic Model
Expert knowledge
What are the objects in the new situation?
How are they related to each other?
Prob.
Inference
Data for New
Situation

12. Two Recent Instantiations
From a relational dataset with objects & links, classify objects and predict relationships:
Target application: Recognize terrorist networks
Actual application: From webpages to database
From raw sensor data to categorized objects
Laser data acquired by robot
Extract objects, with their static & dynamic properties
Discover classes of similar objects

13. 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

14. Discriminative Probabilistic Models for Relational Data
Ben Taskar
Stanford University
Joint work with:
Pieter Abbeel
Daphne Koller
Ming-Fai Wong

15. Web  KB Tom Mitchell Professor Project-of WebKB Project Member
Sean Slattery
Student
Advisor-of
Project-of
Member
Of course, data is not always so nicely arranged for us as in a relational database. Let us consider the biggest source of data --- the world wide web. Consider the webpages in a computer science department. Here is one webpage, which links to another. This second webpage links to a third, which links back to the first two. There is also a webpage with a lot of outgoing links to webpages on this site. This is not nice clean data. Nobody labels these webpages for us, and tells us what they are. We would like to learn to understand this data, and conclude from it that we have a “Professor Tom Mitchell” one of whose interests is a project called “WebKB”. “Sean Slattery” is one of the students on the project, and Professor Mitchell is his advisor. Finally, Tom Mitchell is a member of the CS CMU faculty, which contains many other faculty members. How do we get from the raw data to this type of analysis?
[Craven et al.]

16. Undirected PRMs: Relational Markov Nets
Universals: Probabilistic patterns hold for all groups of objects
Locality: Represent local probabilistic dependencies
Address limitations of directed models:
Increase expressive power by removing acyclicity constraint
Improve predictive performance through discriminative training
Course
Reg
Grade
Student
Difficulty
Intelligence
Template potential
Study Group
Student2
Reg2
Grade
Intelligence
The two key ideas that come to our rescue derive from the two approaches that we are trying to combine. From relational logic, we have the notion of universal patterns, which hold for all objects in a class. From Bayesian networks, we have the notion of locality of interaction, which in the relational case has a particular twist: Links give us precisely a notion of “interaction”, and thereby provide a roadmap for which objects can interact with each other.
In this example, we have a template, like a universal quantifier for a probabilistic statement. It tells us: “For any registration record in my database, the grade of the student in the course depends on the intelligence of that student and the difficulty of that course.” This dependency will be instantiated for every object (of the right type) in our domain. It is also associated with a conditional probability distribution that specifies the nature of that dependence. We can also have dependencies over several links, e.g., the satisfaction of a student on the teaching ability of the professor who teaches the course.
[Taskar, Abbeel, Koller ‘02]

17. RMN Semantics Instantiated RMN  MN
variables: attributes of all objects
dependencies: determined by links & RMN
Welcome to
Geo101
Grade
Intelligence
Geo Study Group
Difficulty
George
The two key ideas that come to our rescue derive from the two approaches that we are trying to combine. From relational logic, we have the notion of universal patterns, which hold for all objects in a class. From Bayesian networks, we have the notion of locality of interaction, which in the relational case has a particular twist: Links give us precisely a notion of “interaction”, and thereby provide a roadmap for which objects can interact with each other.
In this example, we have a template, like a universal quantifier for a probabilistic statement. It tells us: “For any registration record in my database, the grade of the student in the course depends on the intelligence of that student and the difficulty of that course.” This dependency will be instantiated for every object (of the right type) in our domain. It is also associated with a conditional probability distribution that specifies the nature of that dependence. We can also have dependencies over several links, e.g., the satisfaction of a student on the teaching ability of the professor who teaches the course.
Welcome to
CS101
CS Study Group
Jane
Jill

18. P(Grades,Intelligence|Difficulty)
Learning RMNs
Parameter estimation is not closed form
Convex problem  unique global maximum
Maximize
L = log
P(Grades,Intelligence|Difficulty)
(Reg1.Grade,Reg2.Grade)
easy / hard
ABC
low / high
Grade
Intelligence
Grade
Intelligence
Grade
Difficulty
Intelligence
Parameters are not independent
Grade
Intelligence
Grade
Difficulty
Intelligence
Grade

19. Web Classification Experiments
WebKB dataset
Four CS department websites
Five categories (faculty,student,project,course,other)
Bag of words on each page
Links between pages
Anchor text for links
Experimental setup
Trained on three universities
Tested on fourth
Repeated for all four combinations

20. Exploiting Links
From-
Page
...
Page
Category
Word1
WordN
To-
Category
...
Classify all pages collectively,
maximizing the joint label probability
Link
Word1
WordN
35.4% relative reduction in error relative to strong flat approach

21. Scalability WebKB data set size Network size / school:
1300 entities
180K attributes
5800 links
Network size / school:
40,000 variables
44,000 edges
Training time: 20 minutes
Classification time: seconds

22. Predicting Relationships
Tom Mitchell
Professor
Advisor-of
Member
WebKB
Project
Sean Slattery
Student
Even more interesting are relationships between objects

23. WebKB++ Four new department web sites: Labeled page type (8 types):
Berkeley, CMU, MIT, Stanford
Labeled page type (8 types):
faculty, student, research scientist, staff, research group, research project, course, organization
Labeled hyperlinks and virtual links (6 types):
advisor, instructor, TA, member, project-of, NONE
Data set size:
11K pages
110K links
2million words

24. Flat Model ... ... ... From- Page To- Page Rel Word1 WordN Word1 WordN
Type
Rel
NONE
advisor
instructor TA
member
project-of
...
LinkWord1
LinkWordN

25. Flat Model
...
...

26. Collective Classification: Links
From-
Page
To-
Page
Category
Category
...
...
Word1
WordN
Word1
WordN
Type
Rel
...
LinkWord1
LinkWordN

27. Link Model
...
...

28. Triad Model
Advisor
Professor
Student
Member
Member
Group

29. Triad Model
Advisor
Professor
Student TA
Instructor
Course

30. Triad Model

31. Link Prediction: Results
...
72.9% relative reduction in error relative to strong flat approach
Error measured over links predicted to be present
Link presence cutoff is at precision/recall break-even point (30% for all models)

32. Summary
Use relational models to recognize entities & relations directly from raw data
Collective classification:
Classifies multiple entities simultaneously
Exploits links & correlations between related entities
Uses web of influence reasoning to reach strong conclusions from weak evidence
Undirected PRMs allow high-accuracy discriminative training & rich graphical patterns

33. Learning Object Maps from Laser Range Data
Dragomir Anguelov
Daphne Koller
Evan Parker
Robotics Lab
Stanford University

34. Occupancy Grid Maps Static world assumption
Inadequate for answering symbolic queries
person
robot

35. Objects Entities with coherent properties: Shape Color
Kinematics (Motion)

36. Object Maps Natural and concise representation
Exploit prior knowledge about object models
Walls are straight, doors open and close
Learn global properties of the environment
Primary orientation of walls, typical door width
Generalize properties across objects
Objects viewed as instances of object classes, parameter sharing

37. Learning Object Maps Define a probabilistic generative model
Suggest object hypotheses
Optimize the object parameters (EM)
Select highest-scoring model
Object Properties
Object
Segmentation

38. Laser Sensor Data

39. Probabilistic Model Data: A set of scans Global Map M
Scan: set of <robot position, laser beam reading> tuples
Each scan is associated with a static map M t
Global Map M
A set of objects {1, …, J}
Each object i = {S[i ], Dt[i ]}
S[i ] – static parameters
Dt[i ] – dynamic parameters
Non-static environment – dynamic parameters vary only between static maps
Fully dynamic environment – dynamic parameters vary between scans

40. Probabilistic Model - II
General map M
Static maps M1… MT
Objects i
Robot positions sit
Laser beams zit
Correspondence variables Cit

41. Generative Model Specification
Sensor model
Object models
Particular instantiation:
Walls
Doors
Model score

42. Sensor Model Modeling occlusion: Why we should model occlusion:
Reading ztk generated from:
Random model (uniform probability)
First object the beam intersects
Actual object (Gaussian probability)
MaxRange model (Delta function)
Why we should model occlusion:
Realistic sensor model
Helps to infer motion
Improved model search

43. Wall Object Model Wall model i
A line defined by <i, i>, as in
S intervals <1, 2> each denoting a segment along the line
2S + 2 independent parameters
Collinear segments bias

44. Door Object Model Door Model i A pivot p Width w
A set of angles t (t=1,2,…)
Limited rotation (90o)
Arc “center” d
4 static + 1 dynamic parameter

45. Model Score Maximize log-posterior probability of map M, data Z:
increased data likelihood
Increased number of parameters
Maximize log-posterior probability of map M, data Z:
Define structure prior p(M) over possible maps:
|S[M]| — number of static parameters in M
|D[M]| — number of dynamic parameters in M
L[M] — total length of segments in M

46. Learning Model Parameters (EM)
E-step
Compute expectations
M-step
Walls
Optimize line parameters
Optimize segment ends
Doors
Optimize pivot and angles
Optimize door width

47. Suggesting Object Hypotheses
Wall hypotheses
Use Hough transform (histogram-based approach)
Compute preferred direction of the environment
Use both to suggest lines
Door hypotheses
Use temporal differencing of static maps
Check if points along segments in static maps Mt are well explained in the general map M
If not, the segment is a potential door

48. Results for a Single Pass

49. Results for Two Passes

50. Future Work Simultaneous localization & mapping
Object class hierarchies
Dynamic environments
Enrich the object representation:
More sophisticated shape models
Color 3D

51. Hierarchical Object Maps
Learn to recognize object classes  
Come visit our poster!