Representation, Inference and Learning in Relational Probabilistic Languages

1
Representation, Inference and Learning in
Relational Probabilistic Languages

• Lise Getoor
• University of Maryland
• College Park

Avi Pfeffer Harvard University
IJCAI 2005 Tutorial
2
Introduction

• Probability is good
• First-order representations are good
• Variety of approaches that combine them
• We wont cover all of them in detail
• apologies if we leave out your favorite
• We will cover three broad classes of approaches,
  and present exemplars of each approach
• We will highlight common issues, themes, and
  techniques that recur in different approaches

3
Running Example

• There are papers, researchers, citations,
  reviews
• Papers have a quality, and may or may not be
  accepted
• Authors may be smart and good writers
• Papers have topics, and cite other papers which
  may or may not be on the same topic
• Papers are reviewed by reviewers, who have moods
  that are influenced by the quality of the writing

4
Some Queries

• What is the probability that a researcher is
  famous, given that one of her papers was accepted
  despite the fact that a reviewer was in a bad
  mood?
• What is the probability that a paper is accepted,
  given that another paper by the same author is
  accepted?
• What is the probability that a paper is an AI
  paper, given that it is cited by an AI paper?
• What is the probability that a student of a
  famous advisor has seven high quality papers?

5
Sample Domains

• Web Pages and Link Analysis
• Battlespace Awareness
• Epidemiological Studies
• Citation Networks
• Communication Networks (Cellphone Fraud
  Detection)
• Intelligence Analysis (Terrorist Networks)
• Financial Transactions (Money Laundering)
• Computational Biology
• Object Recognition and Scene Analysis
• Natural Language Processing (e.g. Information
  Extraction and Semantic Parsing)

6
Roadmap

• Motivation
• Background Bayesian network inference and
  learning
• Rule-based Approaches
• Frame-based Approaches
• Undirected Relational Approaches
• Programming Language Approaches

7
Bayesian Networks Pearl 87
Smart
Good Writer
Reviewer Mood
Quality
nodes domain variables edges direct causal
influence
Accepted
Review Length
Network structure encodes conditional
independencies I(Review-Length ,
Good-Writer Reviewer-Mood)
8
BN Semantics
conditional independencies in BN structure
local CPTs
full joint distribution over domain

• Compact natural representation
• nodes have ? k parents ?? O(2k n) vs. O(2n)
  params
• natural parameters

9
Reasoning in BNs

• Full joint distribution specifies answer to any
  query P(event evidence)
• Allows combination of different types of
  reasoning
• Causal P(Reviewer-Mood Good-Writer)
• Evidential P(Reviewer-Mood not Accepted)
• Intercausal P(Reviewer-Mood not Accepted,
  Quality)

10
Variable Elimination Zhang Poole 96, Dechter
98

• To compute

factors
A factor is a function from values of variables
to positive real numbers
11
Variable Elimination

• To compute

12
Variable Elimination

• To compute

sum out l
13
Variable Elimination

• To compute

new factor
14
Variable Elimination

• To compute

multiply factors together then sum out w
15
Variable Elimination

• To compute

new factor
16
Variable Elimination

• To compute

17
Some Other Inference Algorithms

• Exact
• Junction Tree Lauritzen Spiegelhalter 88
• Cutset Conditioning Pearl 87
• Approximate
• Loopy Belief Propagation McEliece et al 98
• Likelihood Weighting Shwe Cooper 91
• Markov Chain Monte Carlo eg MacKay 98
• Gibbs Sampling Geman Geman 84
• Metropolis-Hastings Metropolis et al 53,
  Hastings 70
• Variational Methods Jordan et al 98
• etc.

18
Learning in BNs
Parameters only
Structure and Parameters
EM Dempster et al 77 or gradient descent
Russell et al 95
Complete Data
Easy counting
Incomplete Data
Structure search
Structural EM Friedman 97
See Heckerman 98 for a general introduction
19
Parameter Estimation in BNs

• Assume known dependency structure G
• Goal estimate BN parameters q
• entries in local probability models,
• q is good if its likely to generate observed
  data.
• MLE Principle Choose q so as to maximize l
• Alternative incorporate a prior

20
Learning With Complete Data

• Fully observed data data consists of set of
  instances, each with a value for all BN variables
• With fully observed data, we can compute
  number of instances with , and
• and similarly for other counts
• We then estimate

21
Learning with Missing Data Expectation-Maximizati
on (EM)

• Cant compute
• But
• Given parameter values, can compute expected
  counts
• Given expected counts, estimate parameters
• Begin with arbitrary parameter values
• Iterate these two steps
• Converges to local maximum of likelihood

this requires BN inference
22
Structure search

• Begin with an empty network
• Consider all neighbors reached by a search
  operator that are acyclic
• add an edge
• remove an edge
• reverse an edge
• For each neighbor
• compute ML parameter values
• compute score(s)
• Choose the neighbor with the highest score
• Continue until we reach a local maximum

23
Limitations of BNs

• Inability to generalize across collection of
  individuals within a domain
• if you want to talk about multiple individuals in
  a domain, you have to talk about each one
  explicitly, with its own local probability model
• Domains have fixed structure e.g. one author,
  one paper and one reviewer
• if you want to talk about domains with multiple
  inter-related individuals, you have to create a
  special purpose network for the domain
• For learning, all instances have to have the same
  set of entities

24
First Order Approaches

• Advantages of first order probabilistic models
• represent world in terms of individuals and
  relationships between them
• ability to generalize about many instances in
  same domain
• allow compact parameterization
• support reasoning about general classes of
  individuals rather than the individuals
  themselves
• allow representation of high level structure, in
  which objects interact weakly with each other

25
Three Different Approaches

• Rule-based approaches focus on facts
• what is true in the world?
• what facts do other facts depend on?
• 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)?
• Programming language approaches focus on
  processes
• how is the world generated?
• how does one event influence another event?

26
Roadmap

• Motivation
• Background
• Rule-based Approaches
• Basic Approach
• Knowledge-Based Model Construction
• Issues
• First-Order Variable Elimination
• Learning
• Frame-based Approaches
• Undirected Relational Approaches
• Programming Language Approaches

27
Flavors

• Goldman Charniak 93
• Breese 92
• Probabilistic Horn Abduction Poole 93
• Probabilistic Logic Programming Ngo Haddawy
  96
• Relational Bayesian Networks Jaeger 97
• Bayesian Logic Programs Kersting de Raedt 00
• Stochastic Logic Programs Muggleton 96
• PRISM Sato Kameya 97
• CLP(BN) Costa et al. 03
• etc.

28
Intuitive Approach

• In logic programming,
• accepted(P) - author(P,A), famous(A).
• means
• For all P,A if A is the author of P and A is
  famous, then P is accepted
• This is a categorical inference
• But this will not be true in many cases

29
Fudge Factors

• Use
• accepted(P) - author(P,A), famous(A). (0.6)
• This means
• For all P,A if A is the author of P and A is
  famous, then P is accepted with probability 0.6
• But what does this mean when there are other
  possible causes of a paper being accepted?
• e.g. accepted(P) - high_quality(P). (0.8)

30
Intuitive Meaning

• accepted(P) - author(P,A), famous(A). (0.6)
• means
• For all P,A if A is the author of P and A is
  famous, then P is accepted with probability 0.6,
  provided no other possible cause of the paper
  being accepted holds
• If more than one possible cause holds, a
  combining rule is needed to combine the
  probabilities

31
Meaning of Disjunction

• In logic programming
• accepted(P) - author(P,A), famous(A).
• accepted(P) - high_quality(P).
• means
• For all P,A if A is the author of P and A is
  famous, or if P is high quality, then P is
  accepted

32
Intuitive Meaning of Probabilistic Disjunction

• For us
• accepted(P) - author(P,A), famous(A). (0.6)
• accepted(P) - high_quality(P). (0.8)
• means
• For all P,A, if (A is the author of P and A is
  famous successfully cause P to be accepted) or (P
  is high quality successfully causes P to be
  accepted), then P is accepted.
• If A is the author of P and A is famous, they
  successfully cause P to be accepted with
  probability 0.6.
• If P is high quality, it successfully causes P to
  be accepted with probability 0.8.

33
Noisy-Or

• Multiple possible causes of an effect
• Each cause, if it is true, successfully causes
  the effect with a given probability
• Effect is true if any of the possible causes is
  true and successfully causes it
• All causes act independently to produce the
  effect (causal independence)
• Note accepted(P) - author(P,A), famous(A).
  (0.6) may produce multiple possible causes for
  different values of A
• Leak probability effect may happen with no cause
• e.g. accepted(P). (0.1)

34
Noisy-Or
author(p1,alice)
author(p1,bob)
high_quality(p1)
famous(alice)
famous(bob)
0.6
0.6
0.8
accepted(p1)
35
Computing Noisy-Or Probabilities

• What is P(accepted(p1)) given that Alice is an
  author and Alice is famous, and that the paper is
  high quality, but no other possible cause is true?

leak
36
Combination Rules

• Other combination rules are possible
• E.g. max
• In our case,
• P(accepted(p1)) max 0.6,0.8,0.1 0.8
• Harder to interpret in terms of logic program

37
Roadmap

• Motivation
• Background
• Rule-based Approaches
• Basic Approach
• Knowledge-Based Model Construction
• Issues
• First-Order Variable Elimination
• Learning
• Frame-based Approaches
• Undirected Relational Approaches
• Programming Language Approaches

38
Knowledge-Based Model Construction (KBMC)

• Construct a Bayesian network, given a query Q and
  evidence E
• query and evidence are sets of ground atoms,
  i.e., predicates with no variable symbols
• e.g. author(p1,alice)
• Construct network by searching for possible
  proofs of the query and the variables
• Use standard BN inference techniques on
  constructed network

39
KBMC Example

• smart(alice). (0.8)
• smart(bob). (0.9)
• author(p1,alice). (0.7)
• author(p1,bob). (0.3)
• high_quality(P) - author(P,A), smart(A). (0.5)
• high_quality(P). (0.1)
• accepted(P) - high_quality(P). (0.9)
• Query is accepted(p1).
• Evidence is smart(bob).

40
Backward Chaining

• Start with evidence variable smart(bob)

smart(bob)
41
Backward Chaining

• Rule for smart(bob) has no antecedents stop
  backward chaining

smart(bob)
42
Backward Chaining

• Begin with query variable accepted(p1)

smart(bob)
accepted(p1)
43
Backward Chaining

• Rule for accepted(p1) has antecedent
  high_quality(p1) add high_quality(p1) to
  network, and make parent of accepted(p1)

smart(bob)
high_quality(p1)
accepted(p1)
44
Backward Chaining

• All of accepted(p1)s parents have been found
  create its conditional probability table (CPT)

smart(bob)
high_quality(p1)
high_quality(p1)
accepted(p1)
hq
0.7
0.3
accepted(p1)
hq
0
1
45
Backward Chaining

• high_quality(p1) - author(p1,A), smart(A) has
  two groundings Aalice and Abob

smart(bob)
high_quality(p1)
accepted(p1)
46
Backward Chaining

• For grounding Aalice, add author(p1,alice) and
  smart(alice) to network, and make parents of
  high_quality(p1)

smart(bob)
smart(alice)
author(p1,alice)
high_quality(p1)
accepted(p1)
47
Backward Chaining

• For grounding Abob, add author(p1,bob) to
  network. smart(bob) is already in network. Make
  both parents of high_quality(p1)

smart(bob)
smart(alice)
author(p1,alice)
author(p1,bob)
high_quality(p1)
accepted(p1)
48
Backward Chaining

• Create CPT for high_quality(p1) make noisy-or,
  and dont forget leak probability

smart(bob)
smart(alice)
author(p1,alice)
author(p1,bob)
high_quality(p1)
accepted(p1)
49
Backward Chaining

• author(p1,alice), smart(alice) and author(p1,bob)
  have no antecedents stop backward chaining

smart(bob)
smart(alice)
author(p1,alice)
author(p1,bob)
high_quality(p1)
accepted(p1)
50
Backward Chaining

• assert evidence smart(bob) true, and compute
  P(accepted(p1) smart(bob) true)

true
smart(bob)
smart(alice)
author(p1,alice)
author(p1,bob)
high_quality(p1)
accepted(p1)
51
Backward Chaining on Both Query and Evidence

• Necessary, if query and evidence have common
  ancestor
• Sufficient. P(Query Evidence) can be computed
  using only ancestors of query and evidence nodes
• unobserved descendants are irrelevant

Ancestor
Query
Evidence
52
Roadmap

• Motivation
• Background
• Rule-based Approaches
• Basic Approach
• Knowledge-Based Model Construction
• Issues
• First-Order Variable Elimination
• Learning
• Frame-based Approaches
• Undirected Relational Approaches
• Programming Language Approaches

53
The Role of Context

• Context is deterministic knowledge known prior to
  the network being constructed
• May be defined by its own logic program
• Is not a random variable in the BN
• Used to determine the structure of the
  constructed BN
• If a context predicate P appears in the body of a
  rule R, only backward chain on R if P is true

54
Context example

• Suppose author(P,A) is a context predicate,
  author(p1,bob) is true, and author(p1,alice)
  cannot be proven from deterministic KB (and is
  therefore false by assumption)
• Network is

No author(p1,bob) node because it is a context
predicate
smart(bob)
No smart(alice) node because author(p1,alice) is
false
high_quality(p1)
accepted(p1)
55
Basic Assumptions

• No cycles in resulting BN
• If there are cycles, cannot interpret BN as
  definition of joint probability distribution
• Model construction process terminates
• in particular, no function symbols. Consider
• famous(X) - famous(advisor(X)).
• this creates an infinite backwards chain

famous(advisor(advisor(X)))
famous(advisor(X))
famous(X)
56
Resolving Cycles Glesner
Koller 95

• Deal with cycles by introducing time
• E.g.
• famous(X) - coauthor(X,Y), famous(Y).
• becomes
• famous(X,T) - coauthor(X,Y,T-1), famous(Y,T-1).
• Defines a dynamic Bayesian network

57
Resolving Infinite Networks Pfeffer Koller 00

• How to deal with famous(X) - famous(advisor(X)).
• KBMC will construct an infinite network cant
  do
• To compute P(famous(alice))
• Expand network backwards n generations
• Assume uniform distribution at roots
• Compute P n(famous(alice))
• The series P 1(famous(alice)), P
  2(famous(alice)), will converge as n??
• The limit is defined to be P(famous(alice))

58
Reusing Inference

• Iteration 1 compute P 1(famous(X))
• Iteration 2 compute P 2(famous(X))
• requires P 1(famous(advisor(X))) P 1(famous(X))
• already computed reuse result
• Iteration 3 compute P 3(famous(X))
• requires P 2(famous(advisor(X))) P 2(famous(X))
• already computed reuse result
• Constant amount of work per iteration

59
Roadmap

• Motivation
• Background
• Rule-based Approaches
• Basic Approach
• Knowledge-Based Model Construction
• Issues
• First-Order Variable Elimination
• Learning
• Frame-based Approaches
• Undirected Relational Approaches
• Programming Language Approaches

60
Semantics

• Assuming BN construction process terminates,
  conditional probability of any query given any
  evidence is defined by the BN.
• Somewhat unsatisfying because
• meaning of program is query dependent (depends
  on constructed BN)
• meaning is not stated declaratively in terms of
  program but in terms of constructed network
  instead

61
Disadvantage of Intuitive Approach

• Up until now, ground logical atoms have been
  random variables ranging over T,F
• cumbersome to have a different random variable
  for lead_author(p1,alice), lead_author(p1,bob)
  and all possible values of lead_author(p1,A)
• worse, since lead_author(p1,alice) and
  lead_author(p1,bob) are different random
  variables, it is possible for both to be true at
  the same time

62
Bayesian Logic Programs Kersting and de Raedt
00, following Ngo Haddawy 95

• Now, ground atoms are random variables with any
  range (not necessarily Boolean)
• now quality is a random variable, with values
  high, medium, low
• Any probabilistic relationship is allowed
• expressed in CPT
• Semantics of program given once and for all
• not query dependent

63
Meaning of Rules in BLPs

• accepted(P) - quality(P).
• means
• For all P, if quality(P) is a random variable,
  then accepted(P) is a random variable
• Associated with this rule is a conditional
  probability table (CPT) that specifies the
  probability distribution over accepted(P) for any
  possible value of quality(P)

64
Combining Rules for BLPs

• accepted(P) - quality(P).
• accepted(P) - author(P,A), fame(A).
• Before, combining rules combined individual
  probabilities with each other
• noisy-or and max rules made sense
• Now, combining rules combine entire CPTs
• can still define noisy-or and max
• harder to interpret

65
Semantics of BLPs

• Random variables are all ground atoms that have
  finite proofs in logic programs
• assumes acyclicity
• assumes no function symbols
• Can construct BN over all random variables
• parents derived from rules
• CPTs derived using combining rules
• Semantics of BLP joint probability distribution
  over all random variables
• does not depend on query
• Inference in BLP by KBMC

66
An Issue

• How to specify uncertainty over single-valued
  relations?
• Approach 1 make lead_author(P) a random variable
  taking values bob, alice etc.
• we cant say accepted(P) - lead_author(P),
  famous(A) because A does not appear in the rule
  head or in a previous term in the body
• Approach 2 make lead_author(P,A) a random
  variable with values true, false
• we run into the same problems as with the
  intuitive approach (may have zero or many lead
  authors)
• Approach 3 make lead_author a function
• say accepted(P) - famous(lead_author(P))
• BLPs need to specify how to deal with function
  symbols and uncertainty over them

67
First-Order Variable Elimination Poole 03, see
also Braz et al 05

• Generalization of variable elimination to first
  order domains
• Reasons directly about first-order variables,
  instead of at the ground level
• Assumes that the size of the population for each
  type of entity is known

68
FOVE Example

• famous(XPerson) - coauthor(X,Y). (0.2)
• coauthor(XPerson,YPerson) - knows(X,Y). (0.3)
• knows(XPerson,YPerson). (0.01)
• Person 1000
• Evidence knows(alice,bob)
• Query famous(alice)

69
What KBMC Will Produce
knows(a,b)
knows(a,c)
knows(a,d)
1000 times
coauthor(a,b)
coauthor(a,c)
coauthor(a,d)
famous(alice)
70
Better Idea

• Instead of grounding out all variables, reason
  about some of them at the lifted level
• Eliminate entire relations at a time, instead of
  individual ground terms
• Use parameterized variables, e.g. reason directly
  about coauthor(X,Y)
• Use the known population size to quantify over
  populations

71
Parameterized Factors or Parfactors

• Functions from parameterized variables to
  positive real numbers cf. factors in VE
• Plus constraints on parameters

X alice
knows(X,Y)
coauthor(X,Y)
f
f
1
f
t
0
t
f
0.7
t
t
0.3
72
Splitting
knows(X,Y)
coauthor(X,Y)
f
f
1
Split
produces
on
Y bob
f
t
0
t
f
0.7
t
t
0.3
Y ? bob
Y bob
knows(X,Y)
coauthor(X,Y)
knows(X,Y)
coauthor(X,Y)
f
f
1
f
f
1
f
t
0
f
t
0
t
f
0.7
t
f
0.7
t
t
0.3
t
t
0.3
residual
73
Conditioning on Evidence
Condition
produces
on
knows(alice,bob)
X ? alice or Y ? bob
X alice Y bob
coauthor(X,Y)
knows(X,Y)
coauthor(X,Y)
f
f
1
f
0.7
f
t
0
t
0.3
t
f
0.7
t
t
0.3
In reality, constraints are conjunctive. Three
parfactors X alice Y bob, X ? alice
and X alice Y ? bob will be produced
74
Eliminating knows(X,Y)
X ? alice or Y ? bob
knows(X,Y)
coauthor(X,Y)
f
f
1
Multiply
by
produces
f
t
0
t
f
0.7
t
t
0.3
X ? alice or Y ? bob
knows(X,Y)
coauthor(X,Y)
f
f
0.99
f
t
0
t
f
0.007
t
t
0.003
75
Eliminating knows(X,Y)
X ? alice or Y ? bob
knows(X,Y)
coauthor(X,Y)
f
f
0.99
Summing out knows(X,Y) in
produces
f
t
0
t
f
0.007
t
t
0.003
X ? alice or Y ? bob
coauthor(X,Y)
f
0.997
t
0.003
76
Eliminating coauthor(X,Y) Multiplying Multiple
Parfactors

• Use unification to decide which factors to
  multiply,
• and what their constraints will be

X alice
famous(X)
coauthor(X,Y)
f
f
1
f
t
0.8
t
f
0
t
t
0.2
X ? alice or Y ? bob
X alice Y bob
coauthor(X,Y)
coauthor(X,Y)
f
0.7
f
0.997
t
0.3
t
0.003
77
Multiplying Multiple Parfactors

• Multiply each pair of factors that unify, to
  produce

X alice Y ? bob
X alice Y bob
famous(X)
coauthor(X,Y)
famous(X)
coauthor(X,Y)
f
f
0.7
f
f
0.997
f
t
0.24
f
t
0.0024
t
f
0
t
f
0
t
t
0.06
t
t
0.0006
78
Aggregating Over Populations
X alice Y ? bob
famous(X)
coauthor(X,Y)
f
f
0.997
The parfactor
represents a
f
t
0.0024
t
f
0
t
t
0.0006
ground factor for each person in the population
other than bob. These factors combine via noisy
or.
population size - 1
from X alice Y bob parfactor
79
Detail Determining Variables in Product
k(X2,Y2)
f(X2,Y2)
k(X1,Y1)
f
f
1
Multiplying
by
produces
f
0.99
f
t
0
t
0.01
t
f
0.7
t
t
0.3
X1?X2 or Y1 ?Y2
k(X1,Y1)
f(X2,Y2)
k(X2,Y2)
k(X2,Y2)
f(X2,Y2)
f
f
0.99
f
f
f
0.99
f
t
0
f
f
t
0
f
f
0.693
t
t
f
0.007
and
f
t
0.297
t
t
t
0.003
t
f
0.01
f
t
f
0
t
for the case where X1X2 and Y1Y2
t
t
0.007
f
t
t
0.003
t
80
Other details

• When multiplying two parfactors, compute their
  most general unifier (mgu)
• Split the parfactors on the mgu
• Keep the residuals
• Multiply the non-residuals together
• See Poole 03 for more details
• See Braz, Amir and Roth 05 at IJCAI!

81
Roadmap

• Motivation
• Background
• Rule-based Approaches
• Basic Approach
• Knowledge-Based Model Construction
• Issues
• First-Order Variable Elimination
• Learning
• Frame-based Approaches
• Undirected Relational Approaches
• Programming Language Approaches

82
Learning Rule Parameters Koller Pfeffer 97,
Sato Kameya 01

• Problem definition
• Given a skeleton rule base consisting of rules
  without uncertainty parameters
• and a set of instances, each with
• a set of context predicates
• observations about some random variables
• Goal learn parameter values for the rules that
  maximize the likelihood of the data

83
Basic Approach

• Construct a network BNi for each instance i using
  KBMC, backward chaining on all the observed
  variables
• Expectation Maximization (EM)
• exploit parameter sharing

84
Parameter Sharing

• In BNs, all random variables have distinct CPTs
• only share parameters between different
  instances, not different random variables
• In logical approaches, an instance may contain
  many objects of the same kind
• multiple papers, multiple authors, multiple
  citations
• Parameters are shared within instances
• same parameters used across different papers,
  authors, citations
• Parameter sharing allows faster learning, and
  learning from a single instance

85
Rule Parameters and CPT Entries

• In principle, combining rules produce complicated
  relationship between model parameters and CPT
  entries
• With a decomposable combining rule, each node is
  derived from a single rule
• Most natural combining rules are decomposable
• E.g. noisy-or decomposes into set of ands
  followed by or

86
Parameters and Counts

• Each time a node is derived from a rule r, it
  provides one experiment to learn about the
  parameters associated with r
• Each such node should therefore make a separate
  contribution to the count for those parameters
• the parameter associated with
  P(XxParentsXu) when rule r applies
• the number of times a node has value x
  and its parents have value u when rule r applies

87
EM With Parameter Sharing

• Given parameter values, compute expected counts
• where the inner sum is over all nodes derived
  from rule r in BNi
• Given expected counts, estimate
• Iterate these two steps

88
Learning Rule Structure Kersting and De Raedt 02

• Problem definition
• Given a set of instances, each with
• context predicates
• observations about some random variables
• Goal learn
• a skeleton rule base consisting of rules without
  uncertainty parameters
• and parameter values for the rules
• Generalizes BN structure learning
• define legal models
• scoring function same as for BN
• define search operators

89
Legal Models

• Hypothesis space consists of all rule sets using
  given predicates, together with parameter values
• A legal hypothesis
• is logically valid rule set does not draw false
  conclusions for any data cases
• the constructed BN is acyclic for every instance

90
Search operators

• Add a constant-free atom to the body of a single
  clause
• Remove a constant-free atom from the body of a
  single clause

accepted(P) - author(P,A). accepted(P) -
quality(P).
91
Summary Rule-Based Approaches

• Provide an intuitive way to describe how one fact
  depends on other facts
• Incorporate relationships between entities
• Generalizes to many different situations
• Constructed BN for a domain depends on which
  objects exist and what the known relationships
  are between them (context)
• Inference at the ground level via KBMC
• or lifted inference via FOVE
• Both parameters and structure are learnable

92
Roadmap

• Motivation
• Background
• Rule-based Approaches
• Frame-based Approaches
• Undirected Relational Approaches
• Programming Language Approaches

93
Frame-based Approaches

• Probabilistic Relational Models (PRMs)
• Representation Inference Koller Pfeffer 98,
  Pfeffer, Koller, Milch Takusagawa 99, Pfeffer
  00
• Learning Friedman et al. 99, Getoor, Friedman,
  Koller Taskar 01 02, Getoor 01
• Probabilistic Entity Relation Models (PERs)
• Representation Heckerman, Meek Koller 04

94
Roadmap

• Motivation
• Background
• Rule-based Approaches
• Frame-based Approaches
• PRMs w/ Attribute Uncertainty
• Inference in PRMs
• Learning in PRMs
• PRMs w/ Structural Uncertainty
• PRMs w/ Class Hierarchies
• Undirected Relational Models
• Programming Language Approaches

95
Probabilistic Relational Models

• Combine advantages of relational logic Bayesian
  networks
• 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.

96
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

97
Probabilistic Relational Model
Review
Author
Smart
Mood
Good Writer
Length
Paper
Quality
Accepted
98
Probabilistic Relational Model
Review
Author
Smart
Mood
Good Writer
Length
Paper
Quality
Accepted
99
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
100
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

101
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
102
A Portion of the BN
P2.Accepted
P3.Accepted
103
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
104
A Portion of the BN
P2.Accepted
P3.Accepted
105
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
106
PRM Aggregate Dependencies
Paper
Review
Mood
Quality
Length
Accepted
107
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
108
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
109
Roadmap

• Motivation
• Background
• Rule-based Approaches
• Frame-based Models
• PRMs w/ Attribute Uncertainty
• Inference in PRMs
• Learning in PRMs
• PRMs w/ Structural Uncertainty
• PRMs w/ Class Hierarchies
• Undirected Relational Models
• Programming Language Approaches

110
PRM Inference

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

111
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
112
PRM Inference Constructed BN
A1.Smart
A1.Good Writer
113
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

114
PRM Inference Interfaces
Variables pertaining to R2 inputs and internal
attributes
A1.Smart
A1.Good Writer
P1.Quality
P1.Accepted
115
PRM Inference Interfaces
Interface imported and exported attributes
A1.Smart
A1.Good Writer
R2.Mood
P1.Quality
R2.Length
P1.Accepted
116
PRM Inference Encapsulation
R1 and R2 are encapsulated inside P1
A1.Smart
A1.Good Writer
117
PRM Inference Reuse
A1.Smart
A1.Good Writer
118
Structured Variable Elimination
Author 1
A1.Smart
A1.Good Writer
Paper-1
Paper-2
119
Structured Variable Elimination
Author 1
A1.Smart
A1.Good Writer
Paper-1
Paper-2
120
Structured Variable Elimination
Paper 1
A1.Smart
A1.Good Writer
Review-2
Review-1
P1.Quality
P1.Accepted
121
Structured Variable Elimination
Paper 1
A1.Smart
A1.Good Writer
Review-2
Review-1
P1.Quality
P1.Accepted
122
Structured Variable Elimination
Review 2
A1.Good Writer
R2.Mood
R2.Length
123
Structured Variable Elimination
Review 2
A1.Good Writer
R2.Mood
124
Structured Variable Elimination
Paper 1
A1.Smart
A1.Good Writer
Review-2
Review-1
P1.Quality
P1.Accepted
125
Structured Variable Elimination
Paper 1
A1.Smart
A1.Good Writer
Review-1
R2.Mood
P1.Quality
P1.Accepted
126
Structured Variable Elimination
Paper 1
A1.Smart
A1.Good Writer
Review-1
R2.Mood
P1.Quality
P1.Accepted
127
Structured Variable Elimination
Paper 1
A1.Smart
A1.Good Writer
R2.Mood
R1.Mood
P1.Quality
P1.Accepted
128
Structured Variable Elimination
Paper 1
A1.Smart
A1.Good Writer
R2.Mood
R1.Mood
P1.Quality
True
P1.Accepted
129
Structured Variable Elimination
Paper 1
A1.Smart
A1.Good Writer
130
Structured Variable Elimination
Author 1
A1.Smart
A1.Good Writer
Paper-1
Paper-2
131
Structured Variable Elimination
Author 1
A1.Smart
A1.Good Writer
Paper-2
132
Structured Variable Elimination
Author 1
A1.Smart
A1.Good Writer
Paper-2
133
Structured Variable Elimination
Author 1
A1.Smart
A1.Good Writer
134
Structured Variable Elimination
Author 1
A1.Good Writer
135
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

136
Limitations of SVE

• Does not work when encapsulation breaks down
• But when we dont have specific information about
  the connections between objects, we 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
137
Roadmap

• Motivation
• Background
• Rule-based Approaches
• Frame-based Approaches
• PRMs w/ Attribute Uncertainty
• Inference in PRMs
• Learning in PRMs
• PRMs w/ Structural Uncertainty
• PRMs w/ Class Hierarchies
• Undirected Relational Models
• Programming Language Approaches

138
Learning PRMs w/ AU
Author
Database
Paper
Review
PRM
Author
Paper
Review

• Parameter estimation

• Structure selection

Relational Schema
139
ML Parameter Estimation
Review
Mood
Paper
Length
Quality
Accepted
140
ML Parameter Estimation
Review
Mood
Paper
Length
Quality
Accepted
q
141
Structure Selection

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

142
Structure Selection

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

143
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?
144
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
145
Structure Selection

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

146
Structure Selection

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

147
Searching Model Space
Phase 0 consider only dependencies within a class
Author
Review
Paper
148
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
149
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
150
Roadmap

• Motivation
• Background
• Rule-based Approaches
• Frame-based Approaches
• PRMs w/ Attribute Uncertainty
• Inference in PRMs
• Learning in PRMs
• PRMs w/ Structural Uncertainty
• PRMs w/ Class Hierarchies
• Undirected Relational Models
• Programming Language Approaches

151
Reminder 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
152
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?

153
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

154
Citation Relational Schema
Author
Institution
Research Area
Wrote
Paper
Paper
Topic
Topic
Word1
Word1
Word2
Cites

Word2

Citing Paper
WordN
Cited Paper
WordN
155
Attribute Uncertainty
Author
Institution
P( Institution Research Area)
Research Area
Wrote
P( Topic Paper.Author.Research Area
Paper
Topic
P( WordN Topic)
...
Word1
WordN
156
Reference Uncertainty

Bibliography
1. ----- 2. ----- 3. -----
Scientific Paper
Document Collection
157
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

158
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
159
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
160
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
161
PRMs w/ RU Semantics
Paper
Paper
Topic
Topic
Cites
Words
Words
Cited
Citing
PRM RU
162
Learning PRMs w/ RU

• Idea just like in PRMs w/ AU
• define scoring function
• do greedy local structure search
• Issues
• expanded search space
• construct partitions
• new operators

163
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
164
Legal Models
Review
Mood
Paper
Paper
Important
Important
Accepted
Cites
Accepted
Citing
Cited
165
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.
166
Structure Search
Cites
Author
Citing
Institution
Cited
Cited
167
Structure Search New Operators
Cites
Author
Citing
Institution
Cited
Refine on Topic
Cited
?score
Paper
Paper
Paper
Paper
Paper
168
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
169
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

170
Existence Uncertainty
Document Collection
Document Collection
171
PRM w/ Exists Uncertainty
Paper
Paper
Topic
Topic
Cites
Words
Words
Exists
Dependency model for existence of relationship
172
Exists Uncertainty Example
Paper
Paper
Topic
Topic
Cites
Words
Words
Exists
False
True
Cited.Topic
Citer.Topic
173
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
174
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
175
PRMs w/ EU Semantics
Paper
Paper
Topic
Topic
Cites
Words
Words
Exists
PRM EU
176
Learning PRMs w/ EU

• Idea just like in PRMs w/ AU
• define scoring function
• do greedy local structure search
• Issues
• efficiency
• Computation of sufficient statistics for exists
  attribute
• Do not explicitly consider relations that do not
  exist

177
Structure Selection
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
178
Roadmap

• Motivation
• Background
• Rule-based Approaches
• Frame-based Approaches
• PRMs w/ Attribute Uncertainty
• Inference in PRMs
• Learning in PRMs
• PRMs w/ Structural Uncertainty
• PRMs w/ Class Hierarchies
• Undirected Relational Models
• Programming Language Approaches

179
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
180
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)

181
Learning PRM-CHs
Vote
Database
Person
Instance I
PRM-CH
Vote

• Class hierarchy provided

TVProgram
Person
Relational Schema
182
Structure Selection
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
183
Guaranteeing Acyclicity with Subclasses
184
Learning PRM-CH

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

AcceptedPaper
AcceptedJournal
AcceptedConference
AcceptedWorkshop
185
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

186
PRMs w/ Class Hierarchies

• Allows 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
187
PRM-CH Summary

• PRMs with class hierarchies are a natural
  extension of PRMs
• Specialization/Inheritance of CPDs
• Allows new dependency structures
• Provide bridge from class-based to instance-based
  models
• Learning techniques proposed
• Need efficient heuristics
• Empirical validation on real-world domains

188
Summary 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

189
Roadmap

• Motivation
• Background
• Rule-based Approaches
• Frame-based Approaches
• Undirected Relational Approaches
• Background Markov Networks
• Frame-based Approaches
• Rule-based Approaches
• Programming Language Approaches

190
Markov Networks
Author1 Fame
Author2 Fame
Author3 Fame
Author4 Fame
nodes domain variables edges mutual influence
Network structure encodes conditional
independencies I(A1 Fame, A4 Fame A2
Fame, A3 Fame)
191
Markov Network Semantics
F1
F2
conditional independencies in MN structure
local clique potentials
full joint distribution over domain


F3
F4
where Z is a normalizing factor that ensures that
the probabilities sum to 1
Good news no acyclicity constraints Bad news
global normalization (1/Z)
192
Roadmap

• Motivation
• Background
• Rule-based Approaches
• Frame-based Approaches
• Undirected Relational Approaches
• Background Markov Networks
• Frame-based Approaches
• Markov Relational Networks Taskar, Segal
  Koller 01, Taskar, Abbeel Koller 02, Taskar,
  Guestrin, Koller Taskar 04
• Rule-based Approaches
• Programming Language Approaches

193
Advantages of Undirected Models

• Symmetric, non-causal interactions
• Web categories of linked pages are correlated
• Social nets individual correlated with peers
• Cannot introduce direct edges because
• of cycles
• Patterns involving multiple entities
• Web triangle patterns
• Social nets transitive relations

194
Relational Markov Networks

• Locality
• Local probabilistic dependencies given by
  relational links
• Universals
• Same dependencies hold for all objects linked in
  a particular pattern

Template potential
Author1
Paper1