Journal of Machine Learning Research 3 (2002) 679-707 Submitted 12/01; Published 12/02
Learning Probabilistic Models of Link Structure
Lise Getoor GETOOR@CS.UMD.EDU
Computer Science Dept. and UMIACS
University of Maryland
College Park, MD 20742
Nir Friedman NIR@CS.HUJI.AC.IL
School of Computer Sci. & Eng.
Hebrew University
Jerusalem, 91904, Israel
Daphne Koller KOLLER@CS.STANFORD.EDU
Computer Science Dept.
Stanford University
Stanford, CA 94305
Benjamin Taskar BTASKAR@CS.STANFORD.EDU
Computer Science Dept.
Stanford University
Stanford, CA 94305
Abstract
Most real-world data is heterogeneous and richly interconnected. Examples include the Web,
hypertext, bibliometric data and social networks. In contrast, most statistical learning methods
work with “flat” data representations, forcing us to convert our data into a form that loses much of
the link structure. The recently introduced framework of probabilistic relational models (PRMs)
embraces the object-relational nature of structured data by capturing probabilistic interactions be-
tween attributes of related entities. In this paper, we extend this framework by modeling interactions
between the attributes and the link structure itself. An advantage of our approach is a unified gener-
ative model for both content and relational structure. We propose two mechanisms for representing
a probabilistic distribution over link structures: reference uncertainty and existence uncertainty.
We describe the appropriate conditions for using each model and present learning algorithms for
each. We present experimental results showing that the learned models can be used to predict link
structure and, moreover, the observed link structure can be used to provide better predictions for
the attributes in the model.
Keywords: Probabilistic Relational Models, Bayesian Networks, Relational Learning
1. Introduction
In recent years, we have witnessed an explosion in the amount of information that is available to
us in digital form. More and more data is being stored, and more and more data is being made
accessible, through traditional interfaces such as corporate databases and, of course, via the Internet
c
©2002 Lise Getoor, Nir Friedman, Daphne Koller and Benjamin Taskar.

GETOOR, FRIEDMAN, KOLLER AND TASKAR
and the World Wide Web. There is much to be gained by applying machine learning techniques to
these data, in order to extract useful information and patterns.
Most often, the objects in these data do not exist in isolation — there are “links” or relationships
that hold between them. For example, there are links from one web page to another, a scientific
paper cites another paper, and an actor is linked to a movie by the appearance relationship. Most
work in machine learning, however, has focused on “flat” data, where the instances are independent
and identically distributed. The main exception to this rule has been the work on inductive logic
programming (Muggleton, 1992, Lavrac˘ and Dz˘eroski, 1994). This link of work focuses on the
problem of inferring link structure from a logical perspective.
More recently, there has been a growing interest in combining the statistical approaches that have
been so successful when learning from a collection of homogeneous independent instances (typi-
cally represented as a single table in a relational database) with relational learning methods. Slattery
and Craven (1998) were the first to consider combining a statical approach with a first-order rela-
tional learner (FOIL) for the task of web page classification. Chakrabarti et al. (1998) explored
methods for hypertext classifications which used both the content of the current page and informa-
tion from related pages. Popescul et al. (2002) use an approach that uses a relational learner to guide
in the construction of features to be used by a (statistical) propositional learner. Yang et al. (2002)
identify certain categories of relational regularities and explore the conditions under which they can
be exploited to improve classification accuracy.
Here, we propose a unified statistical framework for content and links. Our framework builds on
the recent work on probabilistic relational models (PRMs) (Poole, 1993, Ngo and Haddawy, 1995,
Koller and Pfeffer, 1998). PRMs extend the standard attribute-based Bayesian network representa-
tion to incorporate a much richer relational structure. These models allow properties of an entity to
depend probabilistically on properties of other related entities. The model represents a generic de-
pendence for a class of objects, which is then instantiated for particular sets of entities and relations
between them. Friedman et al. (1999) adapt machinery for learning Bayesian networks from a set
of unrelated homogeneous instances to the task of learning PRMs from structured relational data.
The original PRM framework focused on modeling the distribution over the attributes of the ob-
jects in the model. It took the relational structure itself — the relational links between entities —
to be background knowledge, determined outside the probabilistic model. This assumption implies
that the model cannot be used to predict the relational structure itself. A more subtle yet very impor-
tant point is that the relational structure is informative in and of itself. For example, the links from
and to a web page are very informative about the type of web page (Craven et al., 1998), and the
citation links between papers are very informative about paper topics (Cohn and Hofmann, 2001).
The PRM framework can be naturally extended to address this limitation. By making links first-
class citizens in the model, the PRM language easily allows us to place a probabilistic model directly
over them. In other words, we can extend our framework to define probability distributions over the
presence of relational links between objects in our model. The concept of a probabilistic model
over relational structure was introduced by Koller and Pfeffer (1998) under the name structural
uncertainty. They defined several variants of structural uncertainty, and presented algorithms for
doing probabilistic inference in models involving structural uncertainty.
In this paper, we show how a probabilistic model of relational structure can be learned directly
from data. Specifically, we provide two simple probabilistic models of link structure: The first
is an extension of the reference uncertainty model of Koller and Pfeffer (1998), which makes it
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PROBABILISTIC MODELS OF LINK STRUCTURE
suitable for a learning framework; the second is a new type of structural uncertainty, called existence
uncertainty. We present a clear semantics for these extensions, and propose a method for learning
such models from a relational database. We present empirical results on real-world data, showing
that these models can be used to predict the link structure, as well as use the presence of (observed)
links in the model to provide better predictions about attribute values. Interestingly, these benefits
are obtained even with the very simple models of link structure that we propose in this paper. Thus,
even simplistic models of link uncertainty provide us with increased predictive accuracy.
2. Probabilistic Relational Models
A probabilistic relational model (PRM) specifies a template for a probability distribution over a
database. The template describes the relational schema for the domain, and the probabilistic depen-
dencies between attributes in the domain. A PRM, together with a particular database of objects
and relations, defines a probability distribution over the attributes of the objects and the relations.
2.1 Relational Schema
A schema S for a relational model describes a set of classes, X = X1; : : : ;Xn. Each class is associated
with a set of descriptive attributes and a set of reference slots.1 The set of descriptive attributes of a
class X is denoted A(X). Attribute A of class X is denoted X :A, and its domain of values is denoted
V (X :A). For example, the Actor class might have the descriptive attributes Gender, with domain
fmale, femaleg. For simplicity, we assume in this paper that attribute domains are finite; this is not
a fundamental limitation of our approach.
The set of reference slots of a class X is denoted R (X). We use X :ρ to denote the reference slot ρ of
X . Each reference slot ρ is typed: the domain type of Dom[ρ] = X and the range type Range[ρ] = Y ,
where Y is some class in X . A slot ρ denotes a function from Dom[ρ] = X to Range[ρ] = Y . For
example, we might have a class Role with the reference slots Actor, whose range is the class Actor,
and Movie, whose range is the class Movie. For each reference slot ρ, we can define an inverse slot
ρ−1, which is interpreted as the inverse function of ρ. For example, we can define an inverse slot
for the Actor slot of Role and call it Roles. Note that this is not a one-to-one relation, but returns a
set of Role objects. Finally, we define the notion of a slot chain, which allows us to compose slots,
defining functions from objects to other objects to which they are indirectly related. More precisely,
we define a slot chain τ = ρ1; : : : ;ρk to be a sequence of slots (inverse or otherwise) such that for all
i, Range[ρi] = Dom[ρi+1]. We say that Range[τ] = Range[ρk].
We note that the functional nature of slots does not prevent us from having many-to-many relations
between classes. We simply use a standard transformation where we introduce a class corresponding
to the relationship object. This class will have an object for every tuple of related objects. Each
instance of the relationship has functional reference slots to the objects it relates. The Role class
above is an example of this transformation.
It is useful to distinguish between an entity and a relationship, as in entity-relationship diagrams.
In our language, classes are used to represent both entities and relationships. Thus, a relationship
such as Role, which relates actors to movies, is also represented as a class, with reference slots to
the class Actor and the class Movie. We use XE to denote the set of classes that represent entities,
1. There is a direct mapping between our notion of class and the tables in a relational database: descriptive attributes
correspond to standard table attributes, and reference slots correspond to foreign keys (key attributes of another table).
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GETOOR, FRIEDMAN, KOLLER AND TASKAR
ACTOR
name gender
fred male
ginger female
bing male
MOVIE
name genre
m1 drama
m2 comedy
ROLE
role movie actor role-type
r1 m1 fred hero
r2 m1 ginger heroine
r3 m1 bing villain
r4 m2 bing hero
r5 m2 ginger love-interest
Figure 1: An instantiation of the relational schema for a simple movie domain.
and XR to denote those that represent relationships. We use the generic term object to refer both to
entities and to relationships.
The semantics of this language is straightforward. An complete instantiation I specifies the set of
objects in each class X , and the values for each attribute and each reference slot of each object. Thus,
a complete instantiation I is a set of objects with no missing values and no dangling references. It
describes the set of objects, the relationships that hold between the objects and all the values of the
attributes of the objects. For example, Figure 1 shows an instantiation of our simple movie schema.
It specifies a particular set of actors, movies and roles, along with values for each of their attributes
and references.
As discussed in the introduction, our goal in this paper is to construct probabilistic models over
instantiations. To do so, we need to provide enough background knowledge to circumscribe the set
of possible instantiations. Friedman et al. (1999) assume that the entire relational structure is given
as background knowledge. More precisely, they assume that they are given a relational skeleton,
σr, which specifies the set of objects in all classes, as well as all the relationships that hold between
them; in other words, it specifies the values for all of the reference slots. In our simple movie
example, the relational skeleton would contain all of the information except for the gender of the
actors, the genre of the movies, and the nature of the role.
2.2 Probabilistic Model for Attributes
A probabilistic relational model Π specifies a probability distribution over a set of instantiations I
of the relational schema. More precisely, given a relational skeleton σr, it specifies a distribution
over all complete instantiations I that extend the skeleton σr.
A PRM consists of a qualitative dependency structure, G , and the parameters associated with it,
θG . The dependency structure is defined by associating with each attribute X :A a set of formal
parents Pa(X :A). These correspond to formal parents; they will be instantiated in different ways for
different objects in X . Intuitively, the parents are attributes that are “direct influences” on X :A. The
attribute X :A can depend on another probabilistic attribute B of X . It can also depend on attributes
of related objects. X :τ:B, where τ is a slot chain. In the case where X :τ:B is not single-valued and
(possibly) refers a multi-set of objects, we use an aggregate function and define a dependence on
the computed aggregate value. This is described in greater detail in Getoor (2001).
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PROBABILISTIC MODELS OF LINK STRUCTURE
The quantitative part of the PRM specifies the parameterization of the model. Given a set of
parents for an attribute, we can define a local probability model by associating with it a conditional
probability distribution (CPD). For each attribute we have a CPD that specifies P(X :A j Pa(X :A)).
Each CPD in our PRM is legal, i.e., the entries are positive and sum to 1.
Definition 1 A probabilistic relational model (PRM) Π for a relational schema S defines for each
class X 2 X and each descriptive attribute A 2 A(X), a set of formal parents Pa(X :A), and a
conditional probability distribution (CPD) that represents P(X :A j Pa(X :A)).
Given a relational skeleton σr, a PRM Π specifies a distribution over a set of instantiations I
consistent with σr. This specification is done by mapping the dependencies in the class-level PRM
to the actual objects in the domain. For a class X , we use σr(X) to denote the objects X , as specified
by the relational skeleton σr. (In general we will use the notation σ(X) to refer to the set objects of
each class as defined by any type of domain skeleton.) Let X :A be an attribute in the schema, and
let x be some object in σr(X). The PRM allows two types of formal parents: X :B and X :τ:C. For
a formal parent of the form X :B, the corresponding actual parent of x:A is x:B. For a formal parent
of the form X :τ:C, the corresponding formal parent of x:A is y:C, where y 2 x:τ in σr. Thus, the
class-level dependencies in the PRM are instantiated according to the relational skeleton, to define
object-level dependencies. The parameters specified by the PRM are used for each object in the
skeleton, in the obvious way.
Thus, for a given skeleton, the PRM basically defines a ground Bayesian network. The qualitative
structure of the network is defined via an instance dependency graph Gσr , whose nodes correspond
to descriptive attributes x:A of entities in the skeleton. These are the random variables in our model.
We have a directed edge from y:B to x:A if y:B is an actual parent of x:A, as defined above. The
quantitative parameters of the network are defined by the CPDs in the PRM, with the same CPD
used multiple times in the network. This ground Bayesian network leads to the following chain rule
which defines a distribution over the instantiations compatible with our particular skeleton σr:
P(I j σr;Π) = ∏
X2X
∏
x2σr(X)
∏
A2A(X)
P(x:A j Pa(x:A)) (1)
For this definition to specify a coherent probability distribution over instantiations, we must ensure
that our probabilistic dependencies are acyclic. In particular, we must verify that each random
variable x:A does not depend, directly or indirectly, on its own value. In other words, Gσr must be
acyclic. We say that a dependency structure G is acyclic relative to a relational skeleton σr if the
directed graph Gσr is acyclic.
Theorem 2 (Friedman et al., 1999) Let Π be a PRM with an acyclic instance dependency graph,
and let σr be a relational skeleton. Then, Eq. (1) defines a coherent distributions over instances that
extend σr.
The definition of the instance dependency graph is specific to the particular skeleton at hand: the
existence of an edge from y:B to x:A depends on whether y 2 x:τ, which in turn depends on the
interpretation of the reference slots. Thus, it allows us to determine the coherence of a PRM only
relative to a particular relational skeleton. When we are evaluating different possible PRMs as part
of our learning algorithm, we want to ensure that the dependency structure G we choose results in
coherent probability models for any skeleton.
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GETOOR, FRIEDMAN, KOLLER AND TASKAR






 
 
 






 
Figure 2: Reference uncertainty in a simple citation domain.
For this purpose, we use a class dependency graph, which describes all possible dependencies
among attributes. In this graph, we have an (intra-object) edge X :B ! X :A if X :B is a parent of X :A.
If X :τ:B is a parent of X :A, and Y = Range[τ], we have an (inter-object) edge Y:B ! X :A.
Theorem 3 (Friedman et al., 1999) If the class dependency graph of a PRM Π is acyclic, then the
instance dependency graph for any relational skeleton σr is also acyclic. Hence, Π defines a legal
model for any relational skeleton σr.
In the model described here, all relations between attributes are determined by the relational skele-
ton σr; only the descriptive attributes are uncertain. Thus, Eq. (1) determines the probabilistic model
of the attributes of objects, but does not provide a model for the relations between objects. In the
subsequent sections, we extend our framework to deal with link uncertainty, by extending our prob-
abilistic model to include a distribution over the relational structure itself.
3. Reference Uncertainty
In the previous section, we assumed that the relational structure was not a part of our probabilistic
model, that it was given as background knowledge in the relational skeleton. But by incorporating
the links into the probabilistic model, we can both predict links and more importantly use the links
to help us make predictions about other attributes in the model.
Consider a simple citation domain illustrated in Figure 2. Here we have a document collection.
Each document has a bibliography that references some of the other documents in the collection.
We may know the number of citations made by each document (i.e., it is outside the probabilistic
model). By observing the citations that are made, we can use the links to reach conclusions about
other attributes in the model. For example, by observing the number of citations to papers of various
topics, we may be able to infer something about the topic of the citing paper.
Figure 3(a) shows a simple schema for this domain. We have two classes, Paper and Cites. The
Paper class has information about the topic of the paper and the words contained in the paper. For
now, we simply have an attribute for each word that is true if the word occurs in the page and false
otherwise. The Cites class represents the citation of one paper, the Cited paper, by another paper,
the Citing paper. (In the figure, for readability, we show the Paper class twice.) In this model,
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PROBABILISTIC MODELS OF LINK STRUCTURE
Cites
Cited
Citing
Paper
Topic
Words
Paper
Topic
Words
Paper
P5
Topic
AI
Paper
P4
Topic
Theory
Paper
P2
Topic
Theory
Paper
P3
Topic
AI
Paper
P1
Topic
???
Paper
P5
Topic
AI
Paper
P4
Topic
Theory
Paper
P2
Topic
Theory
Paper
P3
Topic
AI
Paper
P1
Topic
???
RegReg
RegRegCites
(a) (b)
Figure 3: (a) A relational schema for the citation domain. (b) An object skeleton for the citation
domain.
we assume that the set of objects is pre-specified, but relations among them, i.e., reference slots,
are subject to probabilistic choices. Thus, rather than being given a full relational skeleton σr, we
assume that we are given an object skeleton σo. The object skeleton specifies only the objects σo(X)
in each class X 2 X , but not the values of the reference slots. In our example, the object skeleton
specifies the objects in class Paper and the objects in class Cites, but the reference slots of the
Cites relation, Cites.Cited and Cites.Citing are unspecified. In other words, the probabilistic model
does not provide a model of the total number of citation links, but only a distribution over their
“endpoints”. Figure 3 shows an object skeleton for the citation domain.
3.1 Probabilistic model
In the case of reference uncertainty, we must specify a probabilistic model for the value of the
reference slots X :ρ. The domain of a reference slot X :ρ is the set of keys (unique identifiers) of
the objects in the class Y to which X :ρ refers. Thus, we need to specify a probability distribution
over the set of all objects in Y . For example for Cites:Cited, we must specify a distribution over the
objects in class Paper.
A naive approach is to simply have the PRM specify a probability distribution directly over the
objects σo(Y ) in Y . For example for Cites:Cited, we would have to specify a distribution over the
primary keys of Paper. This approach has two major flaws. Most obviously, this distribution would
require a parameter for each object in Y , leading to a very large number of parameters. This is
a problem both from a computational perspective — the model becomes very large, and from a
statistical perspective — we often would not have enough data to make robust estimates for the
parameters. More importantly, we want our dependency model to be general enough to apply over
all possible object skeletons σo; a distribution defined in terms of the objects within a specific object
skeleton would not apply to others.
In order to achieve a general and compact representation, we use the attributes of Y to define the
probability distribution. In this model, we partition the class Y into subsets labeled ψ1; : : : ;ψm ac-
cording to the values of some of its attributes, and specify a probability for choosing each partition,
i.e., a distribution over the partitions. We then select an object within that partition uniformly.
685

GETOOR, FRIEDMAN, KOLLER AND TASKAR
Movie
M5
Genre
foreign
Popularity
low
Decade
1990s
Movie
M4
Genre
thriller
Popularity
low
Decade
1990s
Movie
M3
Genre
foreign
Popularity
high
Decade
1990s
Movie
M2
Genr
thriller
Popularity
high
Decade
1990s
Movie
M1
Genre
thriller
Popularity
high
Decade
1990s
Shows
Theater
 
e
Movie
M5
Genre
foreign
Popularity
low
Decade
1990s
Movie
M3
Genre
foreign
Popularity
high
Decade
1990s
Movie
M4
Genre
thriller
Popularity
low
Decade
1990s
Movie
M2
Genre
thriller
Popularity
high
Decade
1990s
Movie
M1
Genre
thriller
Popularity
high
Decade
1990s
Movie.Genre = foreign Movie.Genre = thriller
M1 M
h  

Te
 at
  M1 M
2  

  

 2    
1
 
        
Type
        
megaplex
art theater
r 
t
Cites
Citing
Cited
Paper
Topic
Words
P1 P2
1.0 9.0
Topic
99.0 01.0
Theory
AI
Paper
P1
Topic
Theory
Paper
P1
Topic
Theory
Paper
P1
Topic
Theory
Paper
P1
Topic
Theory
Paper
P1
Topic
Theory
Paper
P4
Topic
Theory
Paper
P2
Topic
Theory
Paper
P1
Topic
Theory
Paper.Topic = Theory
P2
Paper.Topic = AI
P1Paper
P5
Topic
AI Paper
P3
Topic
AI
(a) (b)
Figure 4: (a) An example of reference uncertainty for a movie theater’s showings. (b) A simple
example of reference uncertainty in the citation domain
For example, consider a description of movie theater showings as in Figure 4(a). For the foreign
key Shows.Movie, we can partition the class Movie by Genre, indicating that a movie theater first
selects the genre of movie it wants to show, and then selects uniformly among the movies with the
selected genre. For example, a movie theater may be much more likely to show a movie which is
a thriller in comparison to a foreign movie. Having selected, for example, to show a thriller, the
theater then selects the actual movie to show uniformly from within the set of thrillers. In addition,
just as in the case of descriptive attributes, the partition choice can depend on other attributes in
our model. Thus, the selector attribute can have parents. As illustrated in the figure, the choice of
movie genre might depend on the type of theater. Consider another example, in our citation domain.
As shown in Figure 4(b), we can partition the class Paper by Topic, indicating that the topic of a
citing paper determines the topics of the papers it cites; and then the cited paper is chosen uniformly
among the papers with the selected topic.
We make this intuition precise by defining, for each slot ρ, a partition function Ψρ. We place
several restrictions on the partition function which are captured in the following definition:
Definition 4 Let X :ρ be a reference slot with range Y . Let Ψρ : Y ! Dom[Ψρ] be a function where
Dom[Ψρ] is a finite set of labels. We say that Ψρ is a partition function for ρ if there is a subset of the
attributes of Y , P [ρ]  A(Y ), such that for any y 2 Y and any y0 2 Y , if the values of the attributes
P [ρ] of y and y0 are the same, i.e., for each A 2 P [ρ], y:A = y0:A, then Ψρ(y) = Ψρ(y0). We refer to
P [ρ] as the partition attributes for ρ.
Thus, the values of the partition attributes are all that is required to determine the partition to which
an object belongs.
In our first example, Ψ
Shows:Movie : Movie !fForeign;Thrillerg and the partition attribute is
P [Shows:Movie] = fGenreg. In the second example, Ψ
Cites:Cited : Paper ! fAI;Theoryg and the
partition attribute is P [Cites:Cited] = fTopicg.
There are a number of natural methods for specifying the partition function. It can be defined
simply by having one partition for each possible combination of values of the partition attributes,
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PROBABILISTIC MODELS OF LINK STRUCTURE
i.e., one partition for each value in the cross product of the partition attribute values. Our examples
above take this approach. In both cases, there is only a single partition attribute, so specifying the
partition function in this manner is not too unwieldy. But for larger collections of partition attributes
or for partition attributes with large domains, this method for defining the partitioning function may
be problematic. A more flexible and scalable approach is to define the partition function using a
decision tree built over the partition attributes. In this case, there is one partition for each of the
leaves in the decision tree.
Each possible value ψ determines a subset of Y from which the value of ρ (the referent) will be
selected. For a particular instantiation I of the database, we use I (Yψ) to represent the set of objects
in I (Y ) that fall into the partition ψ.
We now represent a probabilistic model over the values of ρ by specifying a distribution over
possible partitions, which encodes how likely the reference value of ρ is to fall into one partition
versus another. We formalize our intuition above by introducing a selector attribute Sρ, whose
domain is Dom[Ψρ]. The specification of the probabilistic model for the selector attribute Sρ is the
same as that of any other attribute: it has a set of parents and a CPD. In our earlier example, the
CPD of Show:SMovie might have as a parent Theater:Type. For each instantiation of the parents,
we have a distribution over Dom[Sρ].2 The choice of value for Sρ determines the partition Yψ from
which the reference value of ρ is chosen; the choice of reference value for ρ is uniformly distributed
within this set.
Definition 5 A probabilistic relational model Π with reference uncertainty has the same compo-
nents as in Definition 1. In addition, for each reference slot ρ 2 R (X) with Range[ρ] = Y , we
have:
 a partition function Ψρ with a set of partition attributes P [ρ]  A(Y );
 a new selector attribute Sρ within X which takes on values in the range of Ψρ;
 a set of parents and a CPD for Sρ.
To define the semantics of this extension, we must define the probability of reference slots as well
as descriptive attributes:
P(I j σo;Π) = ∏
X2X
∏
x2σo(X)
∏
A2A(X)
P(x:A j Pa(x:A))
∏
ρ2R (X);y=x:ρ
P(x:Sρ = ψ[y] j Pa(x:Sρ))
jI (Yψ[y])j
(2)
where ψ[y] refers to Ψρ(y) — the partition that the partition function assigns y. Note that the last
term in Eq. (2) depends on I in three ways: the interpretation of x:ρ = y, the values of the attributes
P [ρ] within the object y, and the size of Yψ[y].
2. In the current work, we treat this distribution as a simple multinomial distribution over this value space. In gen-
eral, however, we can represent such a distribution more compactly, e.g., using a Bayesian network. For example,
the genre of movies shown by a movie theater might depend on its type (as above). However, the language of
the movie can depend on the location of the theater. Thus, the partition will be defined by P (Show:Movie) =
fMovie:Genre;Movie:Languageg, and its parents would be Theater:Type and Theater:Location. We can repre-
sent this conditional distribution more compactly by introducing a separate variable S
Movie:Genre, with a parent
Theater:Type, and another S
Movie:Language, with a parent Theater:Location.
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GETOOR, FRIEDMAN, KOLLER AND TASKAR
There is one small problem with this definition: the probability is not well-defined if there are no
objects in a partition, i.e., jI (Yψ)j = 0. First, we require that the referred-to class is non-empty, in
other words, jY j> 0. However, it may still be the case that a partition is empty. In this case, we will
perform a renormalization of the distribution for the selector attribute, removing all probability mass
from the empty partition. We then treat this term, P(x:Sρ=ψ[x:ρ]jPa(x:Sρ))
jYψj , as 0 in the above product. In
other words, an instantiation where x:Sρ = ψ and ψ is an empty partition necessarily has probability
zero.3
3.2 Coherence of the Probabilistic Model
As in the case of PRMs with attribute uncertainty, we must be careful to guarantee that our proba-
bility distribution is in fact coherent. In this case, the object skeleton does not specify which objects
are related to which, and therefore the mapping of formal to actual parents depends on probabilistic
choices made in the model. The associated ground Bayesian network will therefore be cumbersome
and not particularly intuitive. We define our coherence constraints using an instance dependency
graph, relative to our PRM and object skeleton.
Definition 6 The instance dependency graph for a PRM Π and an object skeleton σo is a graph
Gσo with the nodes and edges described below. For each class X and each x 2 σo(X), we have the
following nodes:
 a node x:A for every descriptive attribute X :A;
 a node x:ρ and a node x:Sρ, for every reference slot X :ρ:
The dependency graph contains five types of edges:
 Type I edges: Consider any attribute (descriptive or selector) X :A and formal parent X :B.
We define an edge x:B ! x:A, for every x 2 σo(X).
 Type II edges: Consider any attribute (descriptive or selector) X :A and formal parent X :τ:B
where Range[X :τ] = Y . We define an edge y:B ! x:A, for every x 2 σo(X) and y 2 σo(Y ).
 Type III edges: Consider any attribute X :A and formal parent X :τ:B, where τ = ρ1; : : : ;ρk,
and Dom[ρi] = Xi. We define an edge x:ρ1 ! x:A, for every x 2 σo(X). In addition, for i > 1,
we add an edge xi:ρi ! x:A for for every xi 2 σo(Xi) and for every x 2 σo(X).
 Type IV edges: Consider any slot X :ρ and partition attribute Y:B2P [ρ] where Y = Range[ρ].
We define an edge y:B ! x:Sρ, for every x 2 σo(X) and y 2 σo(Y ).
 Type V edges: Consider any slot X :ρ. We define an edge x:Sρ ! x:ρ, for every x 2 σo(X).
We say that a dependency structure G is acyclic relative to an object skeleton σo if the directed
graph Gσo is acyclic.
Intuitively, type I edges correspond to intra-object dependencies and type II edges to inter-object
dependencies. These are the same edges that we had in the dependency graph for regular PRMs,
except that they also apply to selector attributes. Moreover, there is an important difference in our
3. While an important technical detail, in practice, for the majority of our experimental domains, we did not encounter
empty partitions.
688

PROBABILISTIC MODELS OF LINK STRUCTURE
treatment of type II edges. In this case, the skeleton does not specify the value of x:ρ, and hence we
cannot determine from the skeleton on which object y the attribute x:A actually depends. Therefore,
our instance dependency graph must include an edge from every attribute y:B.
Type III edges represent the fact that the actual choice of parent for x:A depends on the value of
the slots used to define it. When the parent is defined via a slot-chain, the actual choice depends on
the values of all the slots along the chain. Since we cannot determine the particular object from the
skeleton, we must include an edge from every slot xi:ρi potentially included in the chain.
Type IV edges represent the dependency of a selector attribute on the attributes defining the associ-
ated partition. To see why this dependence is required, we observe that our choice of reference value
for x:Sρ depends on the values of the partition attributes P [x:ρ] of all of the different objects y in Y .
Thus, these attributes must be determined before x:Sρ is determined. Finally, type V edges represent
the fact that the actual choice of parent for x:A depends on the value of the selector attributes for
the slots used to define it. In our example, as P [Shows:Movie] = fMovie:Genreg, the genres of all
movies must be determined before we can select the value of the reference slot Shows:Movie.
Based on this definition, we can specify conditions under which Eq. (2) specifies a coherent prob-
ability distribution.
Theorem 7 Let Π be a PRM with reference uncertainty whose dependency structure G is acyclic
relative to an object skeleton σo. Then Π and σo define a coherent probability distribution over
instantiations I that extend σo via Eq. (2).
Proof: The probability of an instantiation I is the joint distribution over a set of random variables
defined via the object skeleton. We have two types of variables:
 We have one random variable x:A for each x 2 σo(X) and each A 2 A(X). Note that this also
includes variables of the form x:Sρ that correspond to selector variables.
 We have one random variable x:ρ for each reference slot ρ 2 R (X) and x 2 σo(X). This
variable denotes the actual object in σo(Range[ρ]) that the slot points to.
Let V1; : : : ;VN be the entire set of variables defined by the object skeleton. Clearly, there is a one-
to-one mapping between joint assignments to these variables and instantiations I of the PRM that
are consistent with σo. Because the instance dependency graph is acyclic, we can assume without
loss of generality that the ordering V1; : : : ;VN is a topological sort of the instance dependency graph;
thus, if Vi = x:A, then all ancestors of Vi in the instance dependency graph appear before it in the
ordering.
Our proof will use the following argument. Assume that we can construct a non-negative function
f (Vi;Vi−1; : : : ;V1) which has the form of a conditional distribution P(Vi j V1; : : : ;Vi−1), i.e., for any
assignment V1; : : : ;Vi−1 to V1; : : : ;Vi−1, we have that
∑
Vi
f (Vi;Vi−1; : : : ;V1) = 1: (3)
Then, by the chain rule for probabilities, we can define
P(V1; : : : ;VN) =
N
∏
i=1
f (Vi;Vi−1; : : : ;V1)
=
N
∏
i=1
P(Vi jV1; : : : ;Vi−1):
689

GETOOR, FRIEDMAN, KOLLER AND TASKAR
If f satisfies Eq. (3), then P is a well-defined joint distribution.
All that remains is to define the function f in a way that it satisfies Eq. (3). Specifically, the
function f will be defined via Eq. (2). We consider the two types of variables in our distribution.
Suppose that Vi is of the form x:A, and consider any parent of X :A. There are two cases:
 If the parent is of the form X :B, then by the existence of type I edges, we have that x:B
precedes x:A in Gσo . Hence, the variable x:B precedes Vi.
 If the parent is of the form X :τ:A, then by the existence of type III edges, for each ρi along
τ xi:ρi precedes x:A in Gσo and hence, by the existence of type V edges all of the x:Sρi also
precede x:A. Furthermore, by the existence of type II edges, for any y 2 x:τ, we have that y:B
precedes x:A.
In both cases, we can define
P(Vi jVi−1; : : : ;V1) = P(x:A j Pa(x:A))
as in Eq. (2). As the right-hand-side is simply a CPD in the PRM, it specifies a well-defined condi-
tional distribution, as required.
Now, suppose that Vi is of the form x:ρ. In this case, the conditional probability depends on
x:Sρ, and the value of the partition attributes of all objects in σo(Range[ρ]). By the existence of
type V edges, x:Sρ precedes x:ρ. Furthermore, by the existence of type IV edges, we have that
y:B precedes x:Sρ for every Y:B 2 P [ρ] and y 2 σo(Range[ρ]). Consequently, the assignment to
V1; : : : ;Vi−1 determines the number of objects in each partition of values of P [ρ] and hence the set
I (Yψ) for every ψ.
Finally, we set
P(Vi = y jVi−1; : : : ;V1) =
(
1
jI (Yψ[y])j
if ψ[y] = x:Sρ
0 otherwise
which is a well defined distribution on the objects in σo(Y ) (if any partitions are empty, we renor-
malize the distribution as described earlier).
This theorem is limited in that it is very specific to the constraints of a given object skeleton. As
in the case of PRMs without relational uncertainty, we want to learn a model in one setting, and
be assured that it will be acyclic for any skeleton we might encounter. We accomplish this goal by
extending our definition of class dependency graph. We do so by extending the class dependency
graph to contain edges that correspond to the edges we defined in the instance dependency graph.
Definition 8 The class dependency graph GΠ for a PRM with reference uncertainty Π has a node
for each descriptive or selector attribute X :A and each reference slot X :ρ, and the following edges:
 Type I edges: For any attribute X :A and formal parent X :B, we have an edge X :B ! X :A.
 Type II edges: For any attribute X :A and formal parent X :τ:B where Range[τ] = Y , we have
an edge Y:B ! X :A.
 Type III edges: For any attribute X :A and formal parent Y:τ:B, where τ = ρ1; : : : ;ρk, and
Dom[ρi] = Xi, we define an edge X :ρ1 ! X :A. In addition, each i > 1, we add an edge
X :ρi ! X :A.
690

PROBABILISTIC MODELS OF LINK STRUCTURE
Shows
Theater
SMovie
Theater
Type
Location
Movie
Genre
Profit
Popularity
Movie
STheater
Type III
Type IVType II
Type IIType III
Type I
Type V
Type V
Type

 

 


Movie
e
e
p

y
 
e

Theater
Te e
(a) (b)
Figure 5: (a) A PRM for the movie theater example. The partition attributes are indicated using
dashed lines. (b) The dependency graph for the movie theater example. The different
edge types are labeled.
 Type IV edges: For any slot X :ρ and partition attribute Y:B where Y = Range[ρ], we have
an edge Y:B ! X :Sρ.
 Type V edges: For any slot X :ρ, we have an edge X :Sρ ! X :ρ.
Figure 5 shows the class dependency graph for our extended movie example.
It is now easy to show that if this class dependency graph is acyclic, then the instance dependency
graph is acyclic.
Lemma 9 If the class dependency graph is acyclic for a PRM with reference uncertainty Π, then
for any object skeleton σo, the instance dependency graph is acyclic.
Proof: Assume by contradiction that there is a cycle
x1:V1 ! x2:V2


xk:Vk ! x1:V1:
Then, because each of these object edges corresponds to an edge in the class dependency graph, we
have the following cycle in the class dependency graph:
X1:V1 ! X2:V2


Xk:Vk ! X1:V1:
This contradicts our hypothesis that the class dependency graph is acyclic.
The following corollary follows immediately:
Corollary 10 Let Π be a PRM with reference uncertainty whose class dependency structure G is
acyclic. For for any object skeleton σo, Π and σo define a coherent probability distribution over
instantiations I that extend σo via Eq. (2).
691

GETOOR, FRIEDMAN, KOLLER AND TASKAR
Paper
P5
Topic
AI
Paper
P4
Topic
Theory
Paper
P2
Topic
Theory
Paper
P3
Topic
AI
Paper
P1
Topic
???
Paper
P5
Topic
AI
Paper
P4
Topic
Theory
Paper
P2
Topic
Theory
Paper
P3
Topic
AI
Paper
P1
Topic
???
???
Cites
Paper
Topic
Words
Paper
Topic
Words
Exists
Citer.Topic Cited.Topic
0.995 0005 Theory Theory
False True
AI Theory 0.999 0001
AI AI 0.993 0008
AI Theory 0.997 0003
(a) (b)
Figure 6: (a) An entity skeleton for the citation domain. (b) A CPD for the Exists attribute of Cites.
4. Existence Uncertainty
The second form of structural uncertainty we introduce is called existence uncertainty. In this case,
we make no assumptions about the number of links that exist. The number of links that exist and
the identity of the links are all part of the probabilistic model and can be used to make inferences
about other attributes in our model. In our citation example above, we might assume that the set
of papers is part of our background knowledge, but we want to provide an explicit model for the
presence or absence of citations. Unlike the reference uncertainty model of the previous section, we
do not assume that the total number of citations is fixed, but rather that each potential citation can
be present or absent.
4.1 Semantics of relational model
The object skeleton used for reference uncertainty assumes that the number of objects in each rela-
tion is known. Thus, if we consider a division of objects into entities and relations, the number of
objects in classes of both types are fixed. Existence uncertainty assumes even less background infor-
mation than specified by the object skeleton. Specifically, we assume that the number of relationship
objects is not fixed in advance.
We assume that we are given only an entity skeleton σe, which specifies the set of objects in our
domain only for the entity classes. Figure 6(a) shows an entity skeleton for the citation example. Our
basic approach is to allow other objects within the model — those in the relationship classes — to
be undetermined, i.e., their existence can be uncertain. In other words, we introduce into the model
all of the objects that can potentially exist in it; with each of them, we associate a special binary
variable that tells us whether the object actually exists or not. We call entity classes determined and
relationship classes undetermined.
To specify the set of potential objects, we note that relationship classes typically represent many-
many relationships; they have at least two reference slots, which refer to determined classes. For
example, our Cite class has the two reference slots Citing and Cited. Thus the potential domain
of the Cites class in a given instantiation I is I (Paper) I (Paper). Each “potential” object x in
692

PROBABILISTIC MODELS OF LINK STRUCTURE
this class has the form Cite[y1;y2]. Each such object is associated with a binary attribute x:E that
specifies whether paper y1 did or did not cite paper y2.
Definition 11 Consider a schema with determined and undetermined classes, and let σe be an entity
skeleton over this schema. We define the induced relational skeleton, σr[σe], to be the relational
skeleton that contains the following objects:
 If X is a determined class, then σr[σe](X) = σe(X).
 Let X be an undetermined class with reference slots ρ1; : : : ;ρk whose range types are Y1; : : : ;Yk
respectively. Then σr[σe](X) contains an object X [y1; : : : ;yk] for all tuples hy1; : : : ;yki 2
σr[σe](Y1)


σr[σe](Yk).
The relations in σr[σe] are defined in the obvious way: Slots of objects of determined classes are
taken from the entity skeleton. Slots of objects of undetermined classes are induced from the object
definition: X [y1; : : : ;yk]:ρi is yi.
To ensure that the semantics of schemas with undetermined classes is well-defined, we need a few
tools. Specifically, we need to ensure that the set of potential objects is well defined and finite. It is
clear that if we allow cyclic references (e.g., an undetermined class with a reference to itself), then
the set of potential objects is not finite. To avoid such situations, we need to put some requirements
on the schema.
Definition 12 A set of classes X is stratified if there exists a partial ordering over the classes 
such that for any reference slot X :ρ with range type Y , Y  X.
Lemma 13 If the set of undetermined classes in a schema is stratified, then given any entity skeleton
σe the number of potential objects in any undetermined class is finite.
Proof: We prove this by induction on the stratification level of the class X . Let ρ1; : : : ;ρk be the set
of reference slots of X , and let Yi = Range[ρi]. Then I (X) = I (Y1)


I (Yk): If X is the first level in
the stratification, it cannot refer to any undetermined classes. Thus, the number of potential objects
in I (X) is simply the product of the number of objects in each determined class Yi as specified in
the entity skeleton σe. Each term is finite, so the product of the terms will be finite.
Next, assume by induction that all of the classes at stratification levels less than i are finite. If X is
at level i in the stratification, the constraints imposed by the stratification constraints imply that the
range type of all of its reference slots are at stratification levels < i. By our induction hypothesis,
each of these classes is finite. Hence, the cross product of the classes of the reference slots is finite
and the number of potential objects in X is finite.
As discussed, each undetermined X has a special existence attribute X :E whose values are V (E) =
ftrue; falseg. For uniformity of notation, we introduce an E attribute for all classes; for classes that
are determined, the E value is defined to be always true. We require that all of the reference slots of
a determined class X have a range type which is also a determined class.
For a PRM with stratified undetermined classes, we define an instantiation to be an assignment of
values to the attributes, including the Exists attribute, of all potential objects.
693

GETOOR, FRIEDMAN, KOLLER AND TASKAR
4.2 Probabilistic model
We now specify the probabilistic model defined by the PRM. By treating the Exists attributes as
standard descriptive attributes, we can essentially build our definition directly on top of the definition
of standard PRMs.
Specifically, the existence attribute for an undetermined class is treated in the same way as a
descriptive attribute in our dependency model, in that it can have parents and children, and has an
associated CPD. Figure 6(b) illustrates a CPD for the Cites.Exists attribute. In this example, the
existence of a citation depends on the topic of the citing paper and the topic of the cited paper; e.g.,
it is more likely that citations will exist between papers with the same topic.4
Using the induced relational skeleton and treating the existence events as descriptive attributes, we
have set things up so that Eq. (1) applies with minor changes. There are two minor changes to the
definition of the distribution:
 We want to enforce that x:E = false if x:ρ:E = false for one of the slots ρ of X . Suppose that
X has the slots ρ1; : : : ;ρk; we define the effective CPD for X :E as follows. Let Pa(X :E) =
Pa(X :E)[fX :ρ1:E; : : : ;X :ρk:Eg, and define
P(X :E j Pa(X :E)) =

P(X :E j Pa(X :E)) if X :ρi:E = true;8i = 1; : : : ;k
0 otherwise
 We want to “decouple” the attributes of non-existent objects from the rest of the PRM. Thus,
if X :A is a descriptive attribute, we define Pa(X :A) = Pa(X :A)[fX :Eg, and
P(X :A j Pa(X :A)) =
(
P(X :A j Pa(X :A)) if X :E = true
1
jV (X :A)j otherwise
It is easy to verify that in both cases P(X :A j Pa(X :A)) is a legal conditional distribution.
In effect, these constraints specify a new PRM Π, in which we treat X :E as a standard descriptive
attribute. For each attribute (including the Exists attribute), we define the parents of X :A in Π to be
Pa(X :A) and the associated CPD to be P(X :A j Pa(X :A).
Given an entity skeleton σe, a PRM with exists uncertainty Π specifies a distribution over a set of
instantiations I consistent with σr[σe]:
P(I j σe;Π) = P(I j σr[σe];Π) = ∏
X2X
∏
x2σr [σe](X)
∏
A2A(x)
P(x:A j Pa(x:A)) (4)
We can similarly define the instance dependency graph and the class dependency graph for a PRM
Π with existence uncertainty using the corresponding notions for the standard PRM Π. As there,
we require that the class dependency graph GΠ is acyclic. One immediate consequence of this
requirement is that the schema is stratified.
Lemma 14 If the class dependency graph GΠ is acyclic, then there is a stratification of the unde-
termined classes.
4. This is similar to the ‘encyclopedic’ links discussed by Ghani et al. (2001). Our exists models can capture each of
the types of links (encyclopedic, co-referencing, and partial co-referencing) defined. Moreover our exists models are
much more general, and can capture much richer patterns in the existence of links.
694

PROBABILISTIC MODELS OF LINK STRUCTURE
Proof: The stratification is given by an ordering of Exists attributes consistent with the class de-
pendency graph. Because the class dependency graph has an edge from Y:E to X :E for every slot
ρ 2 R (X) whose range type is Y , Y will precede X in the constructed ordering. Hence it is a
stratification ordering.
Furthermore, based on this definition, we can now easily prove the following result:
Theorem 15 Let Π be a PRM with existence uncertainty and an acyclic class dependency graph.
Let σe be an entity skeleton. Then Eq. (4) defines a coherent distribution on all instantiations I of
the induced relational skeleton σr[σe].
Proof By Lemma 14 and Lemma 13 we have that σr[σe] is a well-defined relational skeleton. Using
the assumption that GΠ is acyclic, we can apply Theorem 2 to Π and conclude that Π defines a
coherent distribution over instances to σr, and hence so does Π.
One potential shortcoming of our semantics is that it defines probabilities over instances that in-
clude assignment of descriptive attributes to non-existent objects. This potentially presents a prob-
lem. An actual instantiation (i.e., a database) will contain value assignments only for the descriptive
attributes of existing objects. This suggests that in order to compute the likelihood of a database
we need to sum over all possible values of descriptive attributes of potential objects that do not
exist. (As we shall see, such likelihood computations are integral part of our learning procedure.)
Fortunately, we can easily see that the definition of P ensures that if x:E = 0, then variables of the
form x:A are independent of all other variables in the instance. Thus, we can ignore such descriptive
attributes when we compute the likelihood of a database.
The situation with Exists attribute is somewhat more complex. When we observe a database, we
also observe that many potential objects do not exist. The non-existence of these objects can provide
information about other attributes in the model, which is taken into consideration by the correlations
between them and the Exists attributes in the PRM. At first glance, this idea presents computational
difficulties, as there can be a very large number of non-existent objects. However, we note that
the definition of P is such that we need to compute P(x:E = false j Pa(x:E)) only for objects
whose slots refer to existing objects, thereby bounding the number of non-existent objects we have
to consider.
5. Example: Word models
Our two models of structural uncertainty induce simple yet intuitive models for link existence.
We illustrate this by showing a natural connection to two common models of word appearance in
documents. Suppose our domain contains two entity classes: Document, representing the set of doc-
uments in our corpus, and Words, representing the words contained in our dictionary. Documents
may have descriptive attributes such as Topic; dictionary entries have the attribute Word, which is
the word itself, and may also have additional attributes such as the type of word. The relationship
class Appearance represents the appearance of words in documents; it has two slots InDoc and Has-
Word. In this schema, structural uncertainty corresponds to a probabilistic model of the appearance
of words in documents.
In existence uncertainty, the class Appearance is an undetermined class; the potential objects in
this class correspond to document-word pairs (d;w), and the assertion Appearance(d;w):E = true
695

GETOOR, FRIEDMAN, KOLLER AND TASKAR
means that the particular dictionary entry w appears in the particular document d. Now, suppose
that Appearance:E has the parents Appearance.InDoc.Topic and Appearance.HasWord.Word. This
implies, that, for each word w and topic t, we have a parameter pw;t which is the probability that
a word w appears in a document of topic t. Furthermore, the different events Appearance(d;w):E
are conditionally independent given the topic t. It is easy to see that this model is equivalent to
the model often called binary naive Bayes model (McCallum and Nigam, 1998), where the class
variable is the topic and the conditionally independent features are binary variables corresponding
to the appearance of different dictionary entries in the document.
When using reference uncertainty, we can consider several modeling alternatives. The most
straightforward model is to view a document as a bag of words. Now, Appearance also includes
an attribute that designates the position of the word in the document. Thus, a document of n
words has n related Appearance objects. We can provide a probabilistic model of word appear-
ance by using reference uncertainty over the slot Appearance.HasWord. In particular, if we choose
P [Appearance:HasWord] = Words:Word, then we have a multinomial distribution over the words
in the dictionary. If we set Appearance.InDoc.Topic as the parent of the selector variable
Appearance:SHasWord, then we get a different multinomial distribution over words for each topic.
The result is a model where a document is viewed as a sequence of independent samples from a
multinomial distribution over the dictionary, where the sample distribution depends on the docu-
ment topic. This document model is called the multinomial Naive Bayesian model (McCallum and
Nigam, 1998).
Thus, for this simple PRM structure, the two forms of structural uncertainty lead to models that
are well-studied within the statistical NLP community. However, the language of PRMs allows us
to represent more complex structures. Both the existence and reference uncertainty can depend on
properties of words rather than on the exact identity of the word; for example they can also depend
on other attributes, such as the research area of the document’s author.
6. Learning PRMs with Structural Uncertainty
In this section, we briefly review the learning algorithm for the basic PRM framework Friedman
et al. (1999), Getoor et al. (2001) and describe the modifications needed to handle the two extensions
of PRMs proposed in the previous sections. Our aim is to learn such models from data: given a
schema and an instance, construct a PRM that captures the dependencies between objects in the
schema. We stress that basic PRMs and both proposed variants are learned using the same type
of training data: a complete instantiation that describes a set of objects, their attribute values and
their reference slots. However, in each variant, we attempt to learn a somewhat different model
from the data. For basic PRMs, we learn the probability of attributes given other attributes; for
PRMs with reference uncertainty, in addition,we attempt to learn the rules that govern the choice
of slot references; and for PRMs with existence uncertainty, we attempt to learn the probability of
existence of relationship objects.
6.1 Learning basic PRMs
We separate the learning problem into two tasks: evaluating the “goodness” of a candidate structure,
and searching the space of legal candidate structures.
696

PROBABILISTIC MODELS OF LINK STRUCTURE
Model Scoring For scoring candidate structures, we adapt Bayesian model selection Heckerman
(1998). We compute the posterior probability of a PRM Π given an instantiation I . Using Bayes
rule we have that P(Π j I ) ∝ P(I j Π)P(Π). This score is composed of two main parts: the prior
probability of Π, and the probability of the instantiation assuming the PRM is Π. By making fairly
reasonable assumptions about the prior probability of structures and parameters, this term can be
decomposed into a product of terms Friedman et al. (1999). As in Bayesian network learning, each
term in the decomposed form of the score measures how well we predict the values of X :A given
the values of its parents. Moreover, the term for P(X :A j u) depends only on the sufficient statistics
CX :A[v;u], that count the number of entities with x:A = v and Pa(x:A) = u.
Model Search To find a high-scoring structure in basic PRM framework, we use a simple search
procedure that considers operators such as adding, deleting, or reversing edges in the dependency
model Π. The procedure performs greedy hill-climbing search, using the Bayesian score to evaluate
structures. As in Bayesian network search, we can take advantage of score decomposability to
perform the hill-climbing efficiently: after adding or deleting a parent of an attribute, the only steps
that need to be re-scored are other edges that add/delete parents for this attribute.
6.2 Learning with reference uncertainty
The extension to scoring required to deal with reference uncertainty is not a difficult one. Once we
fix the partitions defined by the attributes P [ρ], a CPD for Sρ compactly defines a distribution over
values of ρ. Thus, scoring the success in predicting the value of ρ can be done efficiently using
standard Bayesian methods used for attribute uncertainty (e.g., using a standard Dirichlet prior over
values of ρ).
The extension to search the model space for incorporating reference uncertainty involves expand-
ing our search operators to allow the addition (and deletion) of attributes to partition definition for
each reference slot. Initially, the partition of the range class for a slot X :ρ is not given in the model.
Therefore, we must also search for the appropriate set of attributes P [ρ]. We introduce two new op-
erators refine and abstract, which modify the partition by adding and deleting attributes from P [ρ].
Initially, P [ρ] is empty for each ρ. The refine operator adds an attribute into P [ρ]; the abstract
operator deletes one. As mentioned earlier, we can define the partition simply by looking at the
cross product of the values for each of the partition attributes, or using a decision tree. In the case
of a decision tree, refine adds a split to one of the leaves and abstract removes a split. These newly
introduced operators are treated by the search algorithm in exactly the same way as the standard
edge-manipulation operators: the change in the score is evaluated for each possible operator, and
the algorithm selects the best one to execute.
We note that, as usual, the decomposition of the score can be exploited to substantially speed up
the search. In general, the score change resulting from an operator ω is re-evaluated only after
applying an operator ω0 that modifies the parent or partition set of an attribute that ω modifies. This
is also true when we consider operators that modify the parent of selector attributes.
6.3 Learning with Existence Uncertainty
The extension of the Bayesian score to PRMs with existence uncertainty is straightforward; the ex-
ists attribute is simply a new descriptive attribute. The only new issue is how to compute sufficient
statistics that include existence attributes x:E without explicitly enumerating all the non-existent
697

GETOOR, FRIEDMAN, KOLLER AND TASKAR



























 







  




 


 









 


Figure 7: ΠRU , the PRM learned using reference uncertainty. The reference slots are Role.Movie,
Role.Actor, Vote.Movie, and Vote.Person. Dashed lines indicate attributes used in defin-
ing the partition.
entities. We perform this computation by counting, for each possible instantiation of Pa(X :E),
the number of potential objects with that instantiation, and subtracting the actual number of ob-
jects x with that parent instantiation. Let u be a particular instantiation of Pa(X :E). To compute
CX :E [true;u], we can use standard database query to compute how many objects x 2 σ(X) have
Pa(x:E) = u. To compute CX :E [false;u], we need to compute the number of potential entities. We
can do this without explicitly considering each (x1; : : : ;xk) 2 I (Y1)


I (Yk) by decomposing the
computation as follows: Let ρ be a reference slot of X with Range[ρ] = Y . Let Paρ(X :E) be the
subset of parents of X :E along slot ρ and let uρ be the corresponding instantiation. We count the
number of y consistent with uρ. If Paρ(X :E) is empty, this count is simply jI (Y )j. The product
of these counts is the number of potential entities. To compute CX :E [false;u], we simply subtract
CX :E [true;u] from this number.
No extensions to the search algorithm are required to handle existence uncertainty. We simply
introduce the new attributes X :E , and integrate them into the search space. Our search algorithm
now considers operators that add, delete or reverse edges involving the exist attributes. As usual,
we enforce coherence using the class dependency graph. In addition to having an edge from Y:E to
X :E for every slot ρ 2 R (X) whose range type is Y , when we add an edge from Y:B to X :A, we add
an edge from Y:E to X :E and an edge from Y:E to X :A.
7. Experimental Results
We evaluated our learning algorithms on several real-world data sets. In this section, we describe the
PRM learned for a domain using both of our models for representing structural uncertainty. In each
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PROBABILISTIC MODELS OF LINK STRUCTURE
case, we compare against a simple baseline model. Our experiments used the Bayesian score with a
uniform Dirichlet parameter prior with equivalent sample size α = 2, and a uniform distribution over
structures. The partitions are defined using the cross-product of values of the partition attributes;
they are not represented using a more compact model such as a decision tree.
7.1 Predictive ability
We first tested whether the additional expressive power allows us to better capture regularities in
the domain. Toward this end, we evaluated the likelihood of test data given our learned model.
Unfortunately, we cannot directly compare the likelihood of the EU and RU models, since the
PRMs involve different sets of probabilistic events and condition on varying amounts of background
knowledge. Thus to evaluate our models we compare the PRM with structural uncertainty to a
“baseline” model which incorporates link probabilities, but makes the “null” assumption that the
link structure is uncorrelated with the descriptive attributes. For reference uncertainty, the baseline
has P [ρ] = /0 for each slot. For existence uncertainty, it forces x:E to have no parents in the model.
We evaluated these variants on a dataset that combines information about movies and actors from
the Internet Movie Database5 and information about people’s ratings of movies from the Each Movie
dataset,6 where each person’s demographic information was extended with census information for
their zip code. From these, we constructed five classes (with approximate sizes shown): Movie
(1600), Actor (35,000); Role (50,000), Person (25,000), and Vote (300,000).
We modeled uncertainty about the link structure of the classes Role (relating actors to movies) and
Vote (relating people to movies). For RU this was done by modeling the reference uncertainty of the
slots of these objects. For EU this was done by modeling probability of the existence of such objects.
We evaluated our methods using ten-fold cross validation. To do this, we partitioned our data into ten
subsets. For each subset, we trained on nine-tenths of the data (the data not included in the subset)
and evaluated the log-likelihood of the held-out test subset. We then average the results. In both
cases, the model using structural uncertainty significantly outperformed its “baseline” counterpart.
For RU, we obtained a log-likelihood of −149;705 as compared to −152;280 for the baseline
model. For EU, we obtained a log-likelihood of −210;044 for the EU model, as compared to
−213;798 for the baseline EU model. Thus, we see that the model where the relational structure is
correlated with the attribute values is substantially more predictive than the baseline model that takes
them to be independent: although any particular link is still a low-probability event, our structural
uncertainty models are much more predictive of its presence.
Figure 7 shows the RU model learned. In the RU model we partition each of the movie reference
slots on genre attributes; we partition the actor reference slot on the actor’s gender; and we partition
the person reference of votes on age, gender and education. An examination of the model shows,
for example, that younger voters are much more likely to have voted on action movies and that male
action movies roles are more likely to exist than female roles. Furthermore, the actor reference slot
has Movie:Action as a parent; the CPD encodes the fact that male actors are more likely to have
roles in action movies than female actors.
The EU model learned had an exist attribute for both vote and role. In the model we learned, the
existence of a vote depended on the age of the voter and the movie genre, and the existence of a role
depended on the gender of the actor and the movie genre.
5. c©1990-2000 Internet Movie Database Limited.
6. http://www.research.digital.com/SRC/EachMovie.
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7.2 Classification
While we cannot directly compare the likelihood of the EU and RU models, we can compare the
predictive performance of each model. In this set of experiments, we evaluate the predictive accu-
racy of the different models on two datasets. It is quite interesting to note that we observed that both
models of structural uncertainty significantly increase our prediction accuracy.
We considered the conjecture that by modeling link structure we can improve the prediction of
descriptive attributes. Here, we hide some attribute of a test-set object, and compute the probability
over its possible values given the values of other attributes on the one hand, or the values of other
attributes and the link structure on the other. We tested on two similar domains: Cora (McCallum
et al., 2000) and WebKB (Craven et al., 1998). The Cora dataset contains 4000 machine learning
papers, each with a seven-valued Topic attribute, and 6000 citations. The WebKB dataset contains
approximately 4000 pages from four Computer Science departments, with a five-valued attribute
representing their “type”, and 10,000 links between web pages. In both datasets we also have access
to the content of the document (webpage/paper), which we summarize using a set of attributes that
represent the presence of different words on the page (a binary Naive Bayes model). After stemming
and removing stop words and rare words, the dictionary contains 1400 words in the Cora domain,
and 800 words in the WebKB domain.
In both domains, we compared the performance of models that use only word appearance infor-
mation to predict the category of the document with models that also used probabilistic information
about the link from one document to another. We fixed the dependency structure of the models,
using basically the same structure for both domains. In the Cora EU model, the existence of a ci-
tation depends on the topic of the citing paper and the cited paper. We evaluated two symmetrical
RU models. In the first, we partition the citing paper by topic, inducing a distribution over the
topic of Cites.Citing. The parent of the selector variable is Cites.Cited.Topic. The second model is
symmetrical, using reference uncertainty over the cited paper.
Table 1 shows prediction accuracy on both data sets. We see that both models of structural un-
certainty significantly improve the accuracy scores, although existence uncertainty seems to be
superior. Interestingly, the variant of the RU model that models reference uncertainty over the
citing paper based on the topics of papers cited (or the from webpage based on the categories of
pages to which it points) outperforms the cited variant. However, in all cases, the addition of ci-
tation/hyperlink information helps resolve ambiguous cases that are misclassified by the baseline
model that considers words alone. For example, paper #506 is a Probabilistic Methods paper, but is
classified based on its words as a Genetic Algorithms paper (with probability 0:54). However, the
paper cites two Probabilistic Methods papers, and is cited by three Probabilistic Methods papers,
leading both the EU and RU models to classify it correctly. Paper #1272 contains words such as
rule, theori, refin, induct, decis, and tree. The baseline model classifies it as a Rule Learning paper
(probability 0.96). However, this paper cites one Neural Networks and one Reinforcement Learning
paper, and is cited by seven Neural Networks, five Case-Based Reasoning, fourteen Rule Learning,
three Genetic Algorithms, and seventeen Theory papers. The Cora EU model assigns it probability
0.99 of being a Theory paper, which is the correct topic. The first RU model assigns it a probability
0.56 of being Rule Learning paper, whereas the symmetric RU model classifies it correctly. In this
case, an explanation of this phenomenon is that most of the information for this paper is in the topics
of citing papers; it appears that RU models can make better use of information in the parents of the
selector variable than in the partitioning variables.
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PROBABILISTIC MODELS OF LINK STRUCTURE
Table 1: Prediction accuracy of topic/category attribute of documents in the Cora and WebKB
datasets. Accuracies and reported standard deviations are based on a 10-fold cross val-
idation.
0.65
0.7
0.75
0.8
0.85
0.9
Cora WebKB
Ac
cu
ra
cy
Naive-Baye
RU Citing
RU Cited
Exists
Cora WebKB
baseline 75  2.0 74  2.5
RU Citing 81  1.7 78  2.3
RU Cited 79  1.3 77  1.5
EU 85  0.9 82  1.3
7.3 Collective Classification
In the preceding experiments, the topics of all the linked papers or webpages were observed and we
made a prediction for a single unobserved topic. In a more realistic setting, we will have a whole
collection of unlabeled instances that are linked together. We can no longer make each prediction
in isolation, since for example, the (predicted) topic of one paper influences the (predicted) topics
of the papers it cites.
This task of collective classification requires us to reason about the entire collection of instances at
once. In our framework, this translates into computing the posterior distribution over the unobserved
variables given the data and assigning each unobserved variable its most likely value. This requires
inference over the unrolled network defined by instantiating a PRM for a particular document col-
lection. We cannot decompose this task into separate inference tasks over the objects in the model,
as they are all correlated. In general, the unrolled network can be fairly complex, involving many
documents that are linked in various ways. (In our experiments, the networks involve hundreds of
thousands of nodes.) Exact inference over these networks is clearly impractical, so we must resort
to approximate inference. Following Taskar et al. (2001), we use belief propagation for the task of
inference in PRMs. Belief propagation is a local message passing algorithm introduced by Pearl
(1988). Although a very simple approximation, it has lately been shown to be very effective on a
wide range of probabilistic models Murphy et al. (1999).
We evaluated several existence uncertainty models for the task of collective classification on the
WebKB dataset (Craven et al., 1998). Recall that the WebKB dataset consists of webpages in the
Computer Science departments of four schools: Cornell, University of Texas at Austin, University
of Washington, and University of Wisconsin.
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GETOOR, FRIEDMAN, KOLLER AND TASKAR
...
From  
Category
Word1 WordN
Exists
From
To

 
Hub
o    
Word
   or
Has
...
Category
Word1 WordN
Hub
...
Page#1
Category
Word1 WordN
Exists
From
To
 
#1
Hub
Page#
...
Category
Word1 WordN
Hub
Exists
To
 
#
Page#
Word
   #
Has
...
Category
Word1 WordN
Hub
Word
   #1
Has
Word
   #
From
Has
(a) (b)
Figure 8: (a) PRM Model for WebKB domain; (b) Fragment of unrolled network for WebKB
model.
Figure 8(a) shows a PRM for this domain. For clarity, the Page class is duplicated in the figure,
once as From-Page and once as To-Page. Each page has a category attribute representing the type
of web page which is one of fcourse, professor, student, project, otherg. The text content of the web
page is represented using a set of binary attributes that indicate the presence of different words on
the page. After stemming, removing stop words and rare words, the dictionary contains around 800
words. In addition, each page was labeled with a “category hub” attribute, whose domain is fcourse,
professor, student, project, noneg. A page was labeled with a hub of a particular type (not none)
if it pointed to many pages of that category. The prevalence of such hubs (or directories) on the
web was investigated by Kleinberg (1998), who noted that pages of the same topic or category are
often linked to by hub pages. This insight was utilized in the classification algorithm FOIL-HUBS
of (Slattery and Mitchell, 2000) for the WebKB dataset. Pages in the training set were hand-labeled
with the hub attribute, but the attribute was hidden in the test data. Each school had one hub page of
each category, except for Washington which does not have a project hub page and Wisconsin which
does not have a faculty web page.
The data set also describes the links between the web pages within a given school. In addition, for
each link between pages, the dataset specifies the words (one or more) on the anchor link. There are
approximately 100 possible anchor words.
In these experiments, we included only pages that have at least one out link. The number of
resulting pages for each school are: Cornell (318), Texas (319), Washington (420), and Wisconsin
(465). The number of links for each school are: Cornell (923), Texas (1041), Washington (1534)
and Wisconsin (1823).
Given a particular set of hyperlinked pages, the template is instantiated to produce an “unrolled”
Bayesian network. Figure 8(b) shows a fragment of such a network for three webpages. The two
existing links from page 1 to page 2 and 3 are shown while non-existing links omitted for clarity
(however still play a role in the inference). Also shown are the anchor word for link 1 and two
anchor words for link 2. Note that during classification, existence of links and anchor words in the
links are used as evidence to infer categories of the web pages. Hence, our unrolled Bayes net has
active paths between categories of pages through the v-structures at Link:Exists and Anchor:Word.
These active paths capture exactly the pattern of relational inference we set out to model.
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PROBABILISTIC MODELS OF LINK STRUCTURE
0.54
0.56
0.58
0.6
0.62
0.64
0.66
0.68
0.7
cornell texas wisconsin washington
School
A
cc
u
ra
cy
Naive-Bayes
Anchors
Exists
Ex+Hubs
Ex+Anchors
Ex+Hubs+Anchors
Figure 9: Comparison of accuracy of several models ranging from the simplest model, Naive-Bayes
to the most complex model, Ex+Hubs+Anchors, which incorporates existence uncer-
tainty, hubs and link anchor words. In each case, a model was learned for 3 schools and
tested on the remaining school.
We compared the performance of several models on predicting web page categories. In each case,
we learned a model from three schools, and tested the performance of the learned model on the
remaining school. Our experiments used the Bayesian score with a uniform Dirichlet parameter
prior with equivalent sample size α = 2.
All models we compared can be viewed as a subset of the model in Figure 8(a). Our baseline is
a standard binomial Naive Bayes model that uses only words on the page to predict the category of
the page. We evaluated the following set of models:
1. Naive-Bayes: Our baseline model.
2. Anchors: This model uses both words on the page and anchor words on the links to predict
the category.
3. Exists: This model adds structural uncertainty over the link relationship to the simple baseline
model; the parents of Link.Exists are Link.From-Page.Category and Link.To-Page.Category.
4. Ex+Hubs: This model extends the Exists model with Hubs. In the model Link:Exists depends
on Link.From-Page.Hub in addition to the categories of each of the pages.
5. Ex+Anchors: This model extends the Exists model with includes anchor words (but not
hubs).
6. Ex+Hubs+Anchors: The final model includes existence uncertainty, hubs and anchor words.
Figure 9 compares the accuracy achieved by the different models on each of the schools. The final
model, Ex+Hubs+Anchors, which incorporates structural uncertainty, hubs and anchor words, con-
703

GETOOR, FRIEDMAN, KOLLER AND TASKAR
sistently outperforms the Naive-Bayes model by a significant amount. In addition, it outperforms
any of the simpler variants.
Our algorithm was fairly successful at identifying the hubs in the test set: it recognized 8 out of
14 hubs correctly. However, precision was not very high: 38 pages were mislabeled as hubs. The
pages mislabeled as hubs often pointed to many pages that had been labeled as Other web pages.
However, on further inspection, these hub pages often were directories pointing to pages that were
likely to be researcher home pages or course home pages and seemed to have been mislabeled in the
training set as Other. We investigated how much these misclassifications hurt the performance by
revealing the labels of the hub attribute in the test data. The improvement in classification accuracy
of Ex+Hubs+Anchors model was roughly 2%.
We also experimented with classification using models where the category of each page has a
(multinomial) logistic dependence on the presence of content words, i.e., the arrows from Category
to the Wordi’s in Figure 8 are reversed. The logistic CPD is given by
P(Category = c j Word1 = w1; : : : ;Wordn = wn) ∝ expfθc;0 +∑θc;iwig;
where each wi 2 f0;1g indicates the presence of word i. We used a zero-mean independent Gaussian
prior for the parameters given by
P(θc;i) ∝ expf−θ2c;i=2σ2g;
with σ = 0:3 and estimated the CPD using the standard conjugate gradient algorithm. The aver-
age accuracy (averaged over four leave-one-school out experiments) for the flat logistic model was
82.95%, which is significantly higher than all of the models presented so far. This improvement in
performance is not surprising, as discriminatively-trained models such as logistic regression often
achieve significantly higher classification accuracies than their generative counterparts.
Unfortunately, adding the exists edges to this model did not improve accuracy, but in fact lowered
it to 76.42%. Our conjecture is that the evidence about the category of a page from the large number
of exists variables overwhelms the evidence from the content words for the logistic-based models,
while the models that extend Naive Bayes are more balanced. The reason is that, in Naive Bayes,
the posterior over the category (given the words in the document), is usually very sharply peaked
around one value because of the violations of the independence assumption. It takes a significant
amount of evidence from the links to change this posterior. The logistic CPD is much better at
representing the conditional distribution of category given the words. However, as the posterior is
based only on a single CPD, it is not highly skewed, and the large amount of evidence from the
exists variables can easily sway it to a new maximum.
To test this conjecture, we tried to reduce the influence of the exists evidence by modifying the
CPD of the exists variable to be closer to uniform. One way to accomplish this is to “raise the
temperature” or raise the CPD entries to a power between 0 and 1. We discovered that, for values
in the range 0:1–0:2, the model gets accuracies that are 2–3% higher than the logistic baseline.
This interaction of different sources of influence in a globally consistent model warrants further
investigation. One solution to this problem is to discriminatively train the entire model, including
the model for the exists variables, as in the recent work of Taskar et al. (2002). As shown in that
work, this training allows us to gain improvements in accuracy from discriminative training on the
one hand, and from the information flow between related objects on the other.
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PROBABILISTIC MODELS OF LINK STRUCTURE
8. Discussion and Conclusions
In this paper, we proposed two representations for structural uncertainty: reference uncertainty
and existence uncertainty. Reference uncertainty models the process by which reference slots are
selected from a given set. Existence uncertainty provides a model for whether a relation exists
between two objects. We have shown how to integrate them within our learning framework, and
presented results showing that they allow interesting patterns to be learned. The ability to learn
probabilistic models of relational structure has many applications. It allows us to predict whether
two objects with given properties are more likely to be related to each other. More surprisingly,
the link structure also allows us to predict attribute values of interest. For example, we can better
predict the topic of a paper by using the fact that it cites certain types of papers. Both of the models
we propose are relatively simple. They make certain independence assumptions that often will not
hold. Thus the significance of our results is that even these simplistic models of link uncertainty
provide us with increased predictive accuracy.
Several recent works in the literature examine learning from relational data. Kleinberg (1998)
learns a global property of a relation graph (“authority” of web pages) based on local connectivity.
This approach does not generalize beyond the training data, and ignores attributes of the pages
(e.g., words). Slattery and Mitchell (2000) integrate Kleinberg’s authority recognition module with
a first-order rule learner to perform classification that also utilizes the relational structure in the test
set. Their approach is intended purely for classification, and is not a statistical model of the domain.
Furthermore, their approach is not based on a single coherent framework, so that the results of two
different modules are combined procedurally.
A stochastic relational model recently defined by Cohn and Hofmann (2001) introduces a latent
(hidden) variable that describes the “class” of each document. Their model assumes that word oc-
currences and links to other documents are independent given the document’s class. This model is
similar to a PRM model with reference uncertainty, but differs from it in several important ways.
First, it uses a multinomial distribution over specific citations, preventing the model from generaliz-
ing to a different context. Second, each paper is assumed to be independent, so there is no ability to
reach conclusions about the topic of a cited paper from that of a citing paper. Finally, dependencies
between words appearing in the citing document and the presence or absence of a citation cannot
be represented.
The ability to learn probabilistic models of relational structure is an exciting new direction for ma-
chine learning. Our treatment here only scratches the surface of this problem. In particular, although
useful, neither of the representations proposed for structural uncertainty is entirely satisfying as a
generative model. Furthermore, both models are restricted to considering the probabilistic model
of a single relational “link” in isolation. These simple models can be seen as the naive Bayes of
structural uncertainty; in practice, relational patterns involve multiple links, e.g., the concepts of
hubs and authorities. In future work, we hope to provide a unified framework for representing and
learning probabilistic models of relational “fingerprints” involving multiple entities and links.
Acknowledgments
We gratefully thank Eran Segal for many useful discussions and for his help with the collective
classification results. We would also like to thank the editors, Carla Brodley and Andrea Dunyluk,
705

GETOOR, FRIEDMAN, KOLLER AND TASKAR
Matt Geig and the anonymous reviewers for their constructive comments and suggestions. This
work was supported by ONR contract N66001-97-C-8554 under DARPA’s HPKB program, by Air
Force contract F30602-00-2-0598 under DARPA’s EELD program, and by the Sloan foundation. Nir
Friedman was also supported by Israel Science Foundation grant 244/99 and an Alon Fellowship.
References
S. Chakrabarti, B. Dom, and P. Indyk. Enhanced hypertext categorization using hyperlinks. In
Proceedings of ACM SIGMOD International Conference on Management of Data, pages 307–
318, Seattle, Washington, 1998.
D. Cohn and T. Hofmann. The missing link—a probabilistic model of document content and hyper-
text connectivity. In Proceedings of Neural Information Processing Systems 13, pages 430–436,
Vancouver, British Columbia, 2001.
M. Craven, D. DiPasquo, D. Freitag, A. McCallum, T. Mitchell, K. Nigam, and S. Slattery. Learning
to extract symbolic knowledge from the world wide web. In Proceedings of the Fifteenth Confer-
ence of the American Association for Artificial Intelligence, pages 509–516, Madison, Wisconsin,
1998.
N. Friedman, L. Getoor, D. Koller, and A. Pfeffer. Learning probabilistic relational models. In Pro-
ceedings of the Sixteenth International Joint Conference on Artificial Intelligence, pages 1300–
1309, Stockholm, Sweden, 1999.
L. Getoor. Learning Statistical Models from Relational Data. PhD thesis, Stanford University,
2001.
L. Getoor, N. Friedman, D. Koller, and A. Pfeffer. Learning probabilistic relational models. In
S. Dzeroski and N. Lavrac, editors, Relational Data Mining, pages 307–335. Kluwer, 2001.
R. Ghani, S. Slattery, and Y. Yang. Hypertext categorization using hyperlink patterns and meta data.
In Proceedings of the Eighteenth International Conference on Machine Learning, Williamstown,
Massachusetts, 2001.
D. Heckerman. A tutorial on learning with Bayesian networks. In M. I. Jordan, editor, Learning in
Graphical Models. MIT Press, Cambridge, Massachusetts, 1998.
J. Kleinberg. Authoritative sources in a hyperlinked environment. In Proceedings of the Ninth An-
nual ACM-SIAM Symposium on Discrete Algorithms, pages 668–677, San Francisco, California,
1998.
D. Koller and A. Pfeffer. Probabilistic frame-based systems. In Proceedings of the Fifteenth Confer-
ence of the American Association for Artificial Intelligence, pages 580–587, Madison, Wisconsin,
1998.
N. Lavrac˘ and S. Dz˘eroski. Inductive Logic Programming: Techniques and Applications. Ellis
Horwood, New York, New York, 1994.
706

PROBABILISTIC MODELS OF LINK STRUCTURE
A. McCallum and K. Nigam. A comparison of event models for naive bayes text classification.
In Workshop on Learning for Text Categorization at the Fifteenth Conference of the American
Association for Artificial Intelligence, Madison, Wisconsin, 1998.
A. McCallum, K. Nigam, J. Rennie, and K. Seymore. Automating the construction of internet
portals with machine learning. Information Retrieval, 3(2):127–163, 2000.
S. Muggleton, editor. Inductive Logic Programming. Academic Press, 1992.
K. Murphy, Y. Weiss, and M. Jordan. Loopy belief propagation for approximate inference: an em-
pirical study. In Proceedings of the Fifteenth Conference on Uncertainty in Artificial Intelligence,
pages 467–475, Stockholm, Sweden, 1999.
L. Ngo and P. Haddawy. Probabilistic logic programming and bayesian networks. In Algorithms,
Concurrency and Knowledge (Proceedings ACSC95), volume 1023 of Lecture Notes in Computer
Science, pages 286–300. Springer-Verlag, 1995.
J. Pearl. Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann, San Francisco, 1988.
D. Poole. Probabilistic Horn abduction and Bayesian networks. Artificial Intelligence, 64:81–129,
1993.
A. Popescul, L. Ungar, S. Lawrence, and D. Pennock. Towards structural logistic regression: Com-
bining relational and statistical learning. In Proc. KDD-2002 Workshop on Multi-Relational Data
Mining. ACM, 2002.
S. Slattery and M. Craven. Combining statistical and relational methods for learning in hypertext
domains. In David Page, editor, Proceedings of ILP-98, 8th International Conference on Induc-
tive Logic Programming, pages 38–52, Madison, US, 1998. Springer Verlag, Heidelberg, DE.
S. Slattery and T. Mitchell. Discovering test set regularities in relational domains. In Proceedings
of the Seventeenth International Conference on Machine Learning, pages 895–902, Stanford,
California, 2000.
B. Taskar, P. Abbeel, and D. Koller. Discriminative probabilistic models for relational data. In
Proceedings of the Eighteenth Conference on Uncertainty in Artificial Intelligence, Edmonton,
Canada, 2002.
B. Taskar, E. Segal, and D. Koller. Probabilistic classification and clustering in relational data. In
Proceedings of the Seventeenth International Joint Conference on Artificial Intelligence, pages
870–876, Seattle, Washington, 2001.
Y. Yang, S. Slattery, and R. Ghani. A study of approaches to hypertext categorization. Journal of
Intelligent Information Systems, 18(2-3):219–241, 2002.
707