Is this document relevant probably

“Is This Document Relevant? . . . Probably”: A Survey of
Probabilistic Models in Information Retrieval
FABIO CRESTANI, MOUNIA LALMAS, CORNELIS J. VAN RIJSBERGEN, AND
IAIN CAMPBELL
University of Glasgow

This article surveys probabilistic approaches to modeling information retrieval. The
basic concepts of probabilistic approaches to information retrieval are outlined and
the principles and assumptions upon which the approaches are based are
presented. The various models proposed in the development of IR are described,
classified, and compared using a common formalism. New approaches that
constitute the basis of future research are described.
Categories and Subject Descriptors: H.3.3 [Information Storage and Retrieval]:
Information Search and Retrieval—retrieval models; H.3.1 [Information Storage
and Retrieval]: Content Analysis and Indexing—indexing methods
General Terms: Algorithms, Theory
Additional Key Words and Phrases: Information retrieval, probabilistic indexing,
probabilistic modeling, probabilistic retrieval, uncertain inference modeling

1. HISTORY OF PROBABILISTIC
MODELING IN IR

In information retrieval (IR), probabilistic modeling is the use of a model that
ranks documents in decreasing order of
their evaluated probability of relevance
to a user’s information needs. Past and
present research has made much use of
formal theories of probability and of statistics in order to evaluate, or at least
estimate, those probabilities of relevance. These attempts are to be distinguished from looser ones such as the
“vector space model” [Salton 1968] in
which documents are ranked according
to a measure of similarity to the query.
A measure of similarity cannot be directly interpretable as a probability. In
addition, similarity-based models gener-

ally lack the theoretical soundness of
probabilistic models.
The first attempts to develop a probabilistic theory of retrieval were made
over 30 years ago [Maron and Kuhns
1960; Miller 1971], and since then there
has been a steady development of the
approach. There are already several operational IR systems based upon probabilistic or semiprobabilistic models.
One major obstacle in probabilistic or
semiprobabilistic IR models is finding
methods for estimating the probabilities
used to evaluate the probability of relevance that are both theoretically sound
and computationally efficient. The problem of estimating these probabilities is
difficult to tackle unless some simplifying assumptions are made. In the early

Authors’ address: Computing Science Department, University of Glasgow, Glasgow G12 8QQ, Scotland,
UK; email: ͗fabio,mounia,keith,iain͘@dcs.gla.ac.uk.
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ACM Computing Surveys, Vol. 30, No. 4, December 1998

Probabilistic Models in IR
stages of the study of probabilistic modeling in IR, assumptions related to
event independence were employed in
order to facilitate the computations. The
first models to be based upon such assumptions were the “binary independence indexing model” (Section 3.3) and
the “binary independence retrieval
model” (Section 3.2). Recent findings by
Cooper [1995] have shown that these
assumptions are not completely necessary and were, in fact, not actually
made (Section 5).
The earliest techniques that took dependencies into account gave results
that were worse than those given by
techniques based upon the simplifying
assumptions. Moreover, complex techniques that captured dependencies
could only be used at a computational
price regarded as too high with respect
to the value of the results [van Rijsbergen 1977]. One particular research direction aimed at removing the simplifying assumptions has been studied
extensively and much work is being
done.1
Another direction has involved the
application of statistical techniques
used in pattern recognition and regression analysis. These investigations, of
which the Darmstadt indexing approach
(DIA) is a major example [Fuhr 1989;
Fuhr and Buckley 1991] (see Section
3.4), do not make use of independence
assumptions. They are “model-free” in
the sense that the only probabilistic assumptions involved are those implicit in
the statistical regression theory itself.
The major drawback of such approaches
is the degree to which heuristics are
necessary to optimize the description
and retrieval functions.
A theoretical improvement in the DIA
was achieved by using logistic regression instead of standard regression.
Standard regression is, strictly speaking, inappropriate for estimating probabilities of relevance if relevance is considered as a dichotomous event; that is,
1
Please see Fung et al. [1990], Turtle and Croft
[1990], Savoy [1992], and van Rijsbergen [1992].

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529

a document is either relevant to a query
or not. Logistic regression has been specifically developed to deal with dichotomous (or n-dichotomous) dependent
variables. Probabilistic models that
make use of logistic regression have
been developed by Fuhr and Pfeifer
[Fuhr and Buckley 1991] and by Cooper
et al. [1992] (see Sections 3.4 and 3.7).
One area of recent research investigates the use of an explicit network
representation of dependencies. The
networks are processed by means of
Bayesian inference or belief theory, using evidential reasoning techniques
such as those described by Pearl [1988].
This approach is an extension of the
earliest probabilistic models, taking
into account the conditional dependencies present in a real environment.
Moreover, the use of such networks generalizes existing probabilistic models
and allows the integration of several
sources of evidence within a single
framework. Attempts to use Bayesian
(or causal) networks are reported in
Turtle [1990], Turtle and Croft [1991],
and Savoy [1992].
A new stream of research, initiated by
van Rijsbergen [1986] and continued by
him and others,2 aims at developing a
model based upon a nonclassical logic,
specifically a conditional logic where the
semantics are expressed using probability theory. The evaluation can be performed by means of a possible-world
analysis [van Rijsbergen 1989, 1992;
Sembok and van Rijsbergen 1993; Crestani and van Rijsbergen 1995], thus establishing an intentional logic, by using
modal logic [Nie 1988, 1989, 1992;
Amati and Kerpedjiev 1992], by using
situation theory [Lalmas 1992], or by
integrating logic with natural language
processing [Chiaramella and Chevallet
1992]. The area is in its infancy; no
working prototype based on the proposed models has been developed so far,
2

Please see Amati and van Rijsbergen [1995],
Bruza [1993], Bruza and van der Weide [1992],
Huibers [1996], Sebastiani [1994], and Crestani
and van Rijsbergen [1995].


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and the operational validity of these
ideas has still to be confirmed.
2. BACKGROUND

Here we review some general aspects
that are important for a full understanding of the proposed probabilistic
models. We then provide a framework
within which the various models can be
placed for comparison. We assume some
familiarity with principles of probability
theory on the part of the reader.
2.1 Event Space

In general, probabilistic models have as
their event space the set Q ϫ D , where
គ
គ
Q represents the set of all possible queគ
ries and D the set of all documents in
គ
the collection. The differences among
the various models lie in their use of
different representations and descriptions of queries and documents.
In most models, queries and documents are represented by descriptors,
often automatically extracted or manually assigned terms. These descriptors
are represented as binary-valued vectors in which each element corresponds
to a term. More complex models make
use of real-valued vectors or take into
account relationships among terms or
among documents.
A query is an expression of an information need. Here we regard a query as
a unique event; that is, if two users
submit the same query or if the same
query is submitted by the same user on
two different occasions, the two queries
are regarded as different. A query is
submitted to the system, which then
aims to find information relevant to the
information need expressed in the
query. In this article we consider relevance as a subjective user judgment on
a document related to a unique expression of an information need.3
3

The relevance relationship between a query and
a document relies on a user-perceived satisfaction
of his or her information needs. Such a perception
of satisfaction is subjective—different users can
give different relevance judgments to a given query-


A document is any object carrying information: a piece of text, an image, a
sound, or a video. However, most all
current IR systems deal only with text,
a limitation resulting from the difficulty
of finding suitable representations for
nontextual objects. We thus consider
here only text-based IR systems.
Some assumptions common to all retrieval models are:
—Users’ understanding of their information needs changes during a search
session, is subject to continuous refinement, and is expressed by different queries.
—Retrieval is based only upon representations of queries and documents,
not upon the queries and documents
themselves.
—The representation of IR objects is
“uncertain.” For example, the extraction of index terms from a document
or a query to represent the document
or query information content is a
highly uncertain process. As a consequence, the retrieval process becomes
uncertain.
It is particularly this last assumption
that led to the study of probabilistic
retrieval models. Probability theory
[Good 1950] is, however, only one way of
dealing with uncertainty.4 Also, earlier
models were largely based on classical
probability theory, but recently new approaches to dealing with uncertainty
have been applied to IR. Sections 3 and
4 present traditional and new approaches to probabilistic retrieval.
2.2 A Conceptual Model

The importance of conceptual modeling
is widely recognized in fields such as
database management systems and in-

document pair. Moreover, this relevance relationship depends on time, so that the same user could
give a different relevance judgment on the same
query-document pair on two different occasions.
4
Other approaches are based, for example, on
fuzzy logic [Zadeh 1987] and Dempster–Shafer’s
theory of evidence [Shafer 1976].


Figure 1.

•

531

The underlying conceptual model.

formation systems. Here we use the conceptual model proposed by Fuhr
[1992b], which has the advantage of being both simple and general enough to
be a conceptual basis for all the probabilistic models presented in this survey,
although some of them predate it.
The model is shown in Figure 1. The
basic objects of an IR system are: a
finite set of documents D (e.g., books,
គ
articles, images) and a finite set of queries Q (e.g., information needs). We conគ
sider a set of queries and not a single
query alone because a single user may
have varying information needs. If we
consider ᏾ a finite set of possible relevance judgments, for example, in the
៮
binary case ᏾ ϭ {R, R } (i.e., a document
can either be relevant to a query or not)
then the IR system’s task is to map
every query-document pair to an element of ᏾. Unfortunately, IR systems
do not deal directly with queries and
documents but with representations of
them (e.g., a text for a document or a
Boolean expression for a query). It is
mainly the kind of representation technique used that differentiates one IR
indexing model from another.
We denote by ␣ Q the mapping between a set of queries Q and their repគ
resentations Q. For example, a user in
search of information about wine may
express his or her query as follows: “I
am looking for articles dealing with
wine.” Similarly, we denote by ␣ D the
mapping between a set of documents D
គ
and their representations D. For example, in a library, a book is represented
by its author, titles, a summary, the fact
it is a book (and not an article), and
some keywords. These two mappings

can be very different from each other.
Obviously, the better the representation
of queries and documents, the better the
performance of the IR system.
To make the conceptual model general enough to deal with the most complex IR models, a further mapping is
introduced between representations and
descriptions. For instance, a description
of the preceding query could be the two
terms: “article” and “wine.” The sets of
representations Q and D are mapped to
the sets of descriptions QЈ and DЈ by
means of two mapping functions ␤ Q and
␤ D . Moreover, the need for such additional mapping arises for learning models (see, e.g., Section 3.4) that must
aggregate features to allow large
enough samples for estimation. It is
worth noticing, however, that most
models work directly with the original
document and query representations.
It is common for IR systems to be able
to manage only a poor description of the
representation of the objects (e.g., a set
of stems instead of a text). However,
when representation and description
happen to be the same, it is sufficient to
consider either ␣ Q or ␣ D as an identity
mapping.
Descriptions are taken as the independent variables of the retrieval function r: QЈ ϫ DЈ 3 ᑬ, which maps a
query-document pair onto a set of retrieval status values (RSV) r(qЈ , dЈ)
k
j
[Bookstein and Cooper 1976]. The task
of ranked retrieval IR systems in response to a query q k is to calculate this
value and rank each and every document d j in the collection upon it.
គ
In probabilistic IR the task of the
system is different. If we assume binary

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F. Crestani et al.

relevance judgments (i.e., ᏾ contains
only the two possible judgments R and
៮
R ), then according to the probability
ranking principle (Section 2.4), the task
of an IR system is to rank the documents according to their estimated
probability of being relevant P(R͉q k ,
d j ). This probability is estimated by
គ
P(R͉qЈ , dЈ).
k
j
2.3 On “Relevance” and “Probability of
Relevance”

The concept of relevance is arguably the
fundamental concept of IR. In the preceding model we purposely avoided giving a formal definition of relevance because the notion of relevance has never
been defined precisely in IR. Although
there have been many attempts towards
a definition [Seracevic 1970; Cooper
1971; Mizzaro 1996], there has never
been agreement on a unique precise definition. A treatment of the concept of
relevance is outside the scope of this
article and we do not attempt to formulate a new definition or even accept
some particular already existing one.
What is important for the purpose of
our survey is to understand that relevance is a relationship that may or may
not hold between a document and a user
of the IR system who is searching for
some information: if the user wants the
document in question, then we say that
the relationship holds. With reference
to the preceding model, relevance (᏾) is
a relationship between a document ͑dj ͒
គ

and a user’s information need ͑qk ͒. If
គ
the user wants the document d in relaគ
tion to his information need qk , then dj
គ
គ
is relevant (R).
Most readers will find the concept of
probability of relevance quite unusual.
The necessity of introducing such a
probability arises from the fact that relevance is a function of a large number of
variables concerning the document, the
user, and the information need. It is
virtually impossible to make strict predictions as to whether the relationship
of relevance will hold between a given

document and a given user’s information need. The problem must be approached probabilistically. The preceding model explains what evidence is
available to an IR system to estimate
the probability of relevance P(R͉q k , d j ).
គ
A precise definition of probability of relevance depends on a precise definition
of the concept of relevance, and given a
precise definition of relevance it is possible to define such a probability rigorously. Just as we did not define relevance, we do not attempt to define the
probability of relevance, since every
model presented here uses a somewhat
different definition. We refer the reader
to the treatment given by Robertson et
al. [1982], where different interpretations of the probability of relevance are
given and a unified view is proposed.
2.4 The Probability Ranking Principle

A common characteristic of all the probabilistic models developed in IR is their
adherence to the theoretical justification embodied in the probability ranking principle (PRP) [Robertson 1977].
The PRP asserts that optimal retrieval
performance can be achieved when documents are ranked according to their
probabilities of being judged relevant to
a query. These probabilities should be
estimated as accurately as possible on
the basis of whatever data have been
made available for this purpose.
The principle speaks of “optimal retrieval” as distinct from “perfect retrieval.” Optimal retrieval can be defined
precisely for probabilistic IR because it
can be proved theoretically with respect
to representations (or descriptions) of
documents and information needs. Perfect retrieval relates to the objects of the
IR systems themselves (i.e., documents
and information needs).
The formal definition of the PRP is as
follows. Let C denote the cost of retriev៮
ing a relevant document and C the cost
of retrieving an irrelevant document.
The decision rule that is the basis of the
PRP states that a document d m should
be retrieved in response to a query q k

above any document d i in the collection
if

៮
C ⅐ P ͑ R ͉ q k , d m ͒ ϩ C ⅐ ͑ 1 Ϫ P ͑ R ͉ q k , d m ͒͒
៮
Յ P ͑ R ͉ q k , d i ͒ ϩ C ⅐ ͑ 1 Ϫ P ͑ R ͉ q k , d i ͒͒ .
The decision rule can be extended to
deal with multivalued relevance scales
(e.g., very relevant, possibly relevant,
etc. [Cooper 1971]). In addition, by
means of a continuous cost function, it
is possible to write a decision rule for
approaches where the relevance scale is
assumed to be continuous [Borgogna
and Pasi 1993].
The application of the PRP in probabilistic models involves assumptions.
—Dependencies between documents are
generally ignored. Documents are
considered in isolation, so that the
relevance of one document to a query
is considered independent of that of
other documents in the collection
(nevertheless, see Section 5).
—It is assumed that the probabilities
(e.g., P(R͉q k , d i )) in the decision function can be estimated in the best possible way, that is, accurately enough
to approximate the user’s real relevance judgment and therefore to order the documents accordingly.
Although these assumptions limit the
applicability of the PRP, models based
on it make possible the implementation
of IR systems offering some of the best
retrieval performances currently available [Robertson 1977]. There are, of
course, a number of other retrieval
strategies with high performance levels
that are not consistent with the PRP.
Examples of such strategies are the
Boolean or the cluster model. We are
not concerned here with these models
since they are not probabilistic in nature.
2.5 The Remainder of This Article

In the remainder of this article we survey probabilistic IR models in two main

•

533

categories: relevance models and inference models.
Relevance models, described in Section 3, are based on evidence about
which documents are relevant to a given
query. The problem of estimating the
probability of relevance for every document in the collection is difficult because of the large number of variables
involved in the representation of documents in comparison to the small
amount of document relevance information available. The models differ primarily in the way they estimate this or
related probabilities.
Inference models, presented in Section 4, apply concepts and techniques
originating from areas such as logic and
artificial intelligence. From a probabilistic perspective, the most noteworthy
examples are those that consider IR as
a process of uncertain inference. The
concept of relevance is interpreted in a
different way so that it can be extended
and defined with respect, not only to a
query formulation, but also to an information need.
The models of both categories are presented separately, but using a common
formalism and, as much as possible, at
the same level of detail.
We also note that we are not concerned here with issues related to evaluation. Evaluation is a very important
part of IR research and even a brief
treatment of some of the issues involved
in the area would require an entire paper. The interested reader should look
at the extensive IR literature on this
subject, in particular van Rijsbergen
[1979] and Sparck Jones [1981].
3. PROBABILISTIC RELEVANCE MODELS

The main task of IR systems based upon
relevance models is to evaluate the
probability of a document being relevant. This is done by estimating the
probability P(R͉q k , d i ) for every document d i in the collection, which is a
difficult problem that can be tackled
only by means of simplifying assumptions. Two kinds of approaches have

534

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been developed to deal with such assumptions, model-oriented and description-oriented.
Model-oriented approaches are based
upon some probabilistic independence
assumptions concerning the elements
used in representing5 the documents or
the queries. The probabilities of these
individual representation elements are
estimated, and, by means of the independence assumptions, the probabilities
of the document representations are estimated from them. The binary independence indexing and retrieval models
(Sections 3.3 and 3.2) and the n-Poisson
model (Section 3.8) are examples of this
approach.
Description-oriented approaches are
more heuristic in nature. Given the representation of queries and documents, a
set of features for query-document pairs
is defined (e.g., occurrence frequency information) that allows each query-document pair in the collection to be mapped
onto these features. Then, by means of
some training data containing querydocument pairs together with their corresponding relevance judgments, the
probability of relevance is estimated
with respect to these features. The best
example of the application of this approach is the Darmstadt Indexing model
(Section 3.4). However, a new model
whose experimental results are not yet
known has been proposed by Cooper et
al. [1992]. These models exploit the
mapping between representations and
descriptions introduced in Section 2.2.
3.1 Probabilistic Modeling as a Decision
Strategy

The use of probabilities in IR was advanced in 1960 by Maron and Kuhns
[1960]. In 1976, Robertson and Sparck
Jones went further by showing the powerful contribution of probability theory

5
Depending on the complexity of the models, the
probabilities to be estimated can be with respect
to the representations or the descriptions. For
clarity, we refer to the representations only, unless otherwise stated.


in modeling IR. The probabilistic model
was theoretically finalized by van Rijsbergen [1979, Chapter 6]. The focus of
the model is on its analysis as a decision
strategy based upon a loss or risk function.
Referring to the conceptual model described in Section 2.2, it is assumed
that the representation and the description methods for queries and documents
are the same. Queries and documents
are described by sets of index terms. Let
T ϭ {t 1 , . . . , t n } denote the set of terms
used in the collection of documents. We
represent the query q k with terms belonging to T. Similarly, we represent a
document d j as the set of terms occurring in it. With a binary representation,
d j is represented as the binary vector
x ϭ (x 1 , . . . , x n ) with x i ϭ 1 if t i ʦ d j
ជ
and x i ϭ 0 otherwise. The query q k is
represented in the same manner.
The basic assumption, common to
most models described in Section 3, is
that the distribution of terms within the
document collection provides information about the relevance of a document
to a given query, since it is assumed
that terms are distributed differently in
relevant and irrelevant documents. This
is known as the cluster hypothesis (see
van Rijsbergen [1979, pp. 45–47]). If the
term distribution were the same within
the sets of relevant and irrelevant documents, then it would not be possible to
devise a discrimination criterion between them, in which case a different
representation of the document information content would be necessary.
The term distribution provides information about the “probability of relevance” of a document to a query. If we
assume binary relevance judgments,
then the term distribution provides information about P(R͉q k , d j ).
The quantity P(R͉q k , x ), with x as a
ជ
ជ
binary document representation, cannot
be estimated directly. Instead, Bayes’
theorem is applied [Pearl 1988]:

P ͑ R ͉ q k, x ͒ ϭ
ជ

ជ
P ͑ R ͉ q k ͒ ⅐ P ͑ x ͉ R, q k ͒
P ͑ x ͉ q k͒
ជ

.

Table I. Cost of Retrieving and Not Retrieving a
Relevant and Irrelevant Document

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only those documents for which the following relation holds,

P ͑ d j͉ R ͒
៮
P ͑ d j͉ R ͒

Ͼ ␭,

where
To simplify notation, we omit the q k
on the understanding that evaluations
are with respect to a given query q k .
The previous relation becomes

P͑R͉x͒ ϭ
ជ

P͑R͒ ⅐ P͑x͉R͒
ជ
P͑x͒
ជ

͸ C ͑R, dec͒ ⅐ P͑d ͉R͒ .
j

៮
␭2 ⅐ P͑R͒

␭1 ⅐ P͑R͒

.

3.2 The Binary Independence Retrieval
Model

,

where P(R) is the prior probability of
relevance, P(x ͉R) is the probability of
ជ
observing the description x conditioned
ជ
upon relevance having been observed,
and P(x ) is the probability that x is
ជ
observed. The latter is determined as
the joint probability distribution of the
n terms within the collection. The preceding formula evaluates the “posterior”
probability of relevance conditioned
upon the information provided in the
vector x .
ជ
The provision of a ranking of documents by the PRP can be extended to
provide an “optimal threshold” value.
This can be used to set a cutoff point in
the ranking to distinguish among those
documents that are worth retrieving
and those that are not. This threshold is
determined by means of a decision strategy, whose associated cost function
C j (R, dec) for each document dj is described in Table I.
The decision strategy can be described simply as one that minimizes
the average cost resulting from any decision. This strategy is equivalent to
minimizing the following risk function

᏾ ͑ R, dec ͒ ϭ

␭ϭ

j

d j ʦD

It can be shown (see van Rijsbergen
[1979, pp. 115–117]) that the minimization of that function brings about an
optimal partitioning of the document
collection. This is achieved by retrieving

In the previous section, it remains necessary to estimate the joint probabilities
៮
P(d j ͉R) and P(d j ͉R ), that is, P(x ͉R) and
ជ
P(x ͉R ), if we consider the binary vector
ជ ៮
document representation x .
ជ
In order to simplify the estimation
process, the components of the vector x
ជ
are assumed to be stochastically independent when conditionally dependent
៮
upon R or R . That is, the joint probability distribution of the terms in the document d j is given by the product of the
marginal probability distributions:

͹ P͑ x ͉R͒,
n

ជ
P ͑ d j͉ R ͒ ϭ P ͑ x ͉ R ͒ ϭ

i

iϭ1

៮
͹ P͑ x ͉R͒.
n

៮
ជ ៮
P ͑ d j͉ R ͒ ϭ P ͑ x ͉ R ͒ ϭ

i

iϭ1

This binary independence assumption
is the basis of a model first proposed in
Robertson and Sparck Jones [1976]: the
binary independence retrieval model
(BIR). The assumption has always been
recognized as unrealistic. Nevertheless,
as pointed out by Cooper [1995, Section
5], the assumption that actually underlies the BIR model is not that of binary
independence but the weaker assumption of linked dependence:

ជ
P͑x͉R͒
ជ ៮
P͑x͉R͒

P x R
͹ P͑͑ x ͉͉R͒͒ .
៮
n

ϭ

i

iϭ1

i


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This states that the ratio between the
probabilities of x occurring in relevant
ជ
and irrelevant documents is equal to
the product of the corresponding ratios
of the single terms.
Considering the decision strategy of
the previous section, it is now possible
to obtain a decision strategy by using a
logarithmic transformation to obtain a
linear decision function:

g ͑ d j ͒ ϭ log

P ͑ d j͉ R ͒
៮
P ͑ d j͉ R ͒

Ͼ log ␭ .

To simplify notation, we define p j ϭ
៮
P(x j ϭ 1͉R) and q j ϭ P(x j ϭ 1͉R ), which
represent the probability of the jth term
appearing in a relevant and in an irrelevant document, respectively. Clearly:
1 Ϫ p j ϭ P(x j ϭ 0͉R), and 1 Ϫ q j ϭ
៮
P(x j ϭ 0͉R ). This gives:

͹p
n

ជ
P͑x͉R͒ ϭ

xj
j

⅐ ͑ 1 Ϫ p j ͒ 1Ϫx j,

xj
j

⅐ ͑ 1 Ϫ q j ͒ 1Ϫx j.

jϭ1

͹q
n

ជ ៮
P͑x͉R͒ ϭ

jϭ1

Substituting the preceding gives:

͸ ͩ x ⅐ log p
q
n

g ͑ d i͒ ϭ

j

j

jϭ1

j

ϩ ͑ 1 Ϫ x j ͒ ⅐ log

1 Ϫ pj
1 Ϫ qj

ͪ

͸ c x ϩ C,
n

ϭ

j j

jϭ1

where

c j ϭ log

p j ⅐ ͑ 1 Ϫ q j͒
q j ⅐ ͑ 1 Ϫ p j͒

,

͸ log 1 Ϫ p .
1Ϫq
n

Cϭ

j

jϭ1

j


This formula gives the RSV of document d j for the query under consideration. Documents are ranked according
to their RSV and presented to the user.
The cutoff value ␭ can be used to determine the point at which the display of
the documents is stopped, although the
RSV is generally used only to rank the
entire collection of documents. In a real
IR system, the ordering of documents on
their estimated probability of relevance
to a query matters more than the actual
value of those probabilities. Therefore,
since the value of C is constant for a
specific query, we need only consider
the value of c j . This value, or more often
the value exp(c j ), is called the term relevance weight (TRW) and indicates the
term’s capability to discriminate relevant from irrelevant documents. As can
be seen, in the BIR model term relevance weights contribute “independently” to the relevance of a document.
To apply the BIR model, it is necessary to estimate the parameters p j and
q j for each term used in the query. This
is done in various ways, depending upon
the amount of information available.
The estimation can be retrospective or
predictive. The first is used on test collections where the relevance assessments are known, and the second with
normal collections where parameters
are estimated by means of relevance
feedback from the user.
Another technique, proposed by Croft
and Harper [1979], uses collection information to make estimates and does not
use relevance information. Let us assume that the IR system has already
retrieved some documents for the query
q k . The user is asked to give relevance
assessments for those documents, from
which the parameters of the BIR are
estimated. If we also assume we are
working retrospectively, then we know
the relevance value of all individual
documents in the collection. Let a collection have N documents, R of which are
relevant to the query. Let n j denote the
number of documents in which the term
x j appears, amongst which only r j are
relevant to the query. The parameters

p j and q j can then be estimated:

rj

ˆ
pj ϭ

R

,

ˆ
qj ϭ

nj Ϫ rj
NϪR

•

537

proximated by the “inverse document
frequency” IDF j ϭ log N/n j . This is
widely used in IR to indicate the intuitive discrimination power of a term in a
document collection.

.

These give:

rj
R Ϫ rj

TRW j ϭ

nj Ϫ rj

3.3 The Binary Independence Indexing
Model

.

N Ϫ nj Ϫ R ϩ rj
This approach is possible only if we
have relevance assessments for all documents in the collection, that is, where
we know R and r j . According to Croft
and Harper, given that the only information concerning the relevance of documents is that provided by a user
through relevance feedback, predictive
˜
estimations should be used. Let R denote the number of documents judged
relevant by the user. Furthermore, let r j
˜
be the number of those documents in
which the term x j occurs. We can then
combine this with the estimation technique of Cox [1970] to get

˜
r j ϩ 0.5
˜
TR W j ϭ

˜
˜
R Ϫ r j ϩ 0.5
˜
n j Ϫ r j ϩ 0.5

.

˜
˜
N Ϫ n j Ϫ R ϩ r j ϩ 0.5
Usually, the relevance information
given by a user is limited and is not
sufficiently representative of the entire
collection. Consequently, the resulting
estimates tend to lack precision. As a
partial solution, one generally simplifies by assuming p j to be constant for all
the terms in the indexing vocabulary.
The value p j ϭ 0.5 is often used, which
gives a TRW that can be evaluated easily:

˜
TR W j ϭ

The binary independence indexing
model (BII model) is a variant of the
BIR model. Where the BIR model regards a single query with respect to the
entire document collection, the BII
model regards one document in relation
to a number of queries. The indexing
weight of a term is evaluated as an
estimate of the probability of relevance
of that document with respect to queries
using that term. This idea was first
proposed in Maron and Kuhns’s [1960]
indexing model.
In the BII, the focus is on the query
representation, which we assume to be
a binary vector z . The dimension of the
ជ
vector is given by the set of all terms T
that could be used in a query, and z j ϭ
1 if the term represented by that element is present in the query; z j ϭ 0
otherwise.6 In this model, the term
weights are defined in terms of frequency information derived from queries; that is, an explicit document representation is not required. We assume
only that there is a subset of terms that
can be used to represent any document
and will be given weights with respect
to a particular document.
The BII model seeks an estimate of
the probability P(R͉z , d j ) that the docuជ
ment d j will be judged relevant to the
query represented by z . In the formalជ
ism of the previous section, we use x to
ជ
denote the document representation. So
far this model looks very similar to the
BIR; the difference lies with the appli-

N Ϫ nj
nj

For large N (i.e., large collections of
documents) this expression can be ap-

6
As a consequence, two different information
needs (i.e., two queries) using the same set of
terms produce the same ranking of documents.


538

•

F. Crestani et al.

cation of Bayes’ theorem:

ជ ជ
P͑R͉z, x͒ ϭ

ជ
ជ
ជ
P ͑ R ͉ x ͒ ⅐ P ͑ z ͉ R, x ͒
ជ ជ
P͑z͉x͒

ϭ
.

P(R͉x ) is the probability that the docជ
uments represented by x will be judged
ជ
relevant to an arbitrary query. P(z ͉R, x )
ជ
ជ
is the probability that the document will
be relevant to a query with representation z . As z and x are assumed to be
ជ
ជ
ជ
independent, P(z ͉x ) reduces to the probជ ជ
ability that the query z will be submitជ
ted to the system P(z ).
ជ
To proceed from here, some simplifying assumptions must be made.
(1) The conditional distribution of
terms in all queries is independent.
This is the classic “binary independence assumption” from which the
model’s name arises:

͹ P͑ z ͉R, xជ ͒.
n

ជ
ជ
P ͑ z ͉ R, x ͒ ϭ

i

iϭ1

(2) The relevance of a document with
representation x with respect to a
ជ
query z depends only upon the
ជ
terms used by the query (i.e., those
with z i ϭ 1) and not upon other
terms.
(3) With respect to a specific document,
for each term not used in the document representation, we assume:

ជ
ជ
P͑R͉zi , x͒ ϭ P͑R͉x͒.
Now, applying the first assumption to
P(R͉z , x ), we get:
ជ ជ

ជ ជ
P͑R͉z, x͒ ϭ

ជ
P͑R͉x͒
ជ ជ
P͑z͉x͒

͹ P͑ z ͉R, xជ ͒.
n

⅐

i

iϭ1

By applying the second assumption and
Bayes’ theorem, we get the ranking formula:

ជ ជ
P͑R͉z, x͒ ϭ

͟ iP ͑ z i͒
ជ
P͑z͒

͉
͹ P͑PR͑Rz͉xជ, ͒xជ ͒
n

⅐

i

iϭ1


͟ iP ͑ z i͒
ជ
P͑z͒

ជ
⅐ P͑R͉x͒ ⅐

ជ
P ͑ R ͉ z i ϭ 1, x ͒
ជ
P͑R͉x͒

͹

z i ϭ1

⅐

͹ P͑RP͉z͑Rϭ͉xជ0, xជ ͒ .
͒
i

z i ϭ0

The value of the first fraction is a
constant c for a given query, so there is
no need to estimate it for ranking purposes. In addition, by applying the third
assumption, the third fraction becomes
equal to 1 and we obtain:

ជ ជ
ជ
P͑R͉z, x͒ ϭ c ⅐ P͑R͉x͒ ⅐

͹

ជ ជ
t i ʦz പx

ជ
P͑R͉ti , x͒
ជ
P͑R͉x͒

.

There are a few problems with this
model. The third assumption contrasts
with experimental results reported by
Turtle [1990], who demonstrates the advantage of assigning weights to query
terms not occurring in a document.
Moreover, the second assumption is
called into question by Robertson and
Sparck Jones [1976], who proved experimentally the superiority of a ranking
approach in which the probability of
relevance is based upon both the presence and the absence of query terms in
documents. The results suggest that the
BII model might obtain better results if
it were, for example, used together with
a thesaurus or a set of term–term relations. This would make possible the use
of document terms not present in the
query but related in some way to those
that were.
Fuhr [1992b] pointed out that, in its
present form, the BII model is hardly an
appropriate model because, in general,
not enough relevance information is
available to estimate the probability
ជ
P(R͉t i , x ) for specific term-document
pairs. To overcome this problem in part,
one can assume that a document consists of independent components to
which the indexing weights relate. However, experimental evaluations of this
strategy have shown only average retrieval results [Kwok 1990].
Robertson et al. [1982] proposed a

model that provides a unification of the
BII and BIR models. The proposed
model, simply called Model 3 (as opposed to the BII model called Model 1
and the BIR model called Model 2), lets
us combine the two retrieval strategies
of the BII and the BIR models, thus
providing a new definition of probability
of relevance that unifies those of the BII
and BIR models. In the BII model the
probability of relevance of a document
given a query is computed relative to
evidence consisting of the properties of
the queries for which that document
was considered relevant, whereas in the
BIR model it is computed relative to the
evidence consisting of the properties of
documents considered relevant by that
same query. Model 3 allows us to use
both forms of evidence. Unfortunately, a
computationally tractable estimation
theory fully faithful to Model 3 has not
been proposed. The Model 3 idea was
explored further by Fuhr [1989] and
Wong and Yao [1989] (see Section 3.5).
3.4 The Darmstadt Indexing Model

The basic idea of the DIA is to use
long-term learning of indexing weights
from users’ relevance judgments [Fuhr
and Knowrz 1984; Biebricher et al.
1988; Fuhr and Buckley 1991]. It can be
seen as an attempt to develop indexterm-specific estimates based upon the
use of index terms in the learning sample.
DIA attempts to estimate P(R͉x i , q k )
from a sample of relevance judgments of
query-document
or
term-document
pairs. This approach, when used for indexing, associates a set of heuristically
selected attributes with each term-document pair, rather than estimating the
probability associated with an index
term directly (examples are given in the
following). The use of an attribute set
reduces the amount of training data required and allows the learning to be
collection-specific. However, the degree
to which the resulting estimates are
term-specific depends critically upon
the particular attributes used.

•

539

The indexing performed by the DIA is
divided in two steps: a description step
and a decision step.
In the description step relevance descriptions for term-document pairs
(x i , x ) are formed. These relevance deជ
scriptions s(x i , x ) 7 comprise a set of atជ
tributes considered important for the
task of assigning weights to terms with
respect to documents. A relevance description s(x i , x ) contains values of atជ
tributes of the term x i of the document
(represented by x ) and of their relationជ
ships. This approach does not make any
assumptions about the structure of the
function s or about the choice of attributes. Some possible attributes to be
used by the description function are:
—frequency of occurrence of term x i in
the document;
—inverse frequency of term x i in the
collection;
—information about the location of the
occurrence of term x i in the document;
or
—parameters describing the document,
for example, its length, the number of
different terms occurring in it, and so
on.
In the decision step, a probabilistic
index weight based on the previous data
is assigned. This means that we estimate P(R͉s(x i , x )) and not P(R͉x i , x ). In
ជ
ជ
the latter case, we would have regarded
a single document d j (or x ) with respect
ជ
to all queries containing x i , as in the BII
model. Here, we regard the set of all
query-document pairs in which the
same relevance description s occurs.
The interpretation of P(R͉s(x i , x )) is
ជ
therefore the probability of a document
being judged relevant to an arbitrary
query, given that a term common to
both document and query has a relevance description s(x i , x ).
ជ
The estimates of P(R͉s(x i , x )) are deជ
rived from a learning sample of termdocument pairs with attached relevance
7
These are similar to those used in pattern recognition.


540

•

F. Crestani et al.

judgments derived from the query-document pairs. If we call this new domain
L, we have:

L ʚ DxQxᑬ

or

L ϭ ͕͑ q k , d j , r kj ͖͒ .

By forming relevance descriptions for
the terms common to queries and documents for every query-document pair in
L, we get a multiset of relevance descriptions with relevance judgments:

L x ϭ ͓͑ s ͑ x i , d j ͒ , r kj ͉͒
x i ʦ q k പ d j ٙ ͑ q k , d j , r kj ͒ ʦ L ͔ .
With this set, it would be possible to
estimate P(R͉s(x i , x )) as the relative
ជ
frequency of those elements of L x with
the same relevance description. Nevertheless, the technique used in DIA
makes use of an indexing function, because it provides better estimates
through additional plausible assumptions about the indexing function. In
Fuhr and Buckley [1991], various linear
indexing functions estimated by leastsquares polynomials were used, and in
Fuhr and Buckley [1993] a logistic indexing function estimated by maximum
likelihood was attempted. Experiments
were performed using both a controlled
and a free term vocabulary.
The experimental results on the standard test collections indicate that the
DIA approach is often superior to other
indexing methods. The more recent (but
only partial) results obtained using the
TREC collection [Fuhr and Buckley
1993] tend to support this conclusion.
3.5 The Retrieval with Probabilistic
Indexing Model

The retrieval with probabilistic indexing
(RPI) model described in Fuhr [1989]
takes a different approach from other
probabilistic models. This model assumes that we use not only a weighting
of index terms with respect to the document but also a weighting of query
terms with respect to the query. If we
denote by w mi the weight of index term
x i with respect to the document x m , and
ជ

by v ki the weight of the query term z i ϭ
x i with regard to the query z k , then we
ជ
can evaluate the following scalar product and use it as retrieval function.

ជ ជ
r ͑ x m , z k͒ ϭ

͸

w mi ⅐ v ki .

͕ x m ϭz k ͖

Wong and Yao [1989] give an utilitytheoretic interpretation of this formula
for probabilistic indexing. Assuming we
have a weighting of terms with respect
to documents (similar to those, for example, of BII or DIA), the weight v ki can
be regarded as the utility of the term t i ,
and the retrieval function r(d m , q k ) as
the expected utility of the document
with respect to the query. Therefore,
r(d m , q k ) does not estimate the probability of relevance, but has the same
utility-theoretic justification as the
PRP.
RPI was developed especially for combining probabilistic indexing weighting
with query-term weighting based, for
example, on relevance feedback. As a
result, its main advantage is that it is
suitable for application to different
probabilistic indexing schemes.
3.6 The Probabilistic Inference Model

Wong and Yao [1995] extend the work
reported in Wong and Yao [1989] by
using an epistemological view of probability, from where they proposed a probabilistic inference model for IR. With
the epistemic view of probability theory,
the probabilities under consideration
are defined on the basis of semantic
relationships between documents and
queries. The probabilities are interpreted as degrees of belief.
The general idea of the model starts
with the definition of a concept space,
which can be interpreted as the knowledge space in which documents, index
terms, and user queries are represented
as propositions. For example, the proposition d is the knowledge contained in
the document; the proposition q is the
information need requested; and the
proposition d പ q is the portion of
knowledge common to d and q.

An epistemic probability function P is
defined on the concept space. For example, P(d) is the degree to which the
concept space is covered by the knowledge contained in the document and
P(d പ q) is the degree to which the
concept space is covered by the knowledge common to the document and the
query.
From these probabilities, different
measures can be constructed to evaluate
the relevance of documents to queries;
the measures offer different interpretations of relevance and thus lead to different approaches to model IR. We discuss two of them. The first is:

⌿͑d 3 q͒ ϭ P͑q͉d͒ ϭ

P͑d പ q͒
P͑d͒

.

⌿(q 3 d) is considered a recall index of
the document with respect to the query,
and is defined as the probability that a
relevant document is retrieved. A recalloriented measure should be used when
the user is interested in finding as
many papers as possible on the subject.
Depending on the relationships between concepts, different formulations
of ⌿(d 3 q) and ⌿(q 3 d) are obtained.
For example, suppose that the concept
space is t 1 ഫ . . . ഫ t n where the basic
concepts are (pairwise) disjoint; that is,
ti പ tj ϭ
for i
j. It can be proven
that

⌿͑d 3 q͒ ϭ

͚t P͑d പ q͉t͒ P͑t͒
P͑d͒

P͑q͒

.

Wong and Yao [1995] aim to provide a
probabilistic evaluation of uncertain implications that have been advanced as a
way to measure the relevance of documents to queries (see Section 4.1). Although measuring uncertain implications by a probability function is more
restrictive than, for example, using the
possible-world analysis, the model proposed by Wong and Yao is both expressive and sound. For example, they show
that the Boolean, fuzzy set, vector
space, and probabilistic models are special cases of their model.
3.7 The Staged Logistic Regression Model

P͑q പ d͒
P͑q͒

͚t P͑d പ q͉t͒ P͑t͒

541

.

⌿(d 3 q) can be considered as a
measure of precision of the document
with respect to the query, and is defined
as the probability that a retrieved document is relevant. A precision-oriented
interpretation of relevance should be
used when the user is interested in locating a specific piece of information. A
second measure is:

⌿͑q 3 d͒ ϭ P͑d͉q͒ ϭ

⌿͑q 3 d͒ ϭ

•

,

The staged logistic regression model
(SLR), proposed in Cooper et al. [1992],
is an attempt to overcome some problems in using standard regression methods to estimate probabilities of relevance in IR. Cooper criticizes Fuhr’s
approaches, especially the DIA, which
requires strong simplifying assumptions. He thinks (a longer explanation of
his point of view appears in Section 5)
that these assumptions inevitably distort the final estimate of the probability
of relevance. He advocates a “modelfree” approach to estimation. In addition, a more serious problem lies in the
use of standard polynomial regression
methods. Standard regression theory is
based on the assumption that the sample values taken for the dependent variable are from a continuum of possible
magnitudes. In IR, the dependent variable is usually dichotomous: a document
is either relevant or irrelevant. So standard regression is clearly inappropriate
in such cases.
A more appropriate tool, according to
Cooper, is logistic regression, a statistical method specifically developed for using dichotomous (or discrete) dependent
variables. Related techniques have been
used with some success by other researchers; for example, Fuhr employed

542

•

F. Crestani et al.

it in Fuhr and Pfeifer [1991] and more
recently in Fuhr and Buckley [1993].
The method proposed by Cooper is
based on the guiding notion of treating
composite clues on at least two levels,
an intraclue level at which a predictive
statistic is estimated separately for
each composite clue,8 and an interclue
level in which these separate statistics
are combined to obtain an estimate of
the probability of relevance for a querydocument pair. As this proceeds in
stages, the method is called staged logistic regression. A two-stage SLR
would be as follows.
(1) A statistical simplifying assumption
is used to break down the complex
joint probabilistic distribution of the
composite clues. This assumption is
called linked dependence. For example, assuming that we have only two
clues, a positive real number K exists such that the following conditions hold true.

P ͑ a, b ͉ R ͒ ϭ K ⅐ P ͑ a ͉ R ͒ ⅐ P ͑ b ͉ R ͒
៮
៮
៮
P ͑ a, b ͉ R ͒ ϭ K ⅐ P ͑ a ͉ R ͒ ⅐ P ͑ b ͉ R ͒ .

៮
P ͑ a, b ͉ R ͒

ϭ

P͑a͉R͒ P͑b͉R͒
⅐
.
៮
៮
P͑a͉R͒ P͑b͉R͒

Generalizing this result to the case
of n clues and taking the “log odds,”
we obtain:

LogO ͑ R ͉ a 1 , . . . , a n ͒ ϭ LogO ͑ R ͒

͸ ͑LogO͑R͉a ͒ Ϫ LogO͑R͒͒.
n

ϩ

LogO ͑ R ͉ a i ͒ ϭ LogO ͑ R ͉ h i , . . . , h m ͒
ϭ c 0 ϩ c 1h 1 , ϩ . . .
ϩ c mh m .

It follows that

P ͑ a, b ͉ R ͒

(2) A logistic regression analysis on a
learning sample is used to estimate
the terms on the right-hand side of
the previous equation. Unfortunately, the required learning sample
is often only available within the
environment of test collections, although it could be possible to use
the results of previous good queries
for this purpose.
The estimation of LogO(R) is quite
straightforward using simple proportions. A more complex matter is
the estimation of LogO(R͉a i ), when
there are too few query-document
pairs in the learning set with the
clue a i to yield estimates of P(R͉a i )
៮
and P(R ͉a i ). To go beyond simple
averaging, Cooper uses multiple logistic regression analysis. If we assume that the clue a i is a composite
clue whose elementary attributes
are h i , . . . , h m , then we can estimate LogO(R͉a i ):

i

iϭ1

This is used at retrieval time to
evaluate the log odds of relevance
for each document in the collection
with respect to the query.

8
A simple clue could be, for example, the presence
of an index term in a document. Clues must be
machine-detectable.


To demonstrate how the logistic
function comes into the model, the
probability of relevance of a document can be expressed as

P ͑ R ͉ h i , . . . , h m͒
ϭ

e c 0ϩc 1h 1 ,ϩ. . .ϩc mh m
1 ϩ e c 0ϩc 1h 1 ,ϩ. . .ϩc mh m

.

Taking the log odds of both sides
conveniently reduces this formula to
the previous one.
(3) A second logistic regression analysis, based on the same learning
sample, is used to obtain another
predictive rule to combine the composite clues and correct biases introduced by the simplifying assumption.
The linked dependence assumption
tends to inflate the estimates for

documents near the top of the output ranking whenever the clues on
which the estimates are based are
strongly interdependent. To help
correct this, a second-level logistic
regression analysis is performed on
the results of the first of the following form.

LogO ͑ R ͉ a 1 , . . . , a n ͒
ϭ d 0 ϩ d 1 Z ϩ d 2 n,
n
͚ iϭ1

where Z ϭ
(LogO(R͉a i ) Ϫ
LogO(R)) and n is the number of
composite clues. More elaborate correcting equations might also be considered.
When a query is submitted to the
system and a document is compared
against it, the technique in (2) is applied to evaluate the log odds necessary
to obtain Z. That is then employed in (3)
to adjust the estimate of the log odds of
relevance for the document.
This approach seems flexible enough
to handle almost any type of probabilistic retrieval clues likely to be of interest, and is especially appropriate when
the retrieval clues are grouped or composite. However, the effectiveness of the
methodology remains to be determined
empirically, and its performance compared with other retrieval methods. An
experimental investigation is currently
under way by Cooper, and the use of
logistic regression has also been investigated by Fuhr, as reported in the proceedings of the TREC-1 Conference
[Fuhr and Buckley 1993].
3.8 The N-Poisson Indexing Model

This probabilistic indexing model is an
extension to n dimensions of the 2-Poisson model proposed by Bookstein et al.
[Bookstein and Swanson 1974]. In its
two-dimensional form the model is
based upon the following assumption. If
the number of occurrences of a term
within a document is different depending upon whether the document is relevant, and if the number of occurrences

•

543

of that term can be modeled using a
known distribution, then it is possible
to decide if a term should be assigned to
a document by determining to which of
the two distributions the term belongs.
The 2-Poisson model resulted from a
search for the statistical distribution of
occurrence of potential index terms in a
collection of documents.
We can extend the preceding idea to
the n-dimensional case. We suppose
there are n classes of documents in
which the term x i appears with different frequencies according to the extent
of coverage of the topic related to that
specific term. The distribution of the
term within each class is governed by a
single Poisson distribution. Given a
term x i , a document class for that term
K ij , and the expectation of the number
of occurrences of that term in that class
␭ ij , then the probability that a document contains l occurrences of x i (i.e.,
that tf(x i ) ϭ l), given that it belongs to
the class K ij , is given by

ជ
P ͑ tf ͑ x i ͒ ϭ l ͉ x ʦ K ij ͒ ϭ

␭l
ij
l!

e Ϫ␭ ij.

Extending this result, the distribution
of a certain term within the whole collection of documents is governed by a
sum of Poisson distributions, one for
each class of coverage. In other words, if
we take a document in the collection at
random whose probability of belonging
to class K ij is p ij , then the probability of
having l occurrences of term x i is:

͸pe
n

P ͑ tf ͑ x i ͒ ϭ l ͒ ϭ

ij

jϭ1

Ϫ ␭ ij

␭l
ij
l!

.

This result can be used with a Bayesian inversion to evaluate P(x ʦ K ij ͉tf(x i
ជ
ϭ l)) for retrieval purposes. The parameters ␭ ij and p ij can be estimated without feedback information by applying
statistical techniques to the document
collection.
Experiments have shown that the
performance of this model is not always
consistent. Some experiments by Harter

544

•

F. Crestani et al.

[1975] on a 2-Poisson model showed
that a significant number of “good” index terms were 2-Poisson, but they did
not provide conclusive evidence of the
validity of the n-Poisson model. These
results were covalidated by Robertson
and Walker [1994]. They demonstrated
considerable
performance
improvements by using some effective approximations to the 2-Poisson model on the
TREC collection. Other research investigated the possibility of using a 3-Poisson, and lastly Margulis [1992, 1993]
investigated the generalized n-Poisson
model on several large full-text document collections. His findings were
more encouraging than previous ones.
He determined that over 70% of frequently occurring words were indeed
distributed according to an n-Poisson
distribution. Furthermore, he found
that the distribution of most n-Poisson
words had relatively few single Poisson
components, usually two, three, or four.
He suggests that his study provides
strong evidence that the n-Poisson distribution could be used as a basis for
accurate statistical modeling of large
document collections. However, to date,
the n-Poisson approach lacks work on
retrieval strategies based upon the results so far.
4. UNCERTAIN INFERENCE MODELS

The models presented in this section are
based on the idea that IR is a process of
uncertain inference. Uncertain inference models are based on more complex
forms of relevance than those used in
relevance models, which are based
mainly upon statistical estimates of the
probability of relevance. With uncertain
inference models, information not
present in the query formulation may
be included in the evaluation of the
relevance of a document. Such information might be domain knowledge,
knowledge about the user, user’s relevance feedback, and the like. The estimation of the probabilities P(R͉q k , d i ,
K) involves the representation of the
knowledge K.

Another characteristic of uncertain
inference models is that they are not as
strongly collection-dependent as relevance models. Parameters in relevance
models are valid only for the current
collection, whereas inference models
can use knowledge about the user or the
application domain that can be useful
with many other collections.
This research area is promising in
that it attempts to move away from the
traditional approaches, and it may provide the breakthrough that appears necessary to overcome the limitations of
current IR systems.
There are two main types of uncertain
inference models: one based on nonclassical logic, to which probabilities are
mapped (Section 4.1), and the other
based on Bayesian inferences (Section
4.2).
4.1 A Nonclassical Logic for IR

In 1986, van Rijsbergen proposed a paradigm for probabilistic IR in which IR
was regarded as a process of uncertain
inference [van Rijsbergen 1986]. The
paradigm is based on the assumptions
that queries and documents can be regarded as logical formulae and that to
answer a query, an IR system must
prove the query from the documents.
This means that a document is relevant
to a query only if it implies the query, in
other words, if the logical formula d 3
q can be proven to hold. The proof may
use additional knowledge K; in that
case, the logical formula is then rewritten as (d, K) 3 q.
The introduction of uncertainty comes
from the consideration that a collection
of documents cannot be considered a
consistent and complete set of statements. In fact, documents in the collection could contradict each other in any
particular logic, and not all the necessary knowledge is available. It has been
shown [van Rijsbergen 1986; Lalmas
1997] that classical logic, the most commonly used logic, is not adequate to
represent queries and documents because of the intrinsic uncertainty

present in IR.9 Therefore, van Rijsbergen [1986] proposes the logical uncertainty principle:
Given any two sentences x and y; a
measure of the uncertainty of y 3 x
related to a given data set is determined by the minimal extent to
which we have to add information to
the data set, to establish the truth
of y 3 x.

The principle says nothing about how
“uncertainty” and “minimal” might be
quantified. However, in his paper, van
Rijsbergen suggested an informationtheoretic approach. This idea has been
followed by Nie et al. [1996] and Lalmas
[van Rijsbergen and Lalmas 1996].
However, that work is somewhat beyond the scope of this article.
Nearer to this survey, van Rijsbergen
[1989] later proposed estimating P(d 3
q) by imaging. Imaging formulates
probabilities based on a “possibleworlds” semantics [Stalnaker 1981] according to which a document is represented by a possible world w, that is, a
set of propositions with associated truth
values. Let ␶ denote a logical truth function; then ␶ (w, p) denotes the truth of
the proposition p in the world w. Furthermore, let ␴ (w, p) denote the world
most similar to w where p is true. Then,
y 3 x is true at w if and only if x is true
at ␴ (w, p).
Imaging uses this notion of most similar worlds to estimate P(y 3 x). Every
possible world w has a probability P(w),
and the sum over all possible worlds is
1. P(y 3 x) is computed as follows.

͸ P͑w͒␶͑w, y 3 x͒
ϭ ͸ P ͑ w ͒ ␶ ͑ ␴ ͑ w, y ͒ , y 3 x ͒
ϭ ͸ P ͑ w ͒ ␶ ͑ ␴ ͑ w, y ͒ , x ͒ .

P͑ y 3 x͒ ϭ

w

w

w

9
There are other reasons why classical logic is not
adequate, but these are not relevant to this article
(see Lalmas [1997]).

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545

It remains undetermined how to evaluate the function ␴ on document representations, and furthermore, how to assign a probability P to them. There have
been a few attempts at using imaging in
IR (e.g., Amati and Kerpedjiev [1992]
and Sembok and van Rijsbergen [1993]),
with rather disappointing results. A recent attempt by Crestani and van Rijsbergen [1995] taking the view that “an
index term is a world” obtains better
results.
The concept of relevance is not featured in the preceding framework. In
van Rijsbergen [1992] using Jeffrey’s
conditionalization was proposed to evaluate the probability of relevance P(R͉q k ,
d i ). This conditionalization, described
as “Neo-Bayesianism” by Pearl [1990],
allows conditioning to be based on evidence derived from the “passage of experience,” where the evidence can be
nonpropositional in nature. A comprehensive treatise of Jeffrey’s studies on
probability kinematics (i.e., on how to
revise a probability measure in the light
of uncertain evidence or observation)
can be found in Jeffrey [1965]. By
means of the famous example in that
book of inspecting the color of a piece of
cloth by candlelight, van Rijsbergen introduced a form of conditioning that has
many advantages over Bayesian conditioning. In particular, it makes possible
conditioning on uncertain evidence and
allows order-independent partial assertion of evidence. Such advantages, despite some strong assumptions, convinced van Rijsbergen that this
particular form of conditionalization is
more appropriate for IR than Bayesian
conditionalization. However, despite the
appeal of Jeffrey’s conditionalization,
the evaluation of the probability of relevance involves parameters whose estimation remains problematic.
In the same paper van Rijsbergen
[1992] makes the connection between
Jeffrey’s conditionalization and Dempster–Shafer theory of evidence [Dempster 1968; Shafer 1976]. This theory can
be viewed as a generalization of the
Bayesian method (e.g., it rejects the adACM Computing Surveys, Vol. 30, No. 4, December 1998

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Figure 2.

An inference network for IR.

ditivity rule), and has been used by
some researchers to develop IR models
[Schoken and Hummel 1993; de Silva
and Milidiu 1993].
4.2 The Inference Network Model

When IR is regarded as a process of
uncertain inference, then the calculation of the probability of relevance, and
the general notion of relevance itself,
become more complex. Relevance becomes related to the inferential process
by which we find and evaluate a relation between a document and a query.
A probabilistic formalism for describing inference relations with uncertainty
is provided by Bayesian inference networks, which have been described extensively in Pearl [1988] and Neapolitan [1990]. Turtle and Croft [Turtle
1990; Turtle and Croft 1990, 1991] applied such networks to IR. Figure 2
depicts an example of such a network.
Nodes represent IR entities such as documents, index terms, concepts, queries,
and information needs. We can choose

the number and kind of nodes we wish
to use according to how complex we
want the representation of the document collection or the information needs
to be. Arcs represent probabilistic dependencies between entities. They represent conditional probabilities, that is,
the probability of an entity being true
given the probabilities of its parents
being true.
The inference network is usually
made up of two component networks: a
document network and a query network. The document network, which
represents the document collection, is
built once for a given collection and its
structure does not change. A query network is built for each information need
and can be modified and extended during each session by the user in an interactive and dynamic way. The query network is attached to the static document
network in order to process a query.
In a Bayesian inference network, the
truth value of a node depends only upon
the truth values of its parents. To eval-

uate the strength of an inference chain
going from one document to the query
we set the document node d i to “true”
and evaluate P(q k ϭ true͉d i ϭ true).
This gives us an estimate of P(d i 3 q k ).
It is possible to implement various
traditional IR models on this network
by introducing nodes representing Boolean operators or by setting appropriate
conditional probability evaluation functions within nodes.
One particular characteristic of this
model that warrants exploration is that
multiple document and query representations can be used within the context
of a particular document collection (e.g.,
a Boolean expression or a vector). Moreover, given a single information need, it
is possible to combine results from multiple queries and from multiple search
strategies.
The strength of this model comes from
the fact that most classical retrieval
models can be expressed in terms of a
Bayesian inference network by estimating in different ways the weights in the
inference network [Turtle and Croft
1992a]. Nevertheless, the characteristics of the Bayesian inference process
itself, given that nodes (evidence) can
only be binary (either present or not),
limit its use to where “certain evidence”
[Neapolitan 1990] is available. The approach followed by van Rijsbergen (Section 4.1), which makes use of “uncertain
evidence” using Jeffrey’s conditionalization, therefore appears attractive.
5. EFFECTIVE RESULTS FROM FAULTY
MODELS

Most of the probabilistic models presented in this article use simplifying
assumptions to reduce the complexity
related to the application of mathematical models to real situations. There are
general risks inherent in the use of such
assumptions. One such risk is that
there may be inconsistencies between
the assumptions laid down and the data
to which they are applied.
Another is that there may be a misidentification of the underlying assump-

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547

tions; that is, the stated assumptions
may not be the real assumptions upon
which the derived model or resulting
experiments are actually based. This
risk was noted by Cooper [1995]. He
identified the three most commonly
adopted simplifying assumptions, which
are related to the statistical independence of documents, index terms, and
information needs:
Absolute Independence

P ͑ a, b ͒ ϭ P ͑ a ͒ ⅐ P ͑ b ͒ ;
Conditional Independence

P ͑ a, b ͉ R ͒ ϭ P ͑ a ͉ R ͒ ⅐ P ͑ b ͉ R ͒
៮
៮
៮
P ͑ a, b ͉ R ͒ ϭ P ͑ a ͉ R ͒ ⅐ P ͑ b ͉ R ͒ .
These assumptions are interpreted differently when a and b are regarded as
properties of documents or of users.
Cooper pointed out how the combined
use of the Absolute Independence assumption and either of the Conditional
Independence assumptions yields logical inconsistencies. The combined use of
these assumptions leads to the conclusion that P(a, b, R) Ͼ P(a, b), which is
contrary to the elementary laws of probability theory. Nevertheless, in most
cases where these inconsistencies appeared, the faulty model used as the
basis for experimental work has proved,
on the whole, to be successful. Examples
of this are given in Robertson and
Sparck Jones [1976] and Fuhr and
Buckley [1991].
The conclusion drawn by Cooper is
that the experiments performed were
actually based on somewhat different
assumptions, which were, in fact, consistent. In some cases where the Absolute Independence assumption was used
together with a Conditional Independence assumption, it seems that the required probability rankings could have
been achieved on the basis of the Conditional Independence assumption alone.
This is true of the model proposed by
Maron and Kuhns [1960]. In other
cases, the Conditional Independence asACM Computing Surveys, Vol. 30, No. 4, December 1998

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sumptions could be replaced by the single linked dependence assumption:

P ͑ a, b ͉ R ͒
៮
P ͑ a, b ͉ R ͒

ϭ

P͑a͉R͒ P͑b͉R͒
⅐
.
៮
៮
P͑a͉R͒ P͑b͉R͒

This considerably weaker assumption
is consistent with the Absolute Independence assumption. This is true of the
SLR model presented in Section 3.7 and
of the BIR model (whose name seems to
lose appropriateness in the light of
these results) presented in Section 3.2.
6. FURTHER RESEARCH

In the late ’90s, researchers have come
to realize that there is a leap to be made
towards a new generation of IR systems: towards systems able to cope with
increasingly demanding users, whose
requirements and expectations continue
to outstrip the progress being made in
computing, storage, and transport technology. Faster machines and better interconnectivity make possible access to
enormous amounts of information. This
information is increasing not only in
amount, but also in complexity; for example, structured hypertexts consisting
of multiple media are becoming the
norm. Until recently, research in information retrieval has been confined to
the academic world. Things are changing slowly. The success of the TREC
initiative over the last four years (from
Harman [1993] to Harman [1996]), particularly in terms of the interest shown
by commercial organizations, demonstrates the wider desire to produce sophisticated IR systems. Web searching
engines, which have a high profile in
the wider community, increasingly utilize probabilistic techniques. It can only
be hoped that this increasing awareness
and interest will stimulate new research.
The requirements of the next generation of IR systems include the following.
Multimedia Documents. The difficulty with multimedia document collections lies in the representation of the

nontextual parts of documents such as
sounds, images, and animations. Several approaches have been tried so far:
they can be exemplified in the particular approach of attaching textual descriptions to nontextual parts, and the
derivation of such descriptions by
means of an inference process (e.g.,
Dunlop [1991]). Nevertheless, such
techniques avoid the real issue of handling the media directly. This applies
not only to probabilistic models, but to
all IR models.
Interactive Retrieval. Current IR
systems, even those providing forms of
relevance feedback for the user, are still
based upon the traditional iterative
batch retrieval approach. Even relevance feedback acts upon a previous
retrieval run to improve the quality of
the following run [Harman 1992a,b]. We
need real interactive systems, making
possible a greater variety of interaction
with the user than mere query formulation and relevance feedback [Croft
1987]. User profile information, analysis
of browsing actions, or user modification of probabilistic weights, for example, could all be taken into consideration [Croft and Thompson 1987; Croft
et al. 1988, 1989; Thompson 1989,
1990a,b]. The subjective, contextual,
and dynamic nature of relevance is now
being recognized and incorporated into
probabilistic models [Campbell and van
Rijsbergen 1996].
Integrated Text and Fact Retrieval.
There has been a steady development of
the kinds of information being collected
and stored in databases, notably of text
(unformatted data) and of “facts” (formatted, often numerical, data). Demand
is growing for the availability of systems capable of dealing with all types of
data in a consistent and unified manner
[Fuhr 1992a, 1993; Croft et al. 1992;
Harper and Walker 1992].
Imprecise Data. The use of probabilistic modeling in IR is important not
only for representing the document information content, but also for repre-

senting and dealing with vagueness and
imprecision in the query formulation
and with imprecision and errors in the
textual documents themselves [Fuhr
1990; Turtle and Croft 1992b]. For example, the increasing use of scanners
and OCR in transferring documents
from paper to electronic form inevitably
introduces imprecision (but see Smith
and Stanfill [1988]).
7. CONCLUSIONS

The major concepts and a number of
probabilistic IR models have been described. We are aware that new models
are being developed as we speak. A survey is always a bit dated. However, we
believe we have covered the most important and the most investigated probabilistic models of IR.
It is not easy to draw conclusions
from a survey of 30 years of research. It
is safe to conclude that good results
have been achieved but more research is
required, since there is considerable
room for improvement. Current-generation probabilistic IR systems work quite
well when compared with the Boolean
systems they are replacing. A novice
user using natural language input with
a current-generation probabilistic IR
system gets, on average, better performance than an expert user with a Boolean system on the same collection.
Moreover, theoretically, the probabilistic approach to IR seems inherently
suitable for the representation and processing of the uncertain and imprecise
information that is typical of IR. We
believe that, with development, it will
be capable ultimately of providing an
integrated, holistic, and theoretically
consistent framework for the effective
retrieval of complex information.
ACKNOWLEDGMENTS
We would like to thank the anonymous reviewers
for their interesting and helpful comments.

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Received December 1996; revised October 1997; accepted October 1997


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