Back to the Roots: A Probabilistic Framework for
Query-Performance Prediction
Oren Kurland1 kurland@ie.technion.ac.il
Fiana Raiber1 fiana@tx.technion.ac.il
Anna Shtok 1
annabel@tx.technion.ac.il
David Carmel 2
carmel@il.ibm.com
Shay Hummel1 hummels@tx.technion.ac.il
Ofri Rom1 ofri.rom@gmail.com
1. Faculty of Industrial Engineering and Management, Technion, Haifa 32000, Israel
2. IBM Research Lab, Haifa 31905, Israel
ABSTRACT
The query-performance prediction task is estimating the effectiveness
of a search performed in response to a query when
no relevance judgments are available. Although there exist
many effective prediction methods, these differ substantially
in their basic principles, and rely on diverse hypotheses
about the characteristics of effective retrieval. We present a
novel fundamental probabilistic prediction framework. Using
the framework, we derive and explain various previously
proposed prediction methods that might seem completely
different, but turn out to share the same formal basis. The
derivations provide new perspectives on several predictors
(e.g., Clarity). The framework is also used to devise new prediction
approaches that outperform the state-of-the-art.
Categories and Subject Descriptors: H.3.3 [Information Search
and Retrieval]: Retrieval models
Keywords: query-performance prediction
1. INTRODUCTION
The query-performance prediction task has attracted a
lot of research attention [4]. The goal of this task is to estimate
the effectiveness of a search performed in response to
a query when there is a lack of relevance judgments. The
prediction can be performed before retrieval using the query
and corpus-based information [19, 16]. Post-retrieval prediction,
on the other hand, also uses information induced from
the result list of the most highly ranked documents [4].
Although there exists abundance of effective prediction
methods, these often rely on substantially different principles.
More specifically, various predictors are based on
completely different hypotheses with respect to what results
in effective retrieval. For example, some post-retrieval prediction
methods rely on the premise that a result list that
exhibits high clarity with respect to the corpus indicates
effective retrieval [10, 1, 11, 5, 18]. Other prediction meth-
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ods assume that effective retrieval should be manifested in
a result list that is robust with respect to query perturbations
[37, 42], document perturbations [36, 41], and retrieval
method perturbations [2]. Another class of prediction methods
is based on various analyses of retrieval scores in the
result list [35, 14, 42, 30, 12, 13].
Given the large variety of prediction approaches, and the
underlying hypotheses on which they are based, a few questions
arise. The most fundamental one is whether there is a
unified formal basis (framework) that can help explain various
prediction methods and techniques. A related, albeit
more specific, question is “what formal aspects of prediction
are shared by seemingly different prediction approaches?”.
The operational question that naturally emerges is whether
a formal analysis of the prediction task can give rise to new
(effective) prediction approaches.
To address the questions stated here, we present a novel
query-performance prediction framework that is based on
fundamental probabilistic IR principles. We establish the
framework by starting with a basic question that has not
been explicitly addressed in previous work on prediction:
“What is the probability that this result list is relevant to this
query?”. This question is a generalization to the documentlist
case of the core question of probabilistic retrieval [33, 22,
27]: “What is the probability that this document is relevant
to this query?”.
The framework we present sets the first formal grounds for
integrating pre-retrieval and post-retrieval prediction methods.
We show that these two paradigms target different,
yet complementary, formal aspects of prediction. Empirical
evaluation demonstrates the merits of their integration.
We use the framework to explain and derive various postretrieval
predictors that might seem at first glance to rely on
completely different hypotheses and use different principles.
For example, we derive (explain) the Clarity predictor [10].
This derivation provides a novel perspective about the actual
property of the result list that Clarity quantifies. Furthermore,
we use the framework to show that some predictors
that are based on the result-list robustness notion [37, 36,
41, 42] and on analysis of the retrieval-scores distribution
[14], implicitly share the same formal basis for prediction.
Our framework also provides a formal ground to using, for
prediction, measures of query-independent properties of the
result list (e.g., cohesion and dispersion); and, to integratingthese with query-dependent measures that are the focus of
most previously proposed predictors.
The proposed framework not only provides new insight
into existing predictors, it also gives rise to novel prediction
approaches. Some of these outperform state-of-the-art
methods, as shown by experiments conducted with a large
variety of TREC corpora (including ClueWeb).
To summarize, our main contributions are twofold. The
first, and perhaps most important, is on the formal side. The
prediction framework that we present sets unifying formal
grounds for a variety of prediction methods that might seem
to be completely different. The formal analysis also results
in novel insights about existing, and commonly used, prediction
methods and the connections between them. The
second contribution is using the framework to devise new
prediction approaches that outperform state-of-the-art.
2. RELATED WORK
Pre-retrieval query-performance predictors [19, 29, 25, 16,
39, 28] analyze the query expression, often using corpusbased
information. Post-retrieval predictors also use information
induced from the result list of the most highly
ranked documents [10, 1, 35, 37, 11, 5, 36, 2, 14, 42, 18, 30,
31, 7, 12, 13]. As noted above, the framework we present
sets formal probabilistic grounds to the integration of preretrieval
and post-retrieval prediction. We empirically show
that the integration yields prediction quality that transcends
the state-of-the-art. Furthermore, we show that the framework
provides a unified formal basis that can be used to
explain (derive), and provide new perspectives for, many
previously proposed post-retrieval predictors.
A recently proposed framework [21] explains several postretrieval
predictors. The idea is that an effective result list
is that which is similar to some pseudo effective list and dissimilar
to a pseudo ineffective list. Our framework is shown
to provide a formal probabilistic basis for this framework.
Furthermore, our framework accounts for prediction aspects
not accounted for by this framework [21]. (See Section 3.1.3
for a discussion.) Our framework also provides novel views
of predictors such as Clarity [10] and WIG [42], which substantially
depart from those previously proposed [21].
A conceptual framework for predicting query difficulty [5]
is based on measuring similarities between the result list,
the query, and the corpus. Our framework provides formal
grounds to several aspects of this framework; and, helps explain
several prediction methods and techniques that do not
naturally fit in this framework; e.g, those using reference
document lists [37, 36, 41, 14, 42].
A prediction framework [31], based on statistical decision
theory, uses multiple re-rankings of the result list and
their estimated effectiveness. We show that our framework
provides probabilistic grounds for the underlying prediction
principle of this framework. Furthermore, we use our framework
to explain predictors that cannot be explained in terms
of this approach [31]; e.g., Clarity [10] and WIG [42]. In
addition, the integration of pre-retrieval and post-retrieval
prediction that emerges in our framework was not addressed.
Notions of result list cohesion (dispersion) [37, 36] were argued
to be correlated to some extent with retrieval effectiveness.
We show that the formal prediction aspect targeted
by measures of this query-independent result list property
is complementary to that targeted by measures of querydependent
properties; the latter are the focus of most postretrieval
predictors. We also demonstrate the empirical merits
of the integration of these two types of measures.
Integrating predictors using a linear interpolation of prediction
values was employed with either pre-retrieval predictors
[15] or post-retrieval predictors [36, 14, 42, 31]. In
contrast, our framework provides formal grounds to the integration
of different types of predictors which target different
formal aspects of prediction; namely, pre-retrieval and
post-retrieval prediction, and prediction based on queryindependent
and query-dependent measures of result list
properties. Prediction integration as that proposed in previous
work can be used in our framework (e.g., for improving
pre-retrieval prediction quality) to potentially further improve
overall prediction quality. We leave the exploration of
this direction for future work.
3. PREDICTION FRAMEWORK
The query-performance prediction task is stated as estimating
the effectiveness of a ranking induced by retrieval
method M over a corpus of documents D in response to
query q in lack of relevance judgments [4].
Our first key observation is that the goal of prediction can
be stated, in probabilistic terms, as answering the question:
What is the probability that this result list (Dres), of the most
highly ranked documents, is relevant to this query (q)?
We focus on the result list and its ranking, rather than
address the entire corpus ranking, as this is also the focus of
the most commonly used evaluation measures (e.g., average
precision, precision at top ranks, and NDCG). Furthermore,
the question just stated is a generalization to the document
list case of the question posed by Sparck Jones et al. [33]
with respect to a single document: “What is the probability
that this document is relevant to this query”? This question
directly connects to the probability ranking principle [26]
that states that maximal retrieval effectiveness is attained if
documents are ranked by their relevance probabilities. The
question serves as the basis for the probabilistic retrieval
approach [33] and was used for a parallel derivation of the
language modeling approach [22, 27].
3.1 Probabilistic prediction
In what follows we use the question stated above as the
basis for developing our prediction framework. Let r be the
event of relevance. The query-performance prediction task
is, then, estimating p(r|q, Dres), which can be written as
p(r|q, Dres) =
p(Dres|q, r)p(r|q)
. (1)
p(Dres|q)
The probability p(Dres|q) does not depend on relevance,
but rather on the likelihood of retrieving Dres in response
to q by the given retrieval method M. As a case in point,
if document-query surface-level similarities are used for retrieval,
it is likely that Dres is composed of documents containing
many occurrences of query terms. Furthermore, an
estimate for p(Dres|q) can serve as a normalization factor to
ensure the compatibility of prediction values across retrieval
methods. However, as the evaluation of prediction quality
in prior work was based on fixing the retrieval method (and
varying the queries) [4], such normalization was not calledfor. Accordingly, the expression
Pbase(Dres; q) def
= p(Dres|q, r)p(r|q) (2)
from Equation 1 reflects the core prediction task. It turns
out, as we show below, that Pbase(Dres; q) is the basis for
numerous query-performance prediction methods that might
seem at first glance to be completely different and/or based
on different hypotheses with regard to the characteristics
of effective retrieval. We further show how new prediction
approaches can be devised based on Pbase(Dres; q). Finally,
we note that if Dres contains a single document, then the
prediction task modeled in Equation 2 is based on estimating
the likelihood of the document given the query. This is the
basis of the probabilistic approach to retrieval [33].
3.1.1 Integrating pre- and post- retrieval prediction
Estimating the prior probability of relevance to q in Equation
2, p(r|q), is (in spirit) the goal of pre-retrieval prediction
methods that operate prior to retrieval time [19, 16]. Indeed,
pre-retrieval predictors quantify this query difficulty notion
using information induced only from q and the corpus.
The probability p(Dres|q, r) is the likelihood of the result
list Dres given that a relevance event happens for q; i.e., the
probability that Dres is the list that provides information
pertaining to q. This is the list-based generalization of the
document likelihood principle used in probabilistic retrieval
[33]. Estimating the list likelihood is the implicit goal of
post-retrieval predictors.
Hence, our second key observation is that while by design,
post-retrieval predictors use result-list-based information in
addition to the information used by pre-retrieval predictors
(the query and the corpus), the two types of predictors target
different formal aspects of prediction; that is, the prior
probability of relevance to a query (pre-retrieval) and the
likelihood of a result list given relevance to the query (postretrieval).
To the best of our knowledge, Equation 1, and
consequently Equation 2, set the first formal grounds to
integrating pre-retrieval and post-retrieval prediction. We
demonstrate the empirical merits of the integration in Section
4. In what follows we focus on p(Dres|q, r), i.e., postretrieval
prediction.
3.1.2 Post-retrieval prediction
Our third key observation is that p(Dres|q, r) canbeestimated
by using documents in Dres as Dres’s proxies. We
use ˆp(·) to denote an estimate for p(·). Let p(d|q, r) bethe
probability that d is the document relevant to q; i.e., d’s
likelihood [33]. If we set ˆp(d|q, r) def
= 0 for d ̸∈ Dres, i.e.,
a document not in the result list is not considered relevant;
and, P
ˆp(di|q, r) = 1 holds, that is, ˆp(di|q, r) isa
di∈Dres probability distribution over Dres, and more generally, over
the corpus D, then we can define the following: 1
ˆp(Dres|q, r) def
= X
d i∈Dres
ˆp(Dres|di,r)ˆp(di|q, r). (3)
Assuming that ˆp(r|di) (the prior probability for di’s relevance),
ˆp(di) (the prior probability for di), and ˆp(Dres)
1
As is standard in work on mixture models, Equation 3 is
based on an independence assumption; specifically, Dres is
independent of q given di and r; i.e., we “back off” from Dres
to its proxies (di).
(the prior probability for the result list) are uniformly distributed,
and since ˆp(Dres|di,r)= ˆp(d i|Dres,r)ˆp(r|Dres)ˆp(Dres)
ˆp(r|d i)ˆp(d i)
,
Equation 3 then yields a generic post-retrieval predictor:
Pgeneric(Dres; q) def
= ˆp(r|Dres) X
d i∈Dres
ˆp(di|Dres,r)ˆp(di|q, r).
(4)
Devising an estimate for the probability that Dres is relevant
regardless of a specific query, ˆp(r|Dres), is the implicit
goal of several post-retrieval prediction methods. For example,
some predictors are based on the premise that result
list cohesion indicates effective retrieval, wherein cohesion
can be quantified by the list radius [5] and its clustering
tendency [36] 2 . Conversely, a diversified result list might
be considered as more likely to cover different aspects of
the underlying information need, and thereby assumed to
be relevant [5]. Hence, in Section 4 we empirically contrast
the two hypotheses of list cohesion versus list dispersion as
indicators for list relevance.
The summation in Equation 4 is of the weighted document
likelihood values in Dres (ˆp(di|q, r)); di’s weight is based on
the strength of its “association” with Dres
given that the
latter is relevant (ˆp(di|Dres,r)). For example, ˆp(di|Dres,r)
can be based on the probability that di is generated from
a language model induced from Dres. We next show how
Equation 4 can be used to explain additional predictors.
P
d i∈Dres
Explaining WIG. Let the document-list association strength
in Equation 4, ˆp(di|Dres,r), be a uniform distribution over
Dres; i.e., each document in Dres is considered as its equiimportant
representative. Then, the average document like-
ˆp(di|q, r), where k is the
lihood score in Dres, 1
k
number of documents in Dres, is used for prediction. This
is perhaps the most direct manifestation of the probability
ranking principle; that is, a result list with documents for
which the probability for relevance is high is considered effective
(relevant). 3 Furthermore, this prediction principle is
the basis of the weighted information gain (WIG) predictor
[42]. WIG measures the difference between the average
retrieval score in the list and that of the corpus. The divergence
from the corpus retrieval score serves to ensure
inter-query compatibility of prediction values.
Explaining Clarity. As mentioned above, list-cohesion measures
(e.g., radius) were suggested in prior work as performance
predictors [5, 36]. These implicitly serve for the estimate
ˆp(r|Dres), of the result list relevance prior, in Equation
4. An alternative approach to measuring cohesion is using
the corpus. That is, if a model of the result list is distant
from that of the corpus, which can be viewed as a pseudo
non-relevant document, then the list is considered focused;
hence, the retrieval is presumed to be effective. This is the
hypothesis underlying the Clarity prediction method [10].
However, the method used to compute Clarityexplicitly
depends on the query [10]. Thus, Clarity, in implementation,
cannot be considered as an estimate for p(r|Dres). Rather,
Clarity turns out to be an estimate for the summation in
Equation 4 as we show below.
2
In implementation, the query is used when computing
query-based inter-document similarities [36]. We come back
to this point later on.
3
In the probabilistic retrieval framework [33], for example,
ranking is based on document likelihood values.In Appendix A we show that the standard approach of
computing Clarity can be (re-)written as:
PClarity(Dres; q) = (5)
X
ˆp(di|q, r) ` Sim(di, Dres) − Sim(di, D) ´ ;
d i∈Dres
Sim(di,X), where X is either Dres or D, is the similarity
(based on the cross entropy measure) between a language
model induced from di and that induced from X.
Hence, Clarity is a special case of the generic predictor
from Equation 4 with a uniform ˆp(r|Dres). That is,
Clarity is a weighted sum of document likelihood values,
which in the language modeling case represent normalized
query-likelihood values. (See Appendix A for details.) The
document-list association strength (ˆp(di|Dres,r)) is estimated
in the Clarity case, as is shown in Equation 5, using the
corpus-regularized similarity between the document and the
result list; the corpus-based regularization downplays the
effect of general non-query-related aspects when computing
similarity to the list.
Thus, we attained the following novel perspective about
Clarity. Suppose that the retrieval at hand, whose effectiveness
we want to predict, uses the language-model-based
surface-level similarity between documents and the query.
This is the case for the query likelihood [32] and KL retrieval
[23] methods that are commonly used in work on using Clarity
for prediction. Hence, the induced ranking is essentially
based on the document likelihood scores defined in Appendix
A (i.e., normalized query-likelihood scores). Then, by Equation
5, a result list for which the highest ranked documents
are similar to the entire list — wherein similarity is corpus
regularized — is presumed to be effective (relevant).
3.1.3 Using reference lists
Equation 3 served as the basis for deriving the generic
post-retrieval prediction approach presented in Equation 4.
The underlying idea was to use documents in Dres as its
proxies for estimating the result list likelihood, p(Dres|q, r).
An alternative type of a proxy for Dres is a reference document
list [37, 41, 14, 42, 31]. Formally, a similar formulation
to that in Equation 3 that uses a set Sref of reference document
lists (denoted Dref) is:
Pref(Dres; q) def
= X
ˆp(Dres|Dref,r)ˆp(Dref|q, r). (6)
D ref ∈S ref
Thus, the relevance of Dres is estimated based on its association
with the reference lists, ˆp(Dres|Dref,r), where Dref is
weighted by its presumed relevance, ˆp(Dref|q, r). As we discuss
below, Equation 6 sets the probabilistic formal grounds
to quite a few predictors. These predictors differ by the
choice of reference lists and inter-list association measure.
Explaining QF and autocorrelation. Devising the estimate
ˆp(Dref|q, r) is a prediction problem in its own right,
which is not addressed by most predictors that utilize reference
lists [37, 41, 2, 14, 42]. For example, the query feedback
(QF) predictor [42] uses the overlap at top ranks between
Dres and a (single) list Dref for ˆp(Dres|Dref,r); Dref is retrieved
from the corpus using a (relevance) language model
induced from Dres. However, the presumed relevance of
Dref, ˆp(Dref|q, r), is not accounted for. Thus, it is not surprising
that QF was found to be a highly effective predictor
for the effectiveness of the reference result list (Dref) and
not only to that of the original result list (Dres) [31]. Indeed,
this finding can be formally explained using Equation
6. That is, the effectiveness (relevance) of the reference list
Dref can be estimated using Dres as its proxy, by switching
their roles in Equation 6. (The inter-list association estimate,
overlap at top ranks, is symmetric.)
Autocorrelation [14] is another example of a predictor that
does not account for the presumed relevance of a single reference
list used for prediction. The reference list, Dref,
is obtained by re-ranking Dres using score regularization;
and, Pearson correlation between retrieval scores serves for
ˆp(Dres|Dref,r). In Section 4 we show that the autocorrelation
predictor does not only predict the effectiveness of
Dres, as originally reported, but also the effectiveness of
Dref. This finding provides further support to the prediction
principle presented in Equation 6.
Other predictors that use reference lists without accounting
for their presumed relevance include those using query
perturbations [37], document perturbations [41], and retrieval
method perturbations [2, 14], to induce reference lists.
Explaining additional prediction approaches. The UEF
predictor [31] does estimate the presumed relevance of the
reference lists used. Specifically, each reference list is created
by re-ranking Dres using a relevance language model. The
presumed relevance of the reference list is estimated using
previously proposed predictors. Thus, while UEF is based
on statistical decision theory, it can be directly explained by
Equation 6.
A prediction framework recently proposed [21] relies on
the premise that the result list Dres is relevant to the extent
that it is similar to a pseudo relevant result list and dissimilar
to a pseudo non-relevant result list. The framework was
used to explain several post-retrieval predictors. In contrast
to the framework we present in this work, this framework
does not arise from a probabilistic analysis of the prediction
task. Moreover, it does not account for several formal
aspects of prediction that emerged in the development of
our framework. These include integrating pre-retrieval and
post-retrieval prediction and integrating query dependent
and independent measures of result list properties. Yet, in
Appendix B we show that Equation 1, which served as the
basis for deriving our framework, can be used to provide
formal probabilistic grounds to this framework [21].
4. EVALUATION
We present an empirical exploration of three formal aspects
that emerged in the development of our prediction
framework. These give rise to new prediction methods and
shed new light on existing ones.
We study the merits of the integration of pre-retrieval and
post-retrieval prediction in Section 4.2.1. In Section 4.2.2 we
explore the use of measures of query-independent properties
— specifically, cohesion and dispersion — of the result list
for prediction. Finally, in Section 4.2.3 we focus on using
reference lists for prediction. We begin by describing the
experimental setup we used for evaluation in Section 4.1.
4.1 Experimental setup
Table 1 presents the eight TREC-based experimental settings
used for evaluation. TREC5 and ROBUST are composed
primarily of newswire documents. WT10G is a smallCollection Data Num Docs Topics
TREC5 Disks 2,4 524,929 251-300
ROBUST Disks 4,5-CR 528,155 301-450,
601-700
WT10G WT10g 1,692,096 451-550
GOV2 GOV2 25,205,179 701-850
Clue09 ClueWeb (Category B) 50,220,423 1-50
Clue09+SpamRm ClueWeb (Category B) 50,220,423 1-50
Clue10 ClueWeb (Category B) 50,220,423 51-100
Clue10+SpamRm ClueWeb (Category B) 50,220,423 51-100
Table 1: TREC settings used for experiments.
Web collection that contains some pages of low quality (e.g.,
spam). GOV2 is a much larger Web collection that is a crawl
of the .GOV domain and hence contains mainly well edited
pages. We also use the ClueWeb corpus (category B) [6],
which is a large scale noisy Web collection, with the queries
used in TREC 2009 (Clue09) and TREC 2010 (Clue10).
Retrieval effectiveness for ClueWeb can be significantly
influenced by spam documents [8]. Thus, in what follows we
explore the potential effects of spam on query-performance
prediction. Specifically, we also perform evaluation where
documents“suspected”as spam that are initially highly ranked
are removed from the result list. To that end, we scan the
initial ranking from top to bottom and remove documents
that are assigned by Waterloo’s spam classifier [8] a score
below 50 [8, 3], until 1000 documents are accumulated. The
score reflects the presumed percentage of documents in the
ClueWeb English collection (category A) that are “spammier”
than the document at hand. Clue09+SpamRm and
Clue10+SpamRm denote the resulting (ranked) corpora for
Clue09 and Clue10, respectively.
The titles of TREC topics serve as queries. We applied
Porter stemming and stopword removal, using the INQUERY
list, to documents and queries using the Lemur/Indri toolkit 4 .
To measure prediction quality, we follow the standard
practice in work on query-performance prediction [4]. Specifically,
we report the Pearson correlation between the prediction
values assigned by a predictor to the set of queries per
setting, and the ground-truth average precision (AP@1000)
values for these queries; the ground truth is determined
based on TREC’s relevance judgments. 5
Statistically significant
differences of prediction quality (between Pearson
correlations) are determined at the 95% confidence level [34].
As in many previous reports of work on predicting query
performance [10, 11, 41, 14, 16, 30, 17, 31], we use the query
likelihood (QL) model [32] for the retrieval method. The
goal of the predictors we study is to estimate the retrieval
effectiveness of this standard language-model-based retrieval
approach. The QL retrieval score assigned to document d
in response to query q is ScoreQL(q; d) def
= log Q
qi∈q p(qi|d),
where p(w|d) is the probability assigned to term w by a language
model induced from d; qi is a query term. Unless otherwise
stated, we use Dirichlet-smoothed unigram language
models with the smoothing parameter set to 1000 [38].
Post-retrieval predictors. We use the Clarity [10], query
feedback (QF) [42], and weighted information gain (WIG)
4
www.lemurproject.org
5
Prediction-quality patterns similar to those we report are
observed if Kendall’s-τ is used for evaluation [4]. Actual
numbers are omitted to avoid cluttering the presentation.
[42] predictors in the following. These methods represent the
three classes of post-retrieval prediction approaches mentioned
in Section 1; these include, measuring the clarity of
the result list with respect to the corpus (Clarity), quantifying
the robustness of the result list (QF), and analyzing
properties of retrieval scores in the result list (WIG). WIG
and QF were shown to yield state-of-the-art prediction quality
[42, 40, 31]. In Section 4.2.3 we address the autocorrelation
predictor [14], which uses regularized retrieval scores.
The predictors just mentioned, as other previously proposed
post-retrieval predictors, do not provide direct estimates
for the likelihood of the result list.
(Refer back to
Equation 2 and the accompanying discussion. 6 )Rather,the
assigned prediction values are correlated with this likelihood
as implied by the attained prediction quality. We hasten to
point out, however, that the purpose of using these predictors
in the evaluation to follow is exploring aspects and principles
that emerged in the development of our framework;
specifically, the integration of different types of prediction
approaches, namely, pre-retrieval and post-retrieval methods
and query-independent and query-dependent measures
of properties of the result list; and, the use of reference lists
for prediction. Devising novel predictors that provide direct
estimates for the result list likelihood, and using them in our
framework, is left to future work.
The post-retrieval predictors analyze the result list Dres
of the k documents that are the highest ranked by the QL retrieval
method. The Clarity predictor [10] measures the KL
divergence between a relevance language model RDres induced
from Dres and a language model induced from the corpus.
(See Appendix A for details.) The QF prediction value
[42] is the number of documents that are both among the
νQF highest ranked by the QL method and among the νQF
highest ranked by retrieval performed over the corpus using
the relevance model RDres; νQF
P
is a free parameter. The
1
prediction value of WIG is √
1
(ScoreQL(q; d) −
|q| k d∈Dres
ScoreQL(q; D)), where |q| is the number of terms in q; i.e.,
the (query-length normalized) difference between the average
retrieval score in Dres and that of the corpus 7 .
One of the goals of the exploration we present in Section
4.2 is to study whether formal aspects of prediction that
emerged in the framework described above can be used to
devise prediction methods that outperform state-of-the-art
predictors. Hence, we use highly optimized versions of Clarity,
QF, and WIG as baselines. Specifically, we set the free
parameters of each of these predictors to values maximizing
prediction quality per setting. All three predictors rely on k,
the number of documents in the result list; QF also depends
on νQF . For all settings, except for those for ClueWeb, the
value of k is selected from {5, 10, 50, 100, 150, 200, 300, 500,
700, 1000, 2000, 3000, 4000, 5000}. For ClueWeb we use a
slightly more moderate range of values to alleviate the computational
effort: k ∈{5, 10, 50, 100, 150, 200, 300, 500, 700,
6
There is work on estimating average precision directly from
score distributions [12]. The proposed estimates are still
surrogates for estimates for the result list likelihood.
7
The corpus D is represented by the unsmoothed language
model induced from the concatenation of the documents it
contains; concatenation order has no effect, as we use unigram
language models. While WIG’s original implementation
was with the Markov Random Field model [42], it was
noted [42, 40] and shown [31] that WIG is highly effective for
predicting the effectiveness of the query likelihood model.TREC5 ROBUST WT10G GOV2 Clue09 Clue09+SpamRm Clue10 Clue10+SpamRm
SumIDF 0.053 0.287 0.162 0.281 0.618 0.620 0.202 0.201
AvgIDF 0.208 0.466 0.149 0.266 0.211 0.161 0.175 0.227
MaxIDF 0.269 0.486 0.134 0.294 0.409 0.357 0.233 0.252
SumVarTF.IDF 0.049 0.266 0.269 0.328 0.631 0.637 0.267 0.273
AvgVarTF.IDF 0.203 0.428 0.282 0.360 0.365 0.293 0.247 0.309
MaxVarTF.IDF 0.180 0.443 0.406 0.372 0.590 0.528 0.312 0.341
Clarity 0.431 0.522 0.432 0.456 0.105 −0.016 −0.147 −0.077
SumIDF∧Clarity 0.170 0.392 0.270 0.368 0.687 ∗
0.643 ∗
0.156 ∗
0.225 ∗
AvgIDF∧Clarity 0.310 0.527 0.287 0.367 0.254 0.157 0.093 ∗
0.174 ∗
MaxIDF∧Clarity 0.345 0.545 0.265 0.383 0.457 ∗
0.365 ∗
0.149 ∗
0.222 ∗
SumVarTF.IDF∧Clarity 0.187 0.358 0.358 0.420 0.667 ∗
0.623 ∗
0.207 ∗
0.289 ∗
AvgVarTF.IDF∧Clarity 0.341 0.482 0.401 0.459 0.338 ∗
0.228 ∗
0.139 ∗
0.224 ∗
MaxVarTF.IDF∧Clarity 0.300 0.483 0.450 0.447 0.555 ∗
0.458 ∗
0.206 ∗
0.270 ∗
QF 0.447 0.500 0.483 0.566 0.516 0.616 0.393 0.483
SumIDF∧QF 0.388 0.505 0.487 0.502 0.675 ∗
0.767 ∗
0.496 0.674 ∗
AvgIDF∧QF 0.444 0.602 ∗
0.423 0.518 0.499 0.443 0.520 ∗
0.559 ∗
MaxIDF∧QF 0.489 0.613 ∗
0.482 0.554 0.604 0.595 0.537 ∗
0.610 ∗
SumVarTF.IDF∧QF 0.366 0.496 0.511 0.527 0.689 ∗
0.751 ∗
0.524 ∗
0.684 ∗
AvgVarTF.IDF∧QF 0.409 0.586 ∗
0.432 0.561 0.557 0.488 0.538 ∗
0.579 ∗
MaxVarTF.IDF∧QF 0.390 0.578 ∗
0.496 0.574 0.651 ∗
0.643 0.557 0.603 ∗
WIG 0.297 0.550 0.376 0.486 0.507 0.498 0.415 0.468
SumIDF∧WIG 0.133 0.425 0.314 0.362 0.611 ∗
0.606 ∗
0.233 0.260
AvgIDF∧WIG 0.222 0.540 0.218 0.348 0.246 0.225 0.240 0.302
MaxIDF∧WIG 0.270 0.572 0.261 0.381 0.406 0.385 0.288 0.332
SumVarTF.IDF∧WIG 0.138 0.391 0.404 0.427 0.677 ∗
0.666 ∗
0.277 0.305
AvgVarTF.IDF∧WIG 0.239 0.508 0.334 0.432 0.409 0.372 0.287 0.352
MaxVarTF.IDF∧WIG 0.226 0.524 0.431 0.465 0.587 0.546 0.333 0.372
Table 2: Integrating pre-retrieval and post-retrieval prediction. The best result per setting in a block of a
post-retrieval predictor is boldfaced. The best result in the entire setting (column) is underlined. ’*’ marks
a statistically significant difference with using the post-retrieval predictor alone.
1000}. For all settings, νQF , the additional free parameter of
QF, is set to values in {5, 10, 50, 100, 150, 200, 300, 500, 700,
1000}. Following previous recommendations [31], the number
of terms used by the relevance model, RDres, whichis
utilized by Clarity and QF, is set to 100; and, language
models of documents from which the relevance model is constructed
are not smoothed (i.e., a maximum likelihood estimate
is used).
4.2 Experimental results
4.2.1 Integrating pre- and post- retrieval prediction
Using Equation 2 we showed that pre-retrieval and postretrieval
prediction methods target different, yet complementary,
formal aspects of prediction. Hence, we now turn
to study whether their integration can yield prediction quality
that transcends that of using each alone. To the best of
our knowledge, this is the first such empirical study.
For post-retrieval predictors we use the optimized Clarity,
QF and WIG methods described above. For pre-retrieval
prediction we use two sets of methods, each based on a different
type of statistics computed for a query term. The first
set, referred to as IDF [10, 19, 16], uses the inverse document
frequency (IDF) value of a query term. The second set of
pre-retrieval predictors, denoted VarTF.IDF [39], uses the
variance of the TF.IDF value of a query term in documents
across the corpus in which it appears. Predictors based on
VarTF.IDF were shown to substantially outperform other
pre-retrieval predictors [16]; and, to outperform highly effective
post-retrieval predictors for Clue09 [17].
To aggregate the statistics for the query terms, we use
the sum, average, and maximum operators. Accordingly,
we study the SumIDF, AvgIDF, MaxIDF, SumVarTF.IDF,
AvgVarTF.IDF, and MaxVarTF.IDF predictors. If X is a
pre-retrieval predictor and Y is a post-retrieval predictor,
we use X∧Y to denote their integration, attained by the
multiplication of prediction values following Equation 2.
In what follows we focus on the question of whether the
prediction quality of a highly optimized post-retrieval predictor
can be improved if integrated, as is, with pre-retrieval
predictors. Our experiments show that the prediction quality
of the integration of pre-retrieval and post-retrieval prediction
can be further improved, as expected, by optimizing
the free-parameter values of the post-retrieval predictors to
that end. (The pre-retrieval predictors do not incorporate
free parameters.) Actual numbers are omitted due to space
considerations and as they convey no additional insight.
Our first observation based on Table 2, which is in line
with previous reports [4], is that all post-retrieval predictors,
when used alone, almost always outperform all pre-retrieval
predictors, when used alone, for TREC5, ROBUST, WT10G
and GOV2. For the ClueWeb settings, the pre-retrieval predictors
can in some cases outperform the post-retrieval predictors.
This finding generalizes that presented in a recent
report for (only) the Clue09 setting [17].
Evidently, spam removal for ClueWeb can have considerable
(positive or negative) effect on prediction quality. We
see in Table 2 that in most cases the prediction quality of
the predictors for Clue09+SpamRm is lower than that for
Clue09, while the prediction quality for Clue10+SpamRm is
superior to that for Clue10; the prediction quality differences
can be quite substantial.
Perhaps the most important observation with regard to
Table 2 is the following. In quite a few cases, the integration
of pre-retrieval and post-retrieval prediction can yield
prediction quality that much transcends that of each, evenTREC5 ROBUST WT10G GOV2 Clue09 Clue09+SpamRm Clue10 Clue10+SpamRm
d-dispersion 0.082 0.253 0.158 −0.228 −0.338 −0.082 −0.327 −0.328
d-cohesion −0.084 −0.257 −0.149 0.244 0.153 −0.136 −0.097 −0.104
e-dispersion −0.033 0.139 0.159 −0.100 0.324 0.349 0.391 0.396
e-cohesion 0.028 −0.138 −0.157 0.094 −0.274 −0.323 −0.339 −0.326
Clarity 0.431 0.522 0.432 0.456 0.105 −0.016 −0.147 −0.077
d-dispersion∧Clarity 0.313 0.490 0.418 0.066 −0.229 −0.056 −0.284 −0.248
d-cohesion∧Clarity 0.522 0.460 0.327 0.471 0.130 −0.139 −0.110 −0.118
e-dispersion∧Clarity 0.404 0.522 0.455 0.447 0.353 ∗
0.215 ∗
0.095 ∗
0.155 ∗
e-cohesion∧Clarity 0.454 0.518 0.395 0.446 −0.148 −0.154 −0.301 −0.232
QF 0.447 0.500 0.483 0.566 0.516 0.616 0.393 0.483
d-dispersion∧QF 0.504 0.528 ∗
0.496 0.445 0.465 0.697 0.292 0.373
d-cohesion∧QF 0.368 0.447 0.432 0.544 0.527 0.014 −0.038 −0.021
e-dispersion∧QF 0.448 0.503 0.492 0.565 0.569 ∗
0.679 ∗
0.495 ∗
0.549 ∗
e-cohesion∧QF 0.444 0.496 0.473 0.565 0.450 0.508 0.253 0.367
WIG 0.297 0.550 0.376 0.486 0.507 0.498 0.415 0.468
d-dispersion∧WIG 0.248 0.539 0.359 0.183 0.128 0.293 −0.091 −0.049
d-cohesion∧WIG 0.294 0.481 0.298 0.550 0.430 −0.006 −0.069 −0.072
e-dispersion∧WIG 0.281 0.550 0.394 0.463 0.525 0.514 0.471 0.521
e-cohesion∧WIG 0.313 0.547 0.349 0.500 0.293 0.410 0.104 0.183
Table 3: Integrating query-independent result list measures of cohesion and dispersion with the post-retrieval
predictors that utilize query-dependent list properties; ’*’ marks statistically significant differences between
the integration and using the post-retrieval predictor alone. The best result per setting in a block of a
post-retrieval predictor is boldfaced; the best result in the entire setting (column) is underlined.
in cases where the former is far less effective than the latter;
the improvements over using post-retrieval prediction alone
are often statistically significant for the ClueWeb settings.
For example, QF is the most effective among the three postretrieval
predictors, when these are used alone, for 6 out of
the 8 experimental settings. Yet, the best prediction quality
reported in the QF block in the table (boldfaced numbers),
and more generally, for an entire setting (underlined
numbers), is always attained when integrating QF with a
pre-retrieval predictor. Furthermore, the prediction quality
of the integration in these cases is often much better
than that of QF, although QF’s prediction quality is often
substantially better than that of the pre-retrieval predictor
with which it is integrated. Specifically, MaxIDF∧QF and
SumVarTF.IDF∧QF turn out to be highly effective predictors.
For Clarity and WIG, the best prediction quality is attained
when integrating them with a pre-retrieval predictor
in 7 and 4 out of the 8 experimental settings, respectively.
All in all, these findings attest to the complementary nature
of pre-retrieval and post-retrieval prediction that formally
emerged in Equation 2.
4.2.2 List cohesion versus list dispersion
The generic post-retrieval predictor in Equation 4 uses a
query-independent estimate for the effectiveness (relevance)
of the result list (ˆp(r|Dres)). This estimate plays a complementary
formal role to that addressed by the query-dependent
estimates used in most post-retrieval prediction methods.
Thus, following Equation 4, we turn to study the potential
merits of integrating the measures of query-independent and
query-dependent properties of the result list.
We contrast two hypotheses. The first is that high cohesion
of the result list attests to reduced query drift [36],
and hence, to improved retrieval. The second is that list
dispersion might indicate increased cover of query aspects,
and thereby imply effective retrieval [5]. We employ two
measures that quantify list cohesion and dispersion: the diameter
of the list, adapted in spirit from [5], and the list
entropy, a variant of a measure proposed in [20].
List diameter. We define the result list (Dres) diameteras
dim(Dres) def 1 X
= −
Sim(di,dj);
k(k − 1)
Sim(di,dj) def
“
= −KL p(·|di)
d i,d j ∈Dres;d i̸=d j
”
˛˛ p(·|dj) is the negative KL
divergence between the (unsmoothed) language model induced
from di and the (smoothed 8 ) language model induced
from dj. Increased KL divergence corresponds to a larger
difference between language models (i.e., weaker inter-document
similarity); accordingly, Dres is of greater diameter and consequently
assumed to be less coherent (more dispersed). We
1
use d-cohesion and d-dispersion to refer to
dim(Dres) and
dim(Dres), respectively, that reflect the presumed extent of
cohesion and dispersion, respectively.
List entropy. Let Cent(Dres) be the (arithmetic) centroid
of language models of documents in Dres: p(w|Cent(Dres)) def
P
=
1
p(w|d), where p(w|d) is the probability assigned
k d∈Dres
to term w by an unsmoothed language model induced from
d. We “clip” Cent(Dres) by using the 100 terms w to which
it assigns the highest probability, and re-normalize to yield a
probability distribution. (As noted in Section 4.1, the same
clipping practice was employed to the relevance model used
by the Clarity and QF predictors.)
We define the entropy of the result list Dres as the entropy
of its centroid:
H(Dres) def
= − X
p(w|Cent(Dres)) log p(w|Cent(Dres)).
w
Low entropy amounts to the term distribution in Dres being
focused around a few terms, which potentially implies
increased list cohesion; conversely, high entropy might indicate
increased list dispersion. We use e-cohesion and
e-dispersion to refer to and H(Dres), respectively,
1
H(Dres)
8
The document language model is Jelinek-Mercer smoothed
[38] with a 0.1 weight assigned to corpus-based term counts.which are measures of Dres’s presumed cohesion and dispersion,
respectively.
To integrate the query-independent list cohesion (dispersion)
measures with the optimized query-dependent postretrieval
predictors (Clarity, QF and WIG), we multiply
their assigned values following Equation 4. The cohesion
(dispersion) measures are computed for the k = 100 highest
ranked documents. (Using the top-50 documents yields
similar prediction quality.) Table 3 presents the prediction
quality numbers. (∧ indicates that a list cohesion (dispersion)
measure is integrated with a predictor.)
We can see in Table 3 that in most cases, the queryindependent
result-list cohesion (dispersion) measures yield
very low prediction quality when used alone. The main exception
is the e-dispersion measure, which yields somewhat
higher prediction quality for the ClueWeb settings. This
finding echoes those from previous work [20] about list dispersion
being a potential signal for effective retrieval in the
noisy Web setting.
We also see in Table 3 that for 7 out of the 8 experimental
settings, the best prediction quality (underlined) is
attained by integrating a dispersion measure (d-dispersion
or e-dispersion) with a post-retrieval predictor. For all (optimized)
post-retrieval predictors, Clarity, QF and WIG,
the best prediction quality (boldfaced) is almost always obtained
when integrated with a dispersion measure rather
than when used alone. The improvements over using the
post-retrieval predictor alone can be substantial (e.g., for the
ClueWeb settings) and statistically significant (e.g., for e-
dispersion∧Clarity and e-dispersion∧QF over the ClueWeb
settings). In most cases, integrating a post-retrieval predictor
with the entropy-based dispersion measure (e-dispersion)
outperforms using the post-retrieval predictor alone and integrating
the predictor with the diameter-based dispersion
measure (d-dispersion).
The findings presented above support the merits of the
integration of measures of query-independent and querydependent
result list properties that formally emerged in
Equation 4. Specifically, quantifying list dispersion by measuring
its term-distribution entropy, and integrating this
measure with optimized post-retrieval predictors, yields prediction
quality that can substantially transcend that of using
the predictors alone.
It is important to note that our findings do not contradict
those in previous work [36] with regard to increased list
cohesion being an indicator for effective retrieval. Specifically,
cohesion was shown to be an effective indicator when
measured in a query-dependent manner utilizing documentquery
similarities [36]. In contrast, we measure cohesion
using query-independent estimates.
4.2.3 Using reference lists for prediction
We presented the fundamental approach of using reference
lists for prediction in Equation 6:
Pref(Dres; q) def
= X
ˆp(Dres|Dref,r)ˆp(Dref|q, r).
D ref ∈S ref
The relevance of Dres to q is estimated based on its association
with reference lists Dref (ˆp(Dres|Dref,r)), where each
Dref is weighted by its presumed relevance to q (ˆp(Dref|q, r)).
As discussed in Section 3, this prediction paradigm underlies
several predictors [37, 41, 14, 42, 31]. However, many of
RefList(Uni) RefList(Clarity) Clarity
TREC5 0.414 0.459 0.431
ROBUST 0.520 0.562 c
0.522
WT10G 0.415 0.418 0.432
GOV2 0.377 0.486 u
0.456
Clue09 −0.06 0.019 0.105
Clue09+SpamRm 0.121 0.039 −0.016
Clue10 −0.079 −0.175 −0.147
Clue10+SpamRm 0.047 −0.054 −0.077
Table 4: Using reference lists for prediction. ’c’
and ’u’ mark statistically significant differences with
Clarity and RefList(Uni), respectively. Best result
in a row is boldfaced.
these predictors do not estimate, and consequently utilize,
the presumed relevance of Dref [37, 41, 14, 42].
To further explore the merits, or lack thereof, of the referencelists-based
prediction paradigm, when using both the association
between Dres and Dref and the estimated relevance
of Dref to q, we study the following novel simple predictor,
henceforth referred to as RefList.
In the empirical evaluation presented thus far, we used the
QL retrieval method to induce the corpus ranking used to
create the result list Dres. The document language model
smoothing parameter value was set to 1000.
(See Section
4.1.) To create reference lists, Dref, were-rank Dres using
the QL method with the smoothing parameter set to values
in {100, 200, 500, 800, 1500, 2000, 3000, 4000, 5000, 10000}.
Thus, we get that Sref contains 10 reference lists that are
created by varying the document (language) models.
We set the estimate for the association strength of Dres
and Dref (ˆp(Dres|Dref,r)) to Pearson’s correlation between
the retrieval scores of the documents. The estimate for the
presumed relevance of Dref to q (ˆp(Dref|q, r)) is set to either
the Clarity value of Dref (RefList(Clarity)) orto 1
10
(RefList(Uni)). We employ the predictors upon a result
list, Dres, ofk = 150 documents. The relevance model used
by Clarity is constructed as described in Section 4.1. As a
reference comparison, we use the optimized Clarity predictor
used above and for which the value of k was tuned. Thus,
the RefList predictors are underoptimized with respect to
the Clarity baseline. We hasten to point out that RefList
can also be optimized with respect to k and predictors other
than Clarity can be used by RefList to estimate the relevance
of Dref. Nevertheless, our goal here is to focus on
the underlying principles of using reference lists for prediction
rather than devise the most effective predictor. The
prediction quality numbers are presented in Table 4.
We see in Table 4 that RefList(Clarity) improves — often
substantially, although statistically significantly in a single
case — over RefList(Uni) for all non-ClueWeb settings
(TREC5, ROBUST, WT10G, and GOV2). This finding provides
some support to the importance of using an estimate
for the presumed relevance of the reference list. We also see
that RefList(Clarity) substantially outperforms Clarity for
3 out of these 4 settings, and statistically significantly so for
ROBUST, although RefList(Clarity) is underoptimized with
respect to Clarity. These results support the merits of using
reference lists for prediction, even if the reference lists are
created using a very simple approach as we employ here.
Table 4 shows that for the ClueWeb settings, the RefList
predictors and the optimized Clarity predictor yield very low
prediction quality. This could be attributed to the noisy na-QF
autocorrelation
TREC5 ROBUST WT10G GOV2 Clue09 Clue09+SpamRm Clue10 Clue10+SpamRm
Dres 0.447 0.500 0.483 0.560 0.516 0.616 0.393 0.483
Dref 0.467 0.594 0.482 0.643 0.605 0.543 0.445 0.574
Dres 0.277 0.347 0.328 0.417 0.235 0.505 −0.021 0.213
Dref 0.287 0.427 0.399 0.442 0.105 0.538 −0.080 0.064
Table 5: The prediction quality of QF and autocorrelation. Prediction quality is measured by the success in
predicting the effectiveness of the result list at hand (Dres, as originally proposed) and that of the reference
list used (Dref). The best result in a column within a block of a predictor is boldfaced.
ture of the corpus-based term counts (e.g., due to spam) that
affects Clarity computation and the smoothing of document
language models which is very important in the RefList predictors.
(See [20] for an additional explanation about Clarity’s
low prediction quality for ClueWeb.) We found that
using Clarity (alone) with “term clipping” [18] yields a somewhat
improved prediction for Clue09 and Clue09+SpamRm
but not for Clue10 and Clue10+SpamRm.
On the symmetry in using reference lists. As noted in
Section 3.1.3, the QF [42] and autocorrelation [14] predictors
use a single reference list. However, both predictors do
not use an estimate for the presumed relevance of the reference
list. Hence, these predictors are based solely on the
association between Dres and Dref (ˆp(Dres|Dref,r)). Thus,
if we swap the roles of Dres and Dref in Equation 6, and
given that the inter-list association measure for both predictors
is symmetric, we should get that the predictors do not
only estimate the effectiveness of Dres, but also that of Dref.
This hypothesis was shown to hold for QF in some previous
work [31], but the ClueWeb settings were not used. Hence,
we (re-)examine the hypothesis for QF; and, we present a
novel study of the hypothesis for autocorrelation.
We use the optimized QF predictor described in Section
4.1. The optimization was with respect to the prediction
quality for the effectiveness of Dres, which is computed (as
isthecaseforDref) as at the above; that is, by Pearson
correlation with the true AP@1000. We also use an optimized
autocorrelation predictor. Specifically, the number of
nearest neighbors considered when constructing the graph
[14] is selected from {5, 10, 50} (a document is included in
the set of its own nearest neighbors); and, k, thenumberof
documents in Dres (and Dref) is selected from {50, 100} to
optimize prediction quality for Dres with respect to AP@k. 9
Accordingly, we report the prediction quality of autocorrelation
for both Dres and Dref with respect to AP@k. We
do not use AP@1000 since autocorrelation is based on reranking
Dres rather than ranking the entire corpus.
We see in Table 5 that in most settings QF and autocorrelation
post high prediction quality for the effectiveness of
the reference list (Dref); often, the prediction quality transcends
that for the original result list (Dres). Both QF [42]
and autocorrelation [14] were originally stated as targeting
the effectiveness of Dres and they were optimized to this end
as described above. Hence, these findings — which are novel
for autocorrelation — support the hypothesis stated above
with regard to predictors using a single reference list without
utilizing an estimate for its presumed relevance. That
is, prediction is essentially performed for the effectiveness of
9
We used a language-model-based inter-document similarity
measure rather than a vector-space-based measure as in the
original proposal [14].
both the result list and the reference list. These findings
further support the underpinning of the reference-list-based
prediction approach from Equation 6.
5. CONCLUSIONS AND FUTURE WORK
We presented a novel probabilistic query-performance prediction
framework. The importance of the framework is
threefold. First, setting common formal grounds to various
previously proposed prediction methods that might seem to
rely on completely different principles and hypotheses. Second,
providing new insights about commonly used prediction
methods such as Clarity and the connections between
them. Third, giving rise, based on formal arguments, to new
prediction approaches that were empirically shown to improve
over the state-of-the-art. Integrating various prediction
types that emerged in our framework using additional
approaches — e.g., machine-learning-based techniques — is
a future venue we intend to explore.
Acknowledgments We thank the anonymous reviewers,
and Yuval Nardi, for their comments. This paper is based
upon work supported in part by the Israel Science Foundation
under grant no. 557/09, by IBM’s Ph.D. fellowship
and SUR award, by Google’s and Yahoo!’s faculty research
awards, and by Miriam and Aaron Gutwirth Memorial Fellowship.
Any opinions, findings and conclusions or recommendations
expressed here are the authors’ and do not necessarily
reflect those of the sponsors.
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APPENDIX
A. CLARITY
Clarity is defined as the KL divergence between a relevance
language model RDres constructed from Dres and a
language model p(·|D) induced from the corpus [10]:
PClarity(Dres; q) def
= X
p(w|RDres)log
w
p(w|RDres)
; (7)
p(w|D)
w is a term in the vocabulary. RDres is a linear mixture of
language models of documents in Dres [24]:
p(w|RDres) def
= X
p(w|di)p(di|q, r); (8)
d i∈Dres
p(w|di) is the probability assigned to w by a language model
induced from di; and, p(di|q, r) def p(q|di,r) =
is di’s
P
d ′ ∈Dres p(q|d′ ,r)
normalized query-likelihood when using uniform distributions
for p(r|d) andp(d), where d ∈Dres. 10
Using RDres’s definition from Equation 8 in Equation 7,
and applying arithmetic manipulation, we get that:
PClarity(Dres; q) =
X
d i∈Dres
“
p(di|q, r)
X
p(w|di)logp(w|RDres) − X
”
p(w|di)logp(w|D) .
w
w
We define the similarity between di and X (Dres or D) asthe
minus cross entropy between their induced language models:
Sim(di,X) def
= P
def
w
p(w|di)logp(w|X), with p(w|Dres) =
p(w|RDres). Thus, Clarity is:
PClarity(Dres; q) = (9)
X
p(di|q, r) ` Sim(di, Dres) − Sim(di, D) ´ .
d i∈Dres
B. EXPLAINING A PREVIOUSLY PROPOSED
PREDICTION FRAMEWORK
Applying log odds, which is a monotonic transformation,
upon Equation 1 yields:
log O(r|q, Dres) def p(r|q, Dres)
= log = (10)
p(¯r|q, Dres)
log p(Dres|r, q)p(r|q) − log p(Dres|¯r, q)p(¯r|q);
¯r is the non-relevance event.
Suppose we estimate p(Dres|r, q)andp(Dres|¯r, q) based on
Dres’s similarity with a (pseudo) relevant and non-relevant
document lists, respectively; and, use a uniform relevance
prior, p(r|q) =p(¯r|q) = 1
. Then, we obtain the prediction
2
principle underlying the framework proposed in [21].
10
While the common notation used in the language modeling
framework does not include an explicit relevance event
indicator, here we follow Lafferty and Zhai [22] and use it.