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Lecture 5: Introduction to (Robertson/Sp¨arck Jones)
Probabilistic Retrieval
Scribes: Ellis Weng, Andrew Owens
February 11, 2010
1 Introduction
In this lecture, we will introduce our second paradigm for document retrieval: probabilistic retrieval. We
will focus on Roberston and Sp¨arck Jones’ 1976 version, presented in the paperRelevance Weighting of Search
Terms1.
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https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
We
will focus on Roberston and Sp¨arck Jones’ 1976 version, presented in the paperRelevance Weighting of Search
Terms1. This was an influential paper that was published when the Vector Space Model was first being
developed — it is important to keep in mind the differences and similarities between these two models and
the motivations for each. Recall that the Vector Space Model was originally a representation model.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
This was an influential paper that was published when the Vector Space Model was first being
developed — it is important to keep in mind the differences and similarities between these two models and
the motivations for each. Recall that the Vector Space Model was originally a representation model. The retrieval scheme of the
Vector Space Model was empirically-driven and chosen in a fairly atheoretical manner.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Recall that the Vector Space Model was originally a representation model. The retrieval scheme of the
Vector Space Model was empirically-driven and chosen in a fairly atheoretical manner. In contrast, proba-
bilistic retrieval is more principled and theoretically-driven. On the other hand, many statistical estimations
and empirical substitutions will drive the derivation of this paradigm.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Recall that the Vector Space Model was originally a representation model. The retrieval scheme of the
Vector Space Model was empirically-driven and chosen in a fairly atheoretical manner. In contrast, proba-
bilistic retrieval is more principled and theoretically-driven. On the other hand, many statistical estimations
and empirical substitutions will drive the derivation of this paradigm. 2 Notation
2.1 Problem
First, let us assume a binary label set, L, where the documents are either relevant, r, or not relevant, r. (The
binary distinction is not so important at this time but simplifies presentation.)
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
On the other hand, many statistical estimations
and empirical substitutions will drive the derivation of this paradigm. 2 Notation
2.1 Problem
First, let us assume a binary label set, L, where the documents are either relevant, r, or not relevant, r. (The
binary distinction is not so important at this time but simplifies presentation.) L = {r,r}(relevant/irrelevant) (1)
The main goal of probabilistic retrieval is to rank documents by the probability that they are relevant.
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https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
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L = {r,r}(relevant/irrelevant) (1)
The main goal of probabilistic retrieval is to rank documents by the probability that they are relevant. Intuitively, this can be represented in the following way:
Pr(r|d, q), (2)
where d is the document and q is the query.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
L = {r,r}(relevant/irrelevant) (1)
The main goal of probabilistic retrieval is to rank documents by the probability that they are relevant. Intuitively, this can be represented in the following way:
Pr(r|d, q), (2)
where d is the document and q is the query. It is important to note that the score of a document ought to
be a probability between 0 and 1 (inclusive), not a simple binary relevance score of 0 or 1, for this to be a
meaningful scoring function.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
It is important to note that the score of a document ought to
be a probability between 0 and 1 (inclusive), not a simple binary relevance score of 0 or 1, for this to be a
meaningful scoring function. 2.2 Uncertainty
Question: Why is this not 0 or 1? Shouldn’t a document just be either relevant or irrelevant? Answer: There is
uncertainty associated with probabilistic retrieval.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
2.2 Uncertainty
Question: Why is this not 0 or 1? Shouldn’t a document just be either relevant or irrelevant? Answer: There is
uncertainty associated with probabilistic retrieval. Uncertainty can arise from any of the following:
1. Uncertainty with respect to a particular user. A user’s judgment for a document relevancy might change from time to time depending on context.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Shouldn’t a document just be either relevant or irrelevant? Answer: There is
uncertainty associated with probabilistic retrieval. Uncertainty can arise from any of the following:
1. Uncertainty with respect to a particular user. A user’s judgment for a document relevancy might change from time to time depending on context. The resulting sample space for this uncertainty is
1
Q ×D ×L ×F, (3)
where Q is the set of all possible queries (or information need), D is the set of all possible documents,
L is the label set, and F is the set of all other factors that might change your threshold.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
The resulting sample space for this uncertainty is
1
Q ×D ×L ×F, (3)
where Q is the set of all possible queries (or information need), D is the set of all possible documents,
L is the label set, and F is the set of all other factors that might change your threshold. 2. Uncertainty with respect to different users.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
The resulting sample space for this uncertainty is
1
Q ×D ×L ×F, (3)
where Q is the set of all possible queries (or information need), D is the set of all possible documents,
L is the label set, and F is the set of all other factors that might change your threshold. 2. Uncertainty with respect to different users. For a single document-query pair, there is variation in determining if the document is relevant or not
among different users.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
2. Uncertainty with respect to different users. For a single document-query pair, there is variation in determining if the document is relevant or not
among different users. Thus, we have to take different users into account. The underlying sample
space for this uncertainty is
Q ×D ×L ×U, (4)
where U is the set of all possible users. 3. Uncertainty with respect to “lossy” input representation.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Thus, we have to take different users into account. The underlying sample
space for this uncertainty is
Q ×D ×L ×U, (4)
where U is the set of all possible users. 3. Uncertainty with respect to “lossy” input representation. Depending on the representation, a document can either be relevant or not. It is impossible to take
into account all the features of a document.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
The underlying sample
space for this uncertainty is
Q ×D ×L ×U, (4)
where U is the set of all possible users. 3. Uncertainty with respect to “lossy” input representation. Depending on the representation, a document can either be relevant or not. It is impossible to take
into account all the features of a document. Thus, several document-query pairs yield the same rep-
resentation; however, these different documents do not necessarily have to be both relevant or not
relevant.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
3. Uncertainty with respect to “lossy” input representation. Depending on the representation, a document can either be relevant or not. It is impossible to take
into account all the features of a document. Thus, several document-query pairs yield the same rep-
resentation; however, these different documents do not necessarily have to be both relevant or not
relevant. The resulting sample space for this uncertainty is still
Q ×D ×L, (5)
but the “observables” induced from this are different.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Thus, several document-query pairs yield the same rep-
resentation; however, these different documents do not necessarily have to be both relevant or not
relevant. The resulting sample space for this uncertainty is still
Q ×D ×L, (5)
but the “observables” induced from this are different. 4. Uncertainty with respect to to system uncertainty. The retrieval system itself may be a source of uncertainty.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
The resulting sample space for this uncertainty is still
Q ×D ×L, (5)
but the “observables” induced from this are different. 4. Uncertainty with respect to to system uncertainty. The retrieval system itself may be a source of uncertainty. For example, the system might trade off
accuracy for speed by using an approximation algorithm or run on a network that is prone to errors. Question: Why is the type of uncertainty important?
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
4. Uncertainty with respect to to system uncertainty. The retrieval system itself may be a source of uncertainty. For example, the system might trade off
accuracy for speed by using an approximation algorithm or run on a network that is prone to errors. Question: Why is the type of uncertainty important? Answer: The type of uncertainty presumably affects the estima-
tion and derivation of the probabilistic model.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
The retrieval system itself may be a source of uncertainty. For example, the system might trade off
accuracy for speed by using an approximation algorithm or run on a network that is prone to errors. Question: Why is the type of uncertainty important? Answer: The type of uncertainty presumably affects the estima-
tion and derivation of the probabilistic model. The ultimate goal of document retrieval is to find documents that are relevant for a different users.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Question: Why is the type of uncertainty important? Answer: The type of uncertainty presumably affects the estima-
tion and derivation of the probabilistic model. The ultimate goal of document retrieval is to find documents that are relevant for a different users. One can imagine document retrieval systems that were tailored to a specific user’s needs and moods (thus
eliminating type-a uncertainty) or document retrieval systems that captured all the important features of
the documents and queries (thus eliminating type-c uncertainty); however, document retrieval systems are
typically used by more than one user.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
One can imagine document retrieval systems that were tailored to a specific user’s needs and moods (thus
eliminating type-a uncertainty) or document retrieval systems that captured all the important features of
the documents and queries (thus eliminating type-c uncertainty); however, document retrieval systems are
typically used by more than one user. Solving the problem with respect to b-type uncertainty thus seems to
be the goal.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Solving the problem with respect to b-type uncertainty thus seems to
be the goal. Unfortunately, this problem has the worst sample space and seemingly requires extensive user
annotation by many users. Because of these problems, our derivation will be based on representational
uncertainty (type-c uncertainty). (Note: Throughout the derivation, it is worthwhile to ask how this model can apply to these other types
of uncertainty.)
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Unfortunately, this problem has the worst sample space and seemingly requires extensive user
annotation by many users. Because of these problems, our derivation will be based on representational
uncertainty (type-c uncertainty). (Note: Throughout the derivation, it is worthwhile to ask how this model can apply to these other types
of uncertainty.) 2.3 Representation
Assume we have a feature-vector representation (which is not implied by any principles we have men-
tioned so far).
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
(Note: Throughout the derivation, it is worthwhile to ask how this model can apply to these other types
of uncertainty.) 2.3 Representation
Assume we have a feature-vector representation (which is not implied by any principles we have men-
tioned so far). A document can be represented as a set of values of m feature functions. A feature function
fj : D →R describes the important features of the document.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
2.3 Representation
Assume we have a feature-vector representation (which is not implied by any principles we have men-
tioned so far). A document can be represented as a set of values of m feature functions. A feature function
fj : D →R describes the important features of the document. These functions are based on attribute fre-
quency, so 0 must indicate the absence of a particular feature.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
A document can be represented as a set of values of m feature functions. A feature function
fj : D →R describes the important features of the document. These functions are based on attribute fre-
quency, so 0 must indicate the absence of a particular feature. These feature functions are analogous to the
term counts in the VSM, but unlike term counts, these feature functions can, in principle, take more into
account than simply the number of times words appear.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
These functions are based on attribute fre-
quency, so 0 must indicate the absence of a particular feature. These feature functions are analogous to the
term counts in the VSM, but unlike term counts, these feature functions can, in principle, take more into
account than simply the number of times words appear. For example:
f14(d) =the number of times “juice” appears in the document (6)
2
f16(d) =
{
1 if d contains “Cornell” and “best”
0 otherwise (7)
⃗fd =
f1(d)
...
f2(d)
...
fm(d)
(8)
Similarly, query can be represented as a document:
⃗ q=
f1(q)
...
f2(q)
...
fm(q)
(9)
Note that we will sometimes use the notation fd[j] for fj(d) for readability.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Let ⃗F = (F[1], F[2], ..., F[m])T be a feature vector.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Let ⃗F = (F[1], F[2], ..., F[m])T be a feature vector. We can write ⃗F = ⃗fd to mean that a feature vector
matches the description for document d. Under this notation, we can rewrite our probabilistic scoring
function for a given query and a given document d as
Pr(r|⃗F = ⃗fd, ⃗ q) (10)
2.4 Comparison of Scoring Functions
Probabilistic Retrieval Vector Space Model
Pr(r|⃗F = ⃗fd, ⃗ q) ⃗d ·⃗ q
Goal-driven.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
The goal of a probabilistic retrieval
model is clearly to retrieve the documents with
the highest probability of relevance to the given
query. More Arbitrary. The inner product makes intu-
itive sense, but we have no further justification for
this besides empirical evidence. Lack of Data. There seemingly needs to be
data on what documents are relevant (to which
queries) in order to compute these probabilities.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
More Arbitrary. The inner product makes intu-
itive sense, but we have no further justification for
this besides empirical evidence. Lack of Data. There seemingly needs to be
data on what documents are relevant (to which
queries) in order to compute these probabilities. There can potentially be no data in regards to rel-
evance. Computable.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
The inner product makes intu-
itive sense, but we have no further justification for
this besides empirical evidence. Lack of Data. There seemingly needs to be
data on what documents are relevant (to which
queries) in order to compute these probabilities. There can potentially be no data in regards to rel-
evance. Computable. No additional information is
needed in order to compute these values, assum-
ing a typical representation scheme.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Lack of Data. There seemingly needs to be
data on what documents are relevant (to which
queries) in order to compute these probabilities. There can potentially be no data in regards to rel-
evance. Computable. No additional information is
needed in order to compute these values, assum-
ing a typical representation scheme. 3 Computation
Assume the query ⃗ qis fixed.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
There can potentially be no data in regards to rel-
evance. Computable. No additional information is
needed in order to compute these values, assum-
ing a typical representation scheme. 3 Computation
Assume the query ⃗ qis fixed. We want to compute Pr(r|⃗F = ⃗fd, ⃗ q) for each document. There are many
challenges to this computation.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
No additional information is
needed in order to compute these values, assum-
ing a typical representation scheme. 3 Computation
Assume the query ⃗ qis fixed. We want to compute Pr(r|⃗F = ⃗fd, ⃗ q) for each document. There are many
challenges to this computation. One problem is that relevance is unlabeled in our dataset, so we do not
know which documents are relevant for a given query.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
We want to compute Pr(r|⃗F = ⃗fd, ⃗ q) for each document. There are many
challenges to this computation. One problem is that relevance is unlabeled in our dataset, so we do not
know which documents are relevant for a given query. Another problem is the sparse data for ⃗F = ⃗fd.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
There are many
challenges to this computation. One problem is that relevance is unlabeled in our dataset, so we do not
know which documents are relevant for a given query. Another problem is the sparse data for ⃗F = ⃗fd. There may be very few documents d and d′ such that ⃗fd = ⃗fd′ .
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
There are many
challenges to this computation. One problem is that relevance is unlabeled in our dataset, so we do not
know which documents are relevant for a given query. Another problem is the sparse data for ⃗F = ⃗fd. There may be very few documents d and d′ such that ⃗fd = ⃗fd′ . To deal with the first problem, we will use the “strategy of wishful thinking,” a surprisingly powerful
principle.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Another problem is the sparse data for ⃗F = ⃗fd. There may be very few documents d and d′ such that ⃗fd = ⃗fd′ . To deal with the first problem, we will use the “strategy of wishful thinking,” a surprisingly powerful
principle. We will condition on r, as though we had the full labeled dataset! First, we present some probability background material.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
To deal with the first problem, we will use the “strategy of wishful thinking,” a surprisingly powerful
principle. We will condition on r, as though we had the full labeled dataset! First, we present some probability background material. If we have random variables a and b, we can
apply the Bayes Flip
3
Pr(a|b) =Pr(a, b)
Pr(b) = Pr(b|a) Pr(a)
Pr(b) (11)
Terminology: Pr(a, b) is the joint probability of a and b, and Pr(b) is the marginal probabilityof b.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Applying the Bayes flip, we get
Pr( ⃗F = ⃗fd|r, ⃗ q) Pr(r|⃗ q)
Pr( ⃗Fd = ⃗fd|⃗ q)
(12)
Thankfully, we can simplify this equation.Pr(r|⃗ q) is the probability that a randomly-chosen document is
relevant to the user, given his or her query.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Since this value is independent of ⃗fd, the equation is equivalent
under ranking to
Pr( ⃗F = ⃗fd|r, ⃗ q)
Pr( ⃗F = ⃗fd|⃗ q)
(13)
Note that ⃗F = ⃗fd is independent of the the query for systems where the document’s representation is
not affected by the user’s query.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
So, we can rewrite the above as
Pr( ⃗F = ⃗fd|r, ⃗ q)
Pr( ⃗F = ⃗fd)
(14)
Now consider the denominator Pr( ⃗F = ⃗fd). This is the probability that a randomly-chosen document is
represented as fd in our retrieval system, given the query.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
This is the probability that a randomly-chosen document is
represented as fd in our retrieval system, given the query. We could estimate this probability by counting
the fraction of our documents in our database that are represented as fd. Unfortunately, our data is sparse,
so almost always Pr( ⃗F = ⃗fd) = 1/n (where n is the number of documents) because it is unlikely two
documents have the same representation.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
We could estimate this probability by counting
the fraction of our documents in our database that are represented as fd. Unfortunately, our data is sparse,
so almost always Pr( ⃗F = ⃗fd) = 1/n (where n is the number of documents) because it is unlikely two
documents have the same representation. To deal with this issue, we assume (as in Cooper 2) that there exists a constant k >0 such that Pr( ⃗F =
⃗f) = k ∏
j Pr( ⃗F[j] = ⃗fd[j]).
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
To deal with this issue, we assume (as in Cooper 2) that there exists a constant k >0 such that Pr( ⃗F =
⃗f) = k ∏
j Pr( ⃗F[j] = ⃗fd[j]). If k = 1, then we are assuming that each Pr( ⃗F[j] = ⃗fd[j]) is an independent
event.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
If k = 1, then we are assuming that each Pr( ⃗F[j] = ⃗fd[j]) is an independent
event. We run into data sparsity issues in the numerator as well, so we make a similar assumption.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
If k = 1, then we are assuming that each Pr( ⃗F[j] = ⃗fd[j]) is an independent
event. We run into data sparsity issues in the numerator as well, so we make a similar assumption. We assume
there is a k′ > 0 such that Pr( ⃗F = ⃗fd|r, ⃗ q) =k′ ∏
j Pr( ⃗F[j] = ⃗fd[j]|r, ⃗ q).
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
We assume
there is a k′ > 0 such that Pr( ⃗F = ⃗fd|r, ⃗ q) =k′ ∏
j Pr( ⃗F[j] = ⃗fd[j]|r, ⃗ q). If k′ = 1then this is a Naive Bayes
assumption.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
We assume
there is a k′ > 0 such that Pr( ⃗F = ⃗fd|r, ⃗ q) =k′ ∏
j Pr( ⃗F[j] = ⃗fd[j]|r, ⃗ q). If k′ = 1then this is a Naive Bayes
assumption. (Aside: we can avoid making one of the assumptions if we start with a slightly more complex
scoring function.)
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
If k′ = 1then this is a Naive Bayes
assumption. (Aside: we can avoid making one of the assumptions if we start with a slightly more complex
scoring function.) Under these assumptions, we can rewrite the above as
∏
j
Pr( ⃗F[j] = ⃗fd[j]|r, ⃗ q)
Pr( ⃗F[j] = ⃗fd[j])
k′
k (15)
Note that we can remove the constant k′
k and get an equation that is ranking-equivalent to this one.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
At
this point, we still have no information on the relevance of the documents. Now we try to simplify the
equation further by looking at the query, since the terms in the query are the only information we have
regarding relevance. We distinguish between terms that appear in the query and those that don’t.
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https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
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At
this point, we still have no information on the relevance of the documents. Now we try to simplify the
equation further by looking at the query, since the terms in the query are the only information we have
regarding relevance. We distinguish between terms that appear in the query and those that don’t. ∏
j:q[j]̸=0
Pr( ⃗F[j] = ⃗fd[j]|r, ⃗ q)
Pr( ⃗F[j] = ⃗fd[j])
∏
j:q[j]=0
Pr( ⃗F[j] = ⃗fd[j]|r, ⃗ q)
Pr( ⃗F[j] = ⃗fd[j])
(16)
Let us compare the numerator and denominator of the second product.
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https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
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The numerator represents the
probability that a relevant document has a certain feature value for a feature that is not in the query, while
the denominator represents the probability thatany document has a certain value for a feature that is not in
the query. One can argue that these two are similar because words that are left out of a query are usually not
4
intentionally left out by a user.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
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One can argue that these two are similar because words that are left out of a query are usually not
4
intentionally left out by a user. For example, when a user searches for the word ”computer”, the user is not
intentionally leaving out the words “PC”, “Mac”, “Desktop”, etc.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
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One can argue that these two are similar because words that are left out of a query are usually not
4
intentionally left out by a user. For example, when a user searches for the word ”computer”, the user is not
intentionally leaving out the words “PC”, “Mac”, “Desktop”, etc. It is impossible to list all the terms that
are of interest to a user, so the words that are left out do not necessarily indicate that the user thinks they are
irrelevant.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
It is impossible to list all the terms that
are of interest to a user, so the words that are left out do not necessarily indicate that the user thinks they are
irrelevant. So, we don’t know whether these terms occur more in relevant or in general documents. Also
in large corpora, factoring in the features that are not in a query is not practical, because the vocabulary or
set of features of all the documents is massive.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
So, we don’t know whether these terms occur more in relevant or in general documents. Also
in large corpora, factoring in the features that are not in a query is not practical, because the vocabulary or
set of features of all the documents is massive. Thus, one can make an assumption that the words that are
not present in the query are distributed across relevant documents the same as they are distributed across
the whole corpus, since we lack any other information to the contrary.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Thus, one can make an assumption that the words that are
not present in the query are distributed across relevant documents the same as they are distributed across
the whole corpus, since we lack any other information to the contrary. This yields:
∏
j:q[j]̸=0
Pr( ⃗F[j] = ⃗fd[j]|r, ⃗ q)
Pr( ⃗F[j] = ⃗fd[j])
(17)
Although this equation does not seem better than the original probablistic model, there are many terms
that were eliminated using assumptions.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
In the following lectures, we will see the benefits of having the
equation in this form. 4 Questions
Recall that the derivation of the probabilistic model required a number of assumptions and independence
arguments. This exercise will help you understand the meaning of the assumptions and how each of these
assumptions affects the relevance scores of the documents. Suppose we have a set of 10 documents and 5 queries.
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https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
4 Questions
Recall that the derivation of the probabilistic model required a number of assumptions and independence
arguments. This exercise will help you understand the meaning of the assumptions and how each of these
assumptions affects the relevance scores of the documents. Suppose we have a set of 10 documents and 5 queries. Each document can have multiple representations
as a vector of feature functions.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
This exercise will help you understand the meaning of the assumptions and how each of these
assumptions affects the relevance scores of the documents. Suppose we have a set of 10 documents and 5 queries. Each document can have multiple representations
as a vector of feature functions. Assume that the feature functions are binary word counts (the first feature
function is whether the word ”car” is in the document).
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Suppose we have a set of 10 documents and 5 queries. Each document can have multiple representations
as a vector of feature functions. Assume that the feature functions are binary word counts (the first feature
function is whether the word ”car” is in the document). Therefore, each document can be represented as
a vector of 30 feature functions (there are 30 words in this corpus, so there will be one feature function for
each word).
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Assume that the feature functions are binary word counts (the first feature
function is whether the word ”car” is in the document). Therefore, each document can be represented as
a vector of 30 feature functions (there are 30 words in this corpus, so there will be one feature function for
each word). The queries can also be represented as documents (also a vector of 30 feature functions).
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Therefore, each document can be represented as
a vector of 30 feature functions (there are 30 words in this corpus, so there will be one feature function for
each word). The queries can also be represented as documents (also a vector of 30 feature functions). The
following table shows the documents (rows), queries (columns), and whether a document is relevant with
respect to the query (1 or 0).
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
The queries can also be represented as documents (also a vector of 30 feature functions). The
following table shows the documents (rows), queries (columns), and whether a document is relevant with
respect to the query (1 or 0). In practice, we would not have these labels, so it would not be possible to
compute these probabilities. In future lectures we will discuss ways to deal with this problem.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
The
following table shows the documents (rows), queries (columns), and whether a document is relevant with
respect to the query (1 or 0). In practice, we would not have these labels, so it would not be possible to
compute these probabilities. In future lectures we will discuss ways to deal with this problem. car toyota park green
car low
milage
toyota brand
car
this red car is fast and new toyota is a good brand of
car
1 1 0 0 1
this automobile is red and the brand is toyota it is not
very fast but it is cheap
1 1 0 0 1
the green car is fast and the mileage is low 1 0 0 1 0
the car with low mileage is under a tree 1 0 0 0 0
the green tree is in ithe park 0 0 1 0 0
the automobile is green and this car is cheap 1 0 0 0 0
park the car under the tree 1 0 1 0 0
the toyota has low mileage and is very cheap 1 1 0 0 0
the car is not a good brand 1 0 0 0 0
there is a automobile that is red and in the park 1 0 1 0 0
1.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Using the provided table, estimate the relevance of the first document in the first row with the query
“toyota brand car”, using the final equation presented in this lecture:
∏
j:q[j]̸=0
Pr( ⃗F[j] = ⃗fd[j]|r, ⃗ q)
Pr( ⃗F[j] = ⃗fd[j])
5
2.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Estimate the relevance of the first document using the equation without the assumption that the only
significant features are those in the query (i.e. use equation (16) instead of equation (17)).
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Estimate the relevance of the first document using the equation without the assumption that the only
significant features are those in the query (i.e. use equation (16) instead of equation (17)). ∏
j:q[j]̸=0
Pr( ⃗F[j] = ⃗fd[j]|r, ⃗ q)
Pr( ⃗F[j] = ⃗fd[j])
∏
j:q[j]=0
Pr( ⃗F[j] = ⃗fd[j]|r, ⃗ q)
Pr( ⃗F[j] = ⃗fd[j])
Feel free to use a computer program.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
3. Compare and contrast the following two tables. An entry is a relevance score for a given (document,
query) pair. In the first table, the only features considered were those in the query (as in the first
exercise), while the second table uses all of the features (as in the second exercise).
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
3. Compare and contrast the following two tables. An entry is a relevance score for a given (document,
query) pair. In the first table, the only features considered were those in the query (as in the first
exercise), while the second table uses all of the features (as in the second exercise). car toyota park green
car low
milage
toyota brand
car
this red car is fast and new toyota is a good brand of
car
1.11 3.33 0 0 Answer 1
this automobile is red and the brand is toyota it is not
very fast but it is cheap
0.83 3.33 0 0 13.89
the green car is fast and the mileage is low 1.11 0 0 61.73 0
the car with low mileage is under a tree 1.11 0 0 0 0
the green tree is in ithe park 0.83 0 3.33 0 0
the automobile is green and this car is cheap 1.11 0 0 0 0
park the car under the tree 1.11 0 3.33 0 0
the toyota has low mileage and is very cheap 0.83 3.33 0 0 0
the car is not a good brand 1.11 0 0 0 0
there is a automobile that is red and in the park 0.83 0 3.33 0 0
car toyota park green
car low
milage
toyota brand
car
this red car is fast and new toyota is a good brand of
car
3.74 468.17 0 0 Answer 2
this automobile is red and the brand is toyota it is not
very fast but it is cheap
3.18 40782.92 0 0 353160.66
the green car is fast and the mileage is low 1.08 0 0 136672.91 0
the car with low mileage is under a tree 0.85 0 0 0 0
the green tree is in ithe park 0.07 0 2031.53 0 0
the automobile is green and this car is cheap 0.92 0 0 0 0
park the car under the tree 0.35 0 652.99 0 0
the toyota has low mileage and is very cheap 1.45 26.01 0 0 0
the car is not a good brand 1.60 0 0 0 0
there is a automobile that is red and in the park 0.42 0 2571.15 0 0
4.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Why do the tables contain values greater than 1? 5. One problem with the approach we have described is that every time a user performs a query, we
must check whether each document is relevant. This approach does not scale well. It would be nice if
we could process a small subset of the documents instead. We assumed that the features in the query
are the only features that influence the probability calculation.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
5. One problem with the approach we have described is that every time a user performs a query, we
must check whether each document is relevant. This approach does not scale well. It would be nice if
we could process a small subset of the documents instead. We assumed that the features in the query
are the only features that influence the probability calculation. Let’s make an additional assumption
in this same spirit: for any relevant document d, there is a feature j such that fj(d) ̸= 0and fj(q) ̸= 0.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
We assumed that the features in the query
are the only features that influence the probability calculation. Let’s make an additional assumption
in this same spirit: for any relevant document d, there is a feature j such that fj(d) ̸= 0and fj(q) ̸= 0. Is this assumption reasonable?
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
We assumed that the features in the query
are the only features that influence the probability calculation. Let’s make an additional assumption
in this same spirit: for any relevant document d, there is a feature j such that fj(d) ̸= 0and fj(q) ̸= 0. Is this assumption reasonable? Explain how you could build a more efficient information retrieval
system that only looked at a subset of the documents for a given query.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Is this assumption reasonable? Explain how you could build a more efficient information retrieval
system that only looked at a subset of the documents for a given query. 6
5 Answers
1. This equation is only concerned with words that are in the query, so we only have to take the product
of the probabilities of three words.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Is this assumption reasonable? Explain how you could build a more efficient information retrieval
system that only looked at a subset of the documents for a given query. 6
5 Answers
1. This equation is only concerned with words that are in the query, so we only have to take the product
of the probabilities of three words. For the word “toyota”, the probability of seeing the word “toyota”
in the relevant documents is 1.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
6
5 Answers
1. This equation is only concerned with words that are in the query, so we only have to take the product
of the probabilities of three words. For the word “toyota”, the probability of seeing the word “toyota”
in the relevant documents is 1. The probability of seeing the word “toyota” in this corpus is 3/10.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
For the word “toyota”, the probability of seeing the word “toyota”
in the relevant documents is 1. The probability of seeing the word “toyota” in this corpus is 3/10. The word “toyota” contributes 10/3 to the product (note that this is greater than 1).
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
For the word “toyota”, the probability of seeing the word “toyota”
in the relevant documents is 1. The probability of seeing the word “toyota” in this corpus is 3/10. The word “toyota” contributes 10/3 to the product (note that this is greater than 1). For the word
“brand”, the probability of seeing the word “brand” in the relevant documents is 1.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
The probability of seeing the word “toyota” in this corpus is 3/10. The word “toyota” contributes 10/3 to the product (note that this is greater than 1). For the word
“brand”, the probability of seeing the word “brand” in the relevant documents is 1. The probability
of seeing the word “brand” in this corpus is 3/10.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
The word “toyota” contributes 10/3 to the product (note that this is greater than 1). For the word
“brand”, the probability of seeing the word “brand” in the relevant documents is 1. The probability
of seeing the word “brand” in this corpus is 3/10. The word “brand” contributes 10/3 to the product.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
For the word
“brand”, the probability of seeing the word “brand” in the relevant documents is 1. The probability
of seeing the word “brand” in this corpus is 3/10. The word “brand” contributes 10/3 to the product. For the word “car”, the probability of seeing the word “car” in the relevant documents is 1/2.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
The probability
of seeing the word “brand” in this corpus is 3/10. The word “brand” contributes 10/3 to the product. For the word “car”, the probability of seeing the word “car” in the relevant documents is 1/2. The
probability of seeing the word “car” in this corpus is 6/10.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
The word “brand” contributes 10/3 to the product. For the word “car”, the probability of seeing the word “car” in the relevant documents is 1/2. The
probability of seeing the word “car” in this corpus is 6/10. The word “car” contributes 10/12 to the
product. Multiplying these terms yields 9.26. 2.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
For the word “car”, the probability of seeing the word “car” in the relevant documents is 1/2. The
probability of seeing the word “car” in this corpus is 6/10. The word “car” contributes 10/12 to the
product. Multiplying these terms yields 9.26. 2. We used a computer program to compute the result: 64866.24.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
The
probability of seeing the word “car” in this corpus is 6/10. The word “car” contributes 10/12 to the
product. Multiplying these terms yields 9.26. 2. We used a computer program to compute the result: 64866.24. The source code for this program is
included at the end of this document. 3.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
The word “car” contributes 10/12 to the
product. Multiplying these terms yields 9.26. 2. We used a computer program to compute the result: 64866.24. The source code for this program is
included at the end of this document. 3. The scores of the two tables differ greatly, but for the most part, the document ranking has not changed
much for the two tables.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Multiplying these terms yields 9.26. 2. We used a computer program to compute the result: 64866.24. The source code for this program is
included at the end of this document. 3. The scores of the two tables differ greatly, but for the most part, the document ranking has not changed
much for the two tables. The document ranking changed the most with the query “car”.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
2. We used a computer program to compute the result: 64866.24. The source code for this program is
included at the end of this document. 3. The scores of the two tables differ greatly, but for the most part, the document ranking has not changed
much for the two tables. The document ranking changed the most with the query “car”. For example,
in the first table, the second document is tied with being the least relevant document; however, in the
second table, it is the second most relevant document.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
The document ranking changed the most with the query “car”. For example,
in the first table, the second document is tied with being the least relevant document; however, in the
second table, it is the second most relevant document. This particular example shows that terms
not in the query can have different distribution in relevant documents than in documents overall. Also, notice that the word “car” occurs in most of the queries.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
This particular example shows that terms
not in the query can have different distribution in relevant documents than in documents overall. Also, notice that the word “car” occurs in most of the queries. Because the query “car” is so generic,
almost all the documents were relevant to some extent. In the first table, it is impossible to distinguish
between documents with just one common word.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Also, notice that the word “car” occurs in most of the queries. Because the query “car” is so generic,
almost all the documents were relevant to some extent. In the first table, it is impossible to distinguish
between documents with just one common word. In the second table, there is more variation among
the documents’ relevance scores for the “car” column.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
Because the query “car” is so generic,
almost all the documents were relevant to some extent. In the first table, it is impossible to distinguish
between documents with just one common word. In the second table, there is more variation among
the documents’ relevance scores for the “car” column. This variation is due to the fact that we are
using features that were not present in the query.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
In the first table, it is impossible to distinguish
between documents with just one common word. In the second table, there is more variation among
the documents’ relevance scores for the “car” column. This variation is due to the fact that we are
using features that were not present in the query. This implies that sometimes the words in the query
itself are not enough information to decide if a document is relevant or not. 4.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
In the second table, there is more variation among
the documents’ relevance scores for the “car” column. This variation is due to the fact that we are
using features that were not present in the query. This implies that sometimes the words in the query
itself are not enough information to decide if a document is relevant or not. 4. We are no longer computing the probability that the document is relevant.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
This variation is due to the fact that we are
using features that were not present in the query. This implies that sometimes the words in the query
itself are not enough information to decide if a document is relevant or not. 4. We are no longer computing the probability that the document is relevant. Instead, we are computing
a quantity that is equivalent to this probability under rank.
|
https://www.cs.cornell.edu/courses/cs6740/2010sp/guides/lec05.pdf?utm_source=chatgpt.com
|
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