Information Retrieval Models and Processes
Traditional IR models
Jian-Yun Nie
Main IR processes
Main IR processes
Last lecture: Indexing – determine the
important content terms
Next process: Retrieval
◦
How should a retrieval process be done?
Implementation issues: using index (e.g. merge of lists)
(*) What are the criteria to be used?
◦
Ranking criteria
What features?
How should they be combined?
What model to use?
2
Cases
one
-term query:
The documents to be retrieved are those that include
the term
-
Retrieve the inverted list for the term
-
Sort in decreasing order of the weight of the word
Multi-term query?
-
Combining several lists
-
How to interpret the weight?
-
How to interpret the representation with all the
indexing terms for a document?
(IR model)
3
What is an IR model?
Define a way to represent the contents of a
document and a query
Define a way to compare a document
representation to a query representation, so as
to result in a document ranking (score function)
E.g. Given a set of weighted terms for a
document
◦
Should these terms be considered as forming a
Boolean expression? a vector? …
◦
What do the weights mean? a probability, a feature
value, …
◦
What is the associated ranking function?
Plan
This lecture
◦
Boolean model
◦
Extended Boolean models
◦
Vector space model
◦
Probabilistic models
Binary Independent Probabilistic model
Regression models
Next week
◦
Statistical language models
Early IR model – Coordinate
matching score (1960s)
Matching score model
◦
Document D = a set of weighted terms
◦
Query Q = a set of non-weighted terms
Discussion
◦
Simplistic representation of documents and
queries
◦
The ranking score strongly depends on the term
weighting in the document
If the weights are not normalized, then there will be
great variations in
R
6
IR
model -
Boolean model
◦
Document = Logical conjunction of keywords (not
weighted)
◦
Query = any Boolean expression of keywords
◦
R(D, Q) = D
Q
e.g.
D
1
= t
1
t
2
t
3
(the three terms appear in D)
D
2
= t
2
t
3
t
4
t
5
Q = (t
1
t
2
)
(t
3
t
4
)
D
1
Q, thus R(D
1
, Q) = 1.
but
D
2
Q, thus R(D
2
, Q) = 0.
7
/
Properties
Desirable
◦
R(D,Q
∧
Q)=R(D,Q
∨
Q)=R(D,Q)
◦
R(D,D)=1
◦
R(D,Q
∨
¬Q)=1
◦
R(D,Q
∧
¬Q)=0
Undesirable
◦
R(D,Q)=0 or 1
Boolean model
Strengths
◦
Rich expressions for queries
◦
Clear logical interpretation (well studied logical properties)
Each term is considered as a logical proposition
The ranking function is determine by the validity of a logical
implication
Problems:
◦
R
is either 1 or 0 (unordered set of documents)
many documents or few/no documents in the result
No term weighting in document and query is used
◦
Difficulty for end-users for form a correct Boolean query
E.g. documents about
kangaroos
and
koalas
kangaroo
koala ?
kangaroo
koala ?
Specialized application (Westlaw in legal area)
Current status in Web search
◦
Use Boolean model (ANDed terms in query) for a first
step retrieval
◦
Assumption: There are many documents containing all the
query terms
find a few of them
Extensions to Boolean model
(for document ranking)
D = {…, (ti
, w
i
), …}: weighted terms
Interpretation
:
◦
Each term or a logical expression defines a fuzzy set
◦
(t
i
, w
i
):
D is a member of class
t
i
to degree
w
i
.
◦
In terms of fuzzy sets, membership function:
ti
(D)=
w
i
A possible Evaluation
:
R(D, t
i
) =
ti
(D)
∈
[0,1]
R(D, Q
1
Q
2
) =
Q
1
Q
2
(D) =
min(R(D, Q
1
), R(D, Q
2
));
R(D, Q
1
Q
2
) =
Q
1
Q
2
(D) =
max(R(D, Q
1
), R(D, Q
2
));
R(D,
Q
1
) =
Q
1
(D) =
1 - R(D, Q
1
).
10
Recall on fuzzy sets
Classical set
◦
a
belongs to a set
S
:
a
∈
S,
◦
or no:
a
∉
S
Fuzzy set
◦
a
belongs to a set
S
to some degree
(μ
S
(
a
)
∈
[0,1])
◦
E.g. someone is
tall
μ
tall
(
a
)
Recall on fuzzy sets
Combination of concepts
Extension with fuzzy sets
Can take into account term weights
Fuzzy sets are motivated by fuzzy concepts in
natural language (tall, strong, intelligent, fast, slow,
…)
Evaluation reasonable?
◦
min and max are determined by one of the elements
(the value of another element in some range does not
have a direct impact on the final value) -
counterintuitive
◦
Violated logical properties
μ
A
∨
¬A
(.)≠1
μ
A
∧
¬A
(.)≠0
Alternative evaluation in fuzzy sets
R(D, t
i
) =
ti
(D)
∈
[0,1]
R(D, Q
1
Q
2
) = R(D, Q
1
) * R(D, Q
2
);
R(D, Q
1
Q
2
) = R(D, Q
1
) + R(D, Q
2
) - R(D, Q
1
) * R(D, Q
2
);
R(D,
Q
1
) = 1 - R(D, Q
1
).
◦
The resulting value is closely related to both values
◦
Logical properties
μ
A
∨
¬A
(.)≠1
μ
A
∧
¬A
(.)≠0
μ
A
∨
A
(.)≠μ
A
(.)
μ
A
∧
A
(.)≠μ
A
(.)
◦
In practice, better than min-max
◦
Both extensions have lower IR effectiveness than
vector space model
IR model - Vector space model
Assumption: Each term corresponds to a
dimension in a vector space
Vector space = all the keywords encountered
<t
1
, t
2
, t
3
, …, t
n
>
Document
D =
< a
1
, a
2
, a
3
, …, a
n
>
a
i
= weight of t
i
in D
Query
Q =
< b
1
, b
2
, b
3
, …, b
n
>
b
i
= weight of t
i
in Q
R(D,Q) = Sim(D,Q)
15
Matrix representation
t
1
t
2
t
3
…
t
n
D
1
a
11
a
12
a
13
…
a
1n
D
2
a
21
a
22
a
23
…
a
2n
D
3
a
31
a
32
a
33
…
a
3n
…
D
m
a
m1
a
m2
a
m3
…
a
mn
Q
b
1
b
2
b
3
…
b
n
16
Term vector
space
Document space
Some formulas for Sim
Dot product
Cosine
Dice
Jaccard
17
Document-document, document-
query and term-term similarity
t
1
t
2
t
3
…
t
n
D
1
a
11
a
12
a
13
…
a
1n
D
2
a
21
a
22
a
23
…
a
2n
D
3
a
31
a
32
a
33
…
a
3n
…
D
m
a
m1
a
m2
a
m3
…
a
mn
Q
b
1
b
2
b
3
…
b
n
D-D similarity
D-Q similarity
t-t similarity
Euclidean distance
When the vectors are normalized (length
of 1), the ranking is the same as cosine
similarity. (Why?)
Implementation (space)
Matrix is very sparse: a few 100s terms for a document,
and a few terms for a query, while the term space is
large (>100k)
Stored as:
D
1
{(t1
, a
1
), (t
2
,a
2
), …}
t
1
{(D1
,a
1
), …}
(recall possible compressions: ϒ code)
20
Implementation (time)
The implementation of VSM with dot product:
◦
Naïve implementation: Compare Q with each D
◦
O(m*n): m doc. & n terms
◦
Implementation using inverted file:
Given a query = {(t1
,b
1
), (t
2
,b
2
),
(t
3
,b
3
)
}:
1.
find the sets of related documents through inverted file for each
term
2.
calculate the score of the documents to each weighted query term
(t
1
,b
1
)
{(D1
,a
1
*b
1
), …}
3.
combine the sets and sum the weights (
) (in binary tree)
◦
O(|t|*|Q|*log(|Q|)):
|t|<<m (|t|=avg. length of inverted lists),
|Q|*log|Q|<<n (|Q|=length of the query)
21
Pre-normalization
Cosine:
-
use
and
to normalize the
weights after indexing of document and query
-
Dot product
(Similar operations do not apply to Dice and
Jaccard)
22
Best
p
candidates
Can still be too expensive to calculate similarities to all
the documents (Web search)
p
best
Preprocess
: Pre-compute, for each term, its
p
nearest
docs.
◦
(Treat each term as a 1-term query.)
◦
lots of preprocessing.
◦
Result:
“
preferred list
”
for each term.
Search
:
◦
For a
|Q|
-term query, take the union of their
|Q|
preferred
lists – call this set
S,
where
|
S
|
p|Q|
.
◦
Compute cosines from the query to only the docs in
S
, and
choose the top
k
.
◦
If too few results, search in extended index
Need to pick
p>k
to work well empirically.
Discussions on vector space model
Pros:
◦
Mathematical foundation = geometry
Q: How to interpret?
◦
Similarity can be used on different elements
◦
Terms can be weighted according to their importance (in both D and Q)
◦
Good effectiveness in IR tests
Cons
◦
Users cannot specify relationships between terms
world cup
: may find documents on
world
or on
cup
only
A strong term may dominate in retrieval
◦
Term independence assumption (in all classical models)
Comparison with other models
◦
Coordinate matching score – a special case
◦
Boolean model and vector space model: two extreme cases
according to the difference we see between AND and OR
(Gerard Salton, Edward A. Fox, and Harry Wu. 1983.
Extended Boolean information retrieval.
Commun. ACM
26,
11, 1983)
◦
Probabilistic model: can be viewed as a vector space model
with probabilistic weighting.
Why probabilities in IR?
User
Information Need
Documents
Document
Representation
Query
Representation
How to match?
How to match?
In traditional IR systems, matching between each document and
query is attempted in a semantically imprecise space of index terms.
Probabilities provide a principled foundation for uncertain reasoning.
Can we use probabilities to quantify our uncertainties?
Uncertain guess of
whether document has
relevant content
Understanding
of user need is
uncertain
Probabilistic IR topics
Classical probabilistic retrieval model
◦
Probability ranking principle, etc.
(Naïve) Bayesian Text Categorization/classification
Bayesian networks for text retrieval
Language model approach to IR
◦
An important emphasis in recent work
Probabilistic methods are one of the oldest but also one
of the currently hottest topics in IR.
◦
Traditionally: neat ideas, but they
’
ve never won on
performance. It may be different now.
The document ranking problem
We have a collection of documents
User issues a query
A list of documents needs to be returned
Ranking method is core of an IR system:
Ranking method is core of an IR system:
◦
In what order do we present documents to the
In what order do we present documents to the
user?
user?
◦
We want the
“
best
”
document to be first, second
best second, etc….
Idea: Rank by probability of relevance of
Idea: Rank by probability of relevance of
the document w.r.t. information need
the document w.r.t. information need
◦
P(relevant|document
i
, query)
The Probability Ranking Principle
“
If a reference retrieval system's response to each
request is a ranking of the documents in the collection
in order of decreasing probability of relevance to the
user who submitted the request, where the probabilities
are estimated as accurately as possible on the basis of
whatever data have been made available to the system
for this purpose, the overall effectiveness of the system
to its user will be the best that is obtainable on the
basis of those data.
”
[1960s/1970s] S. Robertson, W.S. Cooper, M.E. Maron;
van
Rijsbergen
(1979:113); Manning & Schütze (1999:538)
Recall a few probability basics
For events
a
and
b:
Bayes
’
Rule
Odds:
Posterior
Prior
Probability Ranking Principle
Let
x
be a document in the collection.
Let
R
represent
relevance
of a document w.r.t. given (fixed)
query and let
NR
represent
non-relevance.
p
(
x|R
),
p
(
x|NR
)
-
probability that if a relevant (non-relevant)
document is retrieved, it is
x
.
Need to find
p(
R|x)
- probability that a document
x
is
relevant.
p
(
R
),
p
(
NR
) - prior probability
of retrieving a (non) relevant
document
R={0,1} vs. NR/RProbability Ranking Principle (PRP)
Simple case: no selection costs or other utility
concerns that would differentially weight errors
Bayes
’
Optimal Decision Rule
◦
x
is
relevant
iff
p
(
R
|
x
) >
p
(
NR
|
x
)
PRP in action: Rank all documents by
p
(
R
|
x
)
Theorem:
◦
Using the PRP is optimal, in that it minimizes the loss
(Bayes risk) under 1/0 loss
◦
Provable if all probabilities correct, etc.
[e.g., Ripley
1996]
Probability Ranking Principle
More complex case: retrieval costs.
◦
Let
d
be a document
◦
C
- cost of retrieval of
relevant
document
◦
C
’
- cost of retrieval of
non-relevant
document
Probability Ranking Principle: if
for all
d
’
not yet retrieved
, then
d
is the next
document to be retrieved
We won
’
t further consider loss/utility from
now on
Probability Ranking Principle
How do we compute all those probabilities?
◦
Do not know exact probabilities, have to use
estimates
◦
Binary Independence Retrieval (BIR) – which we
discuss later today – is the simplest model
Questionable assumptions
◦
"
Relevance
"
of each document is independent of
relevance of other documents.
Really, it
’
s bad to keep on returning
duplicates
◦
Boolean model of relevance (relevant or irrelevant)
◦
That one has a single step information need
Seeing a range of results might let user refine query
Probabilistic Retrieval Strategy
Estimate how terms contribute to relevance
◦
How do things like tf, df, and length influence
your judgments about document relevance?
One answer is the Okapi formulae (S. Robertson)
Combine to find document relevance
probability
Order documents by decreasing probability
Probabilistic Ranking
B
a
s
i
c
c
o
n
c
e
p
t
:
"For a given query, if we know some documents that are
relevant, terms that occur in those documents should be
given greater weighting in searching for other relevant
documents.
By making assumptions about the distribution of terms
and applying Bayes Theorem, it is possible to derive
weights theoretically."
Van Rijsbergen
Binary Independence Model
Traditionally used in conjunction with PRP
“
Binary
”
= Boolean
: documents are represented as
binary incidence vectors of terms:
◦
◦
iff
term
i
is present in document
x
.
“
Independence
”
:
terms occur in documents
independently
Different documents can be modeled as same vector
Bernoulli Naive Bayes model (cf. text categorization!)
Binary Independence Model
Queries: binary term incidence vectors
Given query
q
,
◦
for each document
d
need to compute
p
(
R
|
q,d
)
.
◦
replace with computing
p
(
R
|
q,x
)
where
x
is binary
term incidence vector representing
d
Interested only
in ranking
Will use odds and Bayes
’
Rule:
Binary Independence Model
Constant for a
given query
Needs estimation
Binary Independence Model
•
Since
x
i
is either 0 or 1:
Then...
This can be
changed (e.g., in
relevance feedback)
Binary Independence Model
x
i
=1
q
i
=1
Binary Independence Model
Binary Independence Model
•
All boils down to computing RSV.
So, how do we compute
c
i
’
s
from our data ?
Binary Independence Model
•
Estimating RSV coefficients.
•
For each term
i
look at this table of document counts:
Estimation – key challenge
If non-relevant documents are approximated by the
whole collection, then
r
i
(prob. of occurrence in non-
relevant documents for query)
is n/N
and
◦
log (1–
r
i
)/
r
i
= log (N–
n
)/
n
≈
log N/
n
= IDF!
p
i
(probability of occurrence in relevant documents)
can be estimated in various ways:
◦
from relevant documents if know some
Relevance weighting can be used in feedback loop
◦
constant (Croft and Harper combination match) – then
just get idf weighting of terms
◦
proportional to prob. of occurrence in collection
more accurately, to log of this (Greiff, SIGIR 1998)
46
Iteratively estimating
p
i
1.
Assume that
p
i
constant over all
x
i
in query
◦
p
i
= 0.5 (even odds) for any given doc
2.
Determine guess of relevant document set:
◦
V is fixed size set of highest ranked documents
on this model (note: now a bit like tf.idf!)
3.
We need to improve our guesses for
p
i
and
r
i
, so
◦
Use distribution of
x
i
in docs in V. Let V
i
be set
of documents containing
x
i
p
i
= |V
i
| / |V|
◦
Assume if not retrieved then not relevant
r
i
= (n
i
– |V
i
|) / (N – |V|)
4.
Go to 2. until converges then return
ranking
Probabilistic relevance feedback
If user has told us some relevant and some
irrelevant documents, then we can proceed to
build a probabilistic classifier, such as a Naive
Bayes model:
◦
P(
t
k
|R) = |
D
rk
| / |
D
r
|
◦
P(
t
k
|NR) = |
D
nrk
| / |
D
nr
|
t
k
is a term;
D
r
is the set of known relevant
documents;
D
rk
is the subset that contain
t
k
;
D
nr
is
the set of known irrelevant documents;
D
nrk
is the
subset that contain
t
k
.
Probabilistic Relevance Feedback
1.
Guess a preliminary probabilistic
description of
R
and use it to retrieve a first
set of documents V, as above.
2.
Interact with the user to refine the
description: learn some definite members of
R and NR
3.
Reestimate
p
i
and
r
i
on the basis of these
◦
Or can combine new information with original
guess (use Bayesian prior):
4.
Repeat, thus generating a succession of
approximations to
R
.
κ
is
prior
weight
PRP and BIR
Getting reasonable approximations of
probabilities is possible.
Requires restrictive assumptions:
◦
term independence
◦
terms not in query don
’
t affect the outcome
◦
Boolean representation of
documents/queries/relevance
◦
document relevance values are independent
Some of these assumptions can be removed
Problem: either require partial relevance information or
only can derive somewhat inferior term weights
Removing term independence
In general, index terms aren
’
t
independent
Dependencies can be complex
van Rijsbergen (1979)
proposed model of simple tree
dependencies
Each term dependent on one
other
In 1970s, estimation problems
held back success of this model
Food for thought
Think through the differences between
standard tf.idf and the probabilistic
retrieval model in the first iteration
Think through the retrieval process of
probabilistic model similar to vector
space model
Good and Bad News
Standard Vector Space Model
◦
Empirical for the most part; success measured by results
◦
Few properties provable
Probabilistic Model Advantages
◦
Based on a firm theoretical foundation
◦
Theoretically justified optimal ranking scheme
Disadvantages
◦
Making the initial guess to get V
◦
Binary word-in-doc weights (not using term frequencies)
◦
Independence of terms (can be alleviated)
◦
Amount of computation
◦
Has never worked convincingly better in practice
BM25 (Okapi system) – Robertson
et al.
k1, k2, k3, b
: parameters
qtf
: query term frequency
dl
: document length
avdl
: average document length
53
Doc. length
Normalization:
boost short
documents
TF factors
Consider tf, qtf, document length
Pivoted document length
normalization
(Singhal et al. SIGIR’96)
Document length normalization
◦
Weight(t,D) = tf*idf
◦
Cosine normalization: 1/|D|
◦
Normalizatio by max weight:
0.5+0.5*w(t,D)/max{w(t’,D)}55
Document Length Normalization
(Singhal)
Sometimes, additional normalizations e.g. length
to boost longer documents:
Regression models
Extract a set of features from document
(and query)
Define a function to predict the probability
of its relevance
Learn the function on a set of training data
(with relevance judgments)
Probability of Relevance
Document
Query
X1,X2,X3,X4
Probability
of relevance
Ranking Formula
feature vector
Regression model (Berkeley – Chen and Frey)
Relevance Features
Sample Document/Query Feature Vector
Relevance Features
X1
0.0031
0.0429
0.0430
0.0195
0.0856
X2
-2.406
-9.796
-6.342
-9.768
-7.375
X3
-3.223
-15.55
-9.921
-15.096
-12.477
X4
1
8
4
6
5
Relevance value
1
1
1
0
0
Representing one document/query
pair in the training set
Probabilistic Model: Supervised Training
Model: Logistic Regression
Unknown parameters:
b1,b2,b3, b4
Training Data Set:
Document/Query Pairs
with known relevance
value.
Test Data Set
:
New document/query
pairs
1. Model training: estimate the
unknown model parameters using
training data set.
2. Using the estimated parameters
to predict relevance value for a
new pair of document and query.
Logistic Regression Method
Model
: The log odds of the relevance dependent
variable is a linear combination of the independent
feature variables.
Task:
Find the optimal coefficients
Method:
Use statistical software package such as S-plus to
fit the model to a training data set.
relevance
variable
feature
variables
Logistic regression
The function to learn:
f
(
z
):
The variable
z
is usually
defined as
◦
x
i
= feature variables
◦
β
i
=parameters/coefficients
Document Ranking Formula
N is the number of matching terms between document D and
query Q.
Discussions
Usually, terms are considered to be independent
◦
algorithm
independent from
computer
◦
computer architecture:
2 independent dimensions
Different theoretical foundations (assumptions) for IR
◦
Boolean model:
Used in specialized area
Not appropriate for general search alone – often used as a pre-filtering
◦
Vector space model:
Robust
Good experimental results
◦
Probabilistic models:
Difficulty to estimate probabilities accurately
Modified version (BM25) – excellent results
Regression models:
Need training data
Widely used (in a different form) in web search
Learning to rank (a later lecture)
More recent model on statistical language modeling (robust model
relying on a large amount of data – next lecture)
Delve into the world of information retrieval models with a focus on traditional approaches, main processes like indexing and retrieval, cases of one-term and multi-term queries, and the evolution of IR models from boolean to probabilistic and vector space models. Explore the concept of IR models, representation of documents and queries, and ranking functions. Gain insights into early IR models like Coordinate Matching Score and the importance of term weighting.
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Presentation Transcript
Traditional IR models Jian-Yun Nie
Main IR processes Last lecture: Indexing determine the important content terms Next process: Retrieval How should a retrieval process be done? Implementation issues: using index (e.g. merge of lists) (*) What are the criteria to be used? Ranking criteria What features? How should they be combined? What model to use? 2
Cases one-term query: The documents to be retrieved are those that include the term - Retrieve the inverted list for the term - Sort in decreasing order of the weight of the word Multi-term query? - Combining several lists - How to interpret the weight? - How to interpret the representation with all the indexing terms for a document? (IR model) 3
What is an IR model? Define a way to represent the contents of a document and a query Define a way to compare a document representation to a query representation, so as to result in a document ranking (score function) E.g. Given a set of weighted terms for a document Should these terms be considered as forming a Boolean expression? a vector? What do the weights mean? a probability, a feature value, What is the associated ranking function?
Plan This lecture Boolean model Extended Boolean models Vector space model Probabilistic models Binary Independent Probabilistic model Regression models Next week Statistical language models
Early IR model Coordinate matching score (1960s) Matching score model Document D = a set of weighted terms Query Q = a set of non-weighted terms R(D,Q)= w(ti,D) ti Q Discussion Simplistic representation of documents and queries The ranking score strongly depends on the term weighting in the document If the weights are not normalized, then there will be great variations in R 6
IR model - Boolean model Document = Logical conjunction of keywords (not weighted) Query = any Boolean expression of keywords R(D, Q) = D Q D1= t1 t2 t3 D2= t2 t3 t4 t5 Q = (t1 t2) (t3 t4) e.g. (the three terms appear in D) D1 Q, thus R(D1, Q) = 1. but D2 Q, thus R(D2, Q) = 0. / 7
Properties Desirable R(D,Q Q)=R(D,Q Q)=R(D,Q) R(D,D)=1 R(D,Q Q)=1 R(D,Q Q)=0 Undesirable R(D,Q)=0 or 1
Boolean model Strengths Rich expressions for queries Clear logical interpretation (well studied logical properties) Each term is considered as a logical proposition The ranking function is determine by the validity of a logical implication Problems: R is either 1 or 0 (unordered set of documents) many documents or few/no documents in the result No term weighting in document and query is used Difficulty for end-users for form a correct Boolean query E.g. documents about kangaroos and koalas kangaroo koala ? kangaroo koala ? Specialized application (Westlaw in legal area) Current status in Web search Use Boolean model (ANDed terms in query) for a first step retrieval Assumption: There are many documents containing all the query terms find a few of them
Extensions to Boolean model (for document ranking) D = { , (ti, wi), }: weighted terms Interpretation: Each term or a logical expression defines a fuzzy set (ti, wi): D is a member of class tito degree wi. In terms of fuzzy sets, membership function: ti(D)=wi A possible Evaluation: R(D, ti) = ti(D) [0,1] R(D,Q1 Q2) = Q1 Q2(D) = min(R(D,Q1),R(D,Q2)); R(D,Q1 Q2) = Q1( Q2(D) = max(R(D,Q1),R(D,Q2)); R(D, Q1) = Q1(D) = 1 - R(D,Q1). 10
Recall on fuzzy sets Classical set a belongs to a set S: a S, or no: a S Fuzzy set a belongs to a set S to some degree ( S(a) [0,1]) E.g. someone is tall 1.5 tall(a) 1 0.5 0 1.5 1.7 1.9 2.1 2.3
Recall on fuzzy sets Combination of concepts 1.2 1 0.8 Tall Strong Tall&Strong 0.6 0.4 0.2 0 Allan Bret Chris Dan
Extension with fuzzy sets Can take into account term weights Fuzzy sets are motivated by fuzzy concepts in natural language (tall, strong, intelligent, fast, slow, ) Evaluation reasonable? min and max are determined by one of the elements (the value of another element in some range does not have a direct impact on the final value) - counterintuitive Violated logical properties A A(.) 1 A A(.) 0
Alternative evaluation in fuzzy sets R(D, ti) = ti(D) [0,1] R(D,Q1 Q2) = R(D,Q1) * R(D,Q2); R(D,Q1 Q2) = R(D,Q1) + R(D,Q2) - R(D,Q1) * R(D,Q2); R(D, Q1) = 1 - R(D,Q1). The resulting value is closely related to both values Logical properties A A(.) 1 A A(.) 0 A A(.) A(.) A A(.) A(.) In practice, better than min-max Both extensions have lower IR effectiveness than vector space model
IR model -Vector space model Assumption: Each term corresponds to a dimension in a vector space Vector space = all the keywords encountered <t1, t2, t3, , tn> Document D = < a1, a2, a3, , an> ai= weight of tiin D Query Q = < b1,b2,b3, ,bn> bi = weight of tiin Q R(D,Q) = Sim(D,Q) 15
Matrix representation t1 a11 a21 a31 t2 a12 a22 a32 t3 a13 a23 a33 tn a1n a2n a3n Document space Term vector space D1 D2 D3 Dm am1 Q am2 b1 am3 b2 b3 amn bn 16
Some formulas for Sim Dot product i a * = = ( , ) ( i * ) Sim D Q D Q a b i t 3 i D ( ) b i i Cosine = ( , ) Sim D Q Q i i 2 2 * a b i i t 1 i a 2 ( i * ) a b i Dice t2 = ( , ) Sim D Q i i 2 2 + b i i i ( i * ) a b Jaccard i = ( , ) Sim D Q i i i 2 2 + ( i * ) a b a b i i i 17
Document-document, document- query and term-term similarity t1 t2 t3 D1 a11 a12 a13 D2 a21 a22 a23 D3 a31 a32 a33 Dm am1 am2 am3 Q b1 b2 b3 tn a1n a2n a3n D-D similarity amn bn D-Q similarity t-t similarity
Euclidean distance ( ) = n 2 = d d d d , , j k i j i k 1 i When the vectors are normalized (length of 1), the ranking is the same as cosine similarity. (Why?)
Implementation (space) Matrix is very sparse: a few 100s terms for a document, and a few terms for a query, while the term space is large (>100k) Stored as: D1 {(t1, a1), (t2,a2), } t1 {(D1,a1), } (recall possible compressions: code) 20
Implementation (time) The implementation of VSM with dot product: Na ve implementation: Compare Q with each D O(m*n): m doc. & n terms Implementation using inverted file: Given a query = {(t1,b1), (t2,b2),(t3,b3)}: 1. find the sets of related documents through inverted file for each term 2. calculate the score of the documents to each weighted query term (t1,b1) {(D1,a1*b1), } 3. combine the sets and sum the weights ( ) (in binary tree) O(|t|*|Q|*log(|Q|)): |t|<<m (|t|=avg. length of inverted lists), |Q|*log|Q|<<n (|Q|=length of the query) 21
Pre-normalization Cosine: i ( i * ) a b i a b = = ( , ) i i Sim D Q j j j 2* 2 2 2 a b a b i i i j j j j j 2 2 1/ 1/ aj bj - use weights after indexing of document and query - Dot product (Similar operations do not apply to Dice and Jaccard) and to normalize the 22
Best p candidates Can still be too expensive to calculate similarities to all the documents (Web search) p best Preprocess: Pre-compute, for each term, its p nearest docs. (Treat each term as a 1-term query.) lots of preprocessing. Result: preferred list for each term. Search: For a |Q|-term query, take the union of their |Q| preferred lists call this set S, where |S| p|Q|. Compute cosines from the query to only the docs in S, and choose the top k. If too few results, search in extended index Need to pick p>k to work well empirically.
Discussions on vector space model Pros: Mathematical foundation = geometry Q: How to interpret? Similarity can be used on different elements Terms can be weighted according to their importance (in both D and Q) Good effectiveness in IR tests Cons Users cannot specify relationships between terms world cup: may find documents on world or on cup only A strong term may dominate in retrieval Term independence assumption (in all classical models)
Comparison with other models Coordinate matching score a special case Boolean model and vector space model: two extreme cases according to the difference we see between AND and OR (Gerard Salton, Edward A. Fox, and Harry Wu. 1983. Extended Boolean information retrieval. Commun. ACM 26, 11, 1983) Probabilistic model: can be viewed as a vector space model with probabilistic weighting.
Why probabilities in IR? Understanding of user need is uncertain User Query Information Need Representation How to match? Uncertain guess of whether document has relevant content Document Representation Documents In traditional IR systems, matching between each document and query is attempted in a semantically imprecise space of index terms. Probabilities provide a principled foundation for uncertain reasoning. Can we use probabilities to quantify our uncertainties?
Probabilistic IR topics Classical probabilistic retrieval model Probability ranking principle, etc. (Na ve) Bayesian Text Categorization/classification Bayesian networks for text retrieval Language model approach to IR An important emphasis in recent work Probabilistic methods are one of the oldest but also one of the currently hottest topics in IR. Traditionally: neat ideas, but they ve never won on performance. It may be different now.
The document ranking problem We have a collection of documents User issues a query A list of documents needs to be returned Ranking method is core of an IR system: In what order do we present documents to the user? We want the best document to be first, second best second, etc . Idea: Rank by probability of relevance of the document w.r.t. information need P(relevant|documenti, query)
The Probability Ranking Principle If a reference retrieval system's response to each request is a ranking of the documents in the collection in order of decreasing probability of relevance to the user who submitted the request, where the probabilities are estimated as accurately as possible on the basis of whatever data have been made available to the system for this purpose, the overall effectiveness of the system to its user will be the best that is obtainable on the basis of those data. [1960s/1970s] S. Robertson, W.S. Cooper, M.E. Maron; van Rijsbergen (1979:113); Manning & Sch tze (1999:538)
Recall a few probability basics For events a and b: Bayes Rule p(a,b)= p(a b)= p(a|b)p(b)= p(b|a)p(a) p(a |b)p(b)= p(b|a)p(a) p(a|b)=p(b|a)p(a) p(b) Posterior Prior p(b|a)p(a) p(b| x)p(x) x=a,a = Odds: O(a)=p(a) p(a) 1- p(a) p(a)=
Probability Ranking Principle Let x be a document in the collection. Let R represent relevance of a document w.r.t. given (fixed) query and let NR represent non-relevance. R={0,1} vs. NR/R Need to find p(R|x) - probability that a document x is relevant. p(R),p(NR) - prior probability of retrieving a (non) relevant document ( ) | ( + NR p x R p ( | p ) x ( ) p x R p R = ( | ) p R x ( ) ( | ) x ( ) p x NR p NR = ( | ) p NR x = | ) 1 x ( ) p p(x|R), p(x|NR) - probability that if a relevant (non-relevant) document is retrieved, it is x.
Probability Ranking Principle (PRP) Simple case: no selection costs or other utility concerns that would differentially weight errors Bayes x is relevant iff p(R|x) > p(NR|x) Optimal Decision Rule PRP in action: Rank all documents by p(R|x) Theorem: Using the PRP is optimal, in that it minimizes the loss (Bayes risk) under 1/0 loss Provable if all probabilities correct, etc. [e.g., Ripley 1996]
Probability Ranking Principle More complex case: retrieval costs. Let d be a document C - cost of retrieval of relevant document C - cost of retrieval of non-relevant document Probability Ranking Principle: if )) | ( 1 ( ) | ( d R p C d R p C + )) + ( | ) 1 ( ( | C p R d C p R d for all d not yet retrieved, then d is the next document to be retrieved We won t further consider loss/utility from now on
Probability Ranking Principle How do we compute all those probabilities? Do not know exact probabilities, have to use estimates Binary Independence Retrieval (BIR) which we discuss later today is the simplest model Questionable assumptions "Relevance" of each document is independent of relevance of other documents. Really, it s bad to keep on returning duplicates Boolean model of relevance (relevant or irrelevant) That one has a single step information need Seeing a range of results might let user refine query
Probabilistic Retrieval Strategy Estimate how terms contribute to relevance How do things like tf, df, and length influence your judgments about document relevance? One answer is the Okapi formulae (S. Robertson) Combine to find document relevance probability Order documents by decreasing probability
Probabilistic Ranking Basic concept: "For a given query, if we know some documents that are relevant, terms that occur in those documents should be given greater weighting in searching for other relevant documents. By making assumptions about the distribution of terms and applying Bayes Theorem, it is possible to derive weights theoretically." Van Rijsbergen
Binary Independence Model Traditionally used in conjunction with PRP Binary = Boolean: documents are represented as binary incidence vectors of terms: iff term i is present in document x. Independence : terms occur in documents independently Different documents can be modeled as same vector = (1 x 1 = , , ) x nx ix Bernoulli Naive Bayes model (cf. text categorization!)
Binary Independence Model Queries: binary term incidence vectors Given query q, for each document d need to compute p(R|q,d). replace with computing p(R|q,x) where x is binary term incidence vector representing d Interested only in ranking Will use odds and Bayes Rule: ( | ) ( | , ) p R q p x R q ( NR | , ) x p R q x ( | ) p x q = = ( | , ) O R q x ( | ) ( | , ) p NR q p x NR q ( | , ) p q ( | ) p x q
Binary Independence Model ( NR | , ) x ( NR | ) ( x | NR , ) p R q x p R q p x R q = = ( | , ) O R q x ( | , ) ( | ) ( | , ) p q p q p q Constant for a given query Needs estimation Using IndependenceAssumption: n ( | , ) ( | , ) p ( x R q , p ( x R q , = i = i | ) | ) p x NR q p x NR q 1 i n p(xi| R,q) p(xi| NR,q) So : O(R|q,d)=O(R|q) i=1
Binary Independence Model n ( | , ) p ( x R q , = i = ( | , ) ( | ) i O R q d O R q | ) p x NR q 1 i Since xiis either 0 or 1: = | 1 = | 0 ( x | 1 , ) ( x | 0 , ) p x R q p x R q = i x , R = i x NR = ( | , ) ( | ) i i O R q d O R q = = ( , ) ( , ) p NR q p NR q 1 0 i i = | 1 = = | 1 = ( ); ( , ); p p x q r p x q Let i i i i p = r Assume, for all terms not occurring in the query (qi=0) i i This can be changed (e.g., in relevance feedback) Then...
Binary Independence Model 1 p p = i q x = i q = ( | , ) ( | ) i i O R q x O R q 1 r r = 1 0 x i i i = 1 i Non-matching query terms p All matching terms 1 ( i p ) 1 r = i q x = i q = ( | ) i i O R q 1 ( i r ) 1 p r = 1 1 i i i All query terms All matching terms qi=1 xi=1
Binary Independence Model 1 ( i p ) 1 r p = i q x = i q = ( | , ) ( | ) i i O R q x O R q 1 ( i r ) 1 p r = 1 1 i i i Constant for each query Only quantity to be estimated for rankings Retrieval Status Value: 1 ( i p ) 1 ( i p ) r r = q = i q x = = log log i i RSV 1 ( i r ) 1 ( i r ) p p = = 1 x 1 i i i i i
Binary Independence Model All boils down to computing RSV. = = i i q x = = 1 i x 1 ( i p ) 1 ( i p ) r r = q = = log log i i RSV 1 ( i r ) 1 ( i r ) p p = 1 r x 1 i i i i 1 ( i p ) = = ; RSV ic log i c i 1 ( i r ) p i q i So, how do we compute ci s from our data ?
Binary Independence Model Estimating RSV coefficients. For each term i look at this table of document counts: Documens xi=1 xi=0 Total Relevant Non-Relevant Total s n-s n S-s N-n-S+s N-n S N-S N ( N ) n s s pi ri Estimates: s ( ) S ( N S ) S s = ( , , , ) log ci K N n S s Sparck- Jones- Robertson formula + ( ) ( ) n s n S s (si+0.5)/(S-s+0.5) (n-s+0.5)/(N -n-S+s+0.5) ci=log
Estimation key challenge If non-relevant documents are approximated by the whole collection, then ri(prob. of occurrence in non- relevant documents for query) is n/N and log (1 ri)/ri= log (N n)/n log N/n = IDF! pi(probability of occurrence in relevant documents) can be estimated in various ways: from relevant documents if know some Relevance weighting can be used in feedback loop constant (Croft and Harper combination match) then just get idf weighting of terms proportional to prob. of occurrence in collection more accurately, to log of this (Greiff, SIGIR 1998)
Iteratively estimating pi 1. Assume that piconstant over all xiin query pi= 0.5 (even odds) for any given doc 2. Determine guess of relevant document set: V is fixed size set of highest ranked documents on this model (note: now a bit like tf.idf!) 3. We need to improve our guesses for piand ri, so Use distribution of xiin docs in V. Let Vibe set of documents containing xi pi= |Vi| / |V| Assume if not retrieved then not relevant ri= (ni |Vi|) / (N |V|) 4. Go to 2. until converges then return ranking 46
Probabilistic relevance feedback If user has told us some relevant and some irrelevant documents, then we can proceed to build a probabilistic classifier, such as a Naive Bayes model: P(tk|R) = |Drk| / |Dr| P(tk|NR) = |Dnrk| / |Dnr| tkis a term; Dris the set of known relevant documents; Drkis the subset that contain tk; Dnris the set of known irrelevant documents; Dnrkis the subset that contain tk.
Probabilistic Relevance Feedback 1. Guess a preliminary probabilistic description of R and use it to retrieve a first set of documents V, as above. 2. Interact with the user to refine the description: learn some definite members of R and NR 3. Reestimate piand rion the basis of these Or can combine new information with original guess (use Bayesian prior): + | ) 1 ( i | | V p is prior weight = ) 2 ( i i p + | V 4. Repeat, thus generating a succession of approximations to R.
PRP and BIR Getting reasonable approximations of probabilities is possible. Requires restrictive assumptions: term independence terms not in query don t affect the outcome Boolean representation of documents/queries/relevance document relevance values are independent Some of these assumptions can be removed Problem: either require partial relevance information or only can derive somewhat inferior term weights
Removing term independence In general, index terms aren t independent Dependencies can be complex van Rijsbergen (1979) proposed model of simple tree dependencies Each term dependent on one other In 1970s, estimation problems held back success of this model
