Introduction to Information Retrieval
Instructor: Dr. Sumanta Guha
Slide Sources: Introduction to
Information Retrieval book slides from Stanford University, adapted and supplemented
Chapter 11: Probabilistic information retrieval
1

Introduction to Information Retrieval
CS276
Information Retrieval and Web Search
Christopher Manning and Prabhakar Raghavan
Lecture 11: Probabilistic information retrieval

Introduction to Information Retrieval
 Improving search results
 Especially for high recall. E.g., searching for aircraft so it matches with plane; thermodynamics with heat
 Options for improving results…
 Global methods
 Query expansion
 Thesauri
 Automatic thesaurus generation
 Global indirect relevance feedback
 Local methods
 Relevance feedback
 Pseudo relevance feedback

Introduction to Information Retrieval
 Rather than reweighting in a vector space…
 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 subset that contain t k
; D nr is the set of known irrelevant documents; D nrk is the subset that contain t k
.
is the

Introduction to Information Retrieval
User
Information Need
Documents
Query
Representation
Understanding of user need is uncertain
How to match?
Document
Representation
Uncertain guess of whether document has relevant content
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?

Introduction to Information Retrieval
 Classical probabilistic retrieval model
 Probability ranking principle, etc.
 (Naïve) Bayesian Text Categorization
 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.

Introduction to Information Retrieval
 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|document i
, query)

Introduction to Information Retrieval
 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
Bayes’ Rule
( b | p ( b )
Posterior a ) p ( a )
 p ( b
 x
 a , a
| a ) p ( b | p ( a ) x ) p ( x )
Prior
 Odds: O ( a )
 p ( a p ( a )
)

1
 p ( a ) p ( a )
p(b) = p(b|a)p(a) + p(b|a)p(a)

Introduction to Information Retrieval
“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)

Introduction to Information Retrieval
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 | x )
 p ( x | R ) p ( R ) p ( x ) p ( NR | x )
 p ( x | NR ) p ( NR ) p ( x ) p( R) , p( NR ) – prior probability of retrieving a (non) relevant document p ( R | x )
 p ( NR | x )

1 p( x|R ), p( x|NR )
– probability that if a relevant (non-relevant) document is retrieved, it is x .

Introduction to Information Retrieval
 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]

Introduction to Information Retrieval
 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
 That one has a single step information need
 Seeing a range of results might let user refine query

Introduction to Information Retrieval
 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

Introduction to Information Retrieval
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

Introduction to Information Retrieval
 Traditionally used in conjunction with PRP
 “Binary” = Boolean : documents are represented as binary incidence vectors of terms (cf. lecture 1):

 x

( x ,  , x )
 x

1
1 n 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!)

Introduction to Information Retrieval
 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
 Will use odds and Bayes’ Rule:
O ( R | q ,
 x )

R = 1 p ( R p ( NR
|
| q , q ,
 x
 x
)
)
 p ( R p ( NR |
| q ) p (
 x p
|
( q ) p (
 x p
|
(
 x q x
)
| q )
| R , q )
NR , q )
R = 0

Introduction to Information Retrieval
O ( R | q ,
 x )
 p ( R p ( NR
|
| q , q ,
 x
 x
)
)
 p ( R | p ( NR | q ) q )
 p p (
(

 x x
|
| R , q )
NR , q )
O ( R | q ), which is constant for a given query = does not depend on the document
• Using Independence Assumption: p p (
(

 x x
|
| R ,
NR , q ) q )
 i n 

1 p p (
( x i x i
|
| R ,
NR , q ) q )
Needs estimation
• So:
O ( R | q ,
 x )

O ( R | q )
 n 

1 i p ( x i p ( x i
| R , q )
| NR , q )

Introduction to Information Retrieval
O ( R | q , d )

O ( R | q )
 i n 

1 p ( x i p ( x i
• Since x i
O ( R | q , d ) is either 0 or 1:

O ( R | q )
 x i


1 p p (
( x i x i


1
1 |
|
R ,
NR , q ) q )
 x i


0
|
| R ,
NR , q ) q )
So all terms corresponding to q i
= 0 will have equal values in numerator and denominator and, therefore, will vanish.
p p (
( x i x i


0
0
|
| R , q )
NR , q )
• Let p i
 prob ( x i

1 | R , q ); u i
 prob ( x i

1 | NR , q );
Doc Relevant (R = 1) Non-relevant (R = 0)
Term present x i
Term absent x i
= 1 p i
= 0 1 – p i u i
1 – u i
• Assume, for all terms not occurring in the query, i.e., q i
= 0, that p i
= u i
(in other words, non-query terms are equally likely to appear in relevant and non-relevant docs)

Introduction to Information Retrieval
O ( R | q ,
 x )

O ( R | q )
 x i

 q i

1 p i u i
Matching query terms
 x q i i



0
1
1

1
 p i u i
Non-matching query terms

O ( R | q )
 x i

 q i

1 p i u i
 x q i i



1
1
1

1
 u i p i
 x q i i



0
1
1

1
 p i u i
 x q i i



1
1
1

1
 p i u i

O ( R | q )
 x i

 q i

1 u p i
( 1

( 1
 i u i
) p i
)

Insert new terms which cancel!
q i


1
1
1

 p u i i
Matching query terms All query terms

Introduction to Information Retrieval
O ( R | q , x

)

O ( R | q )
 x i

 q i

1 u p i
( 1
( 1

 i u i p i
)
)
 q i


1
1

1
 u i p i
Constant for each query, does not depend on doc
• Retrieval Status Value:
RSV d
 log x i

 q i

1 u p i i
( 1

( 1
 u i
) p i
)
Only quantity to be estimated for rankings
 x i

 q i

1 log u p i i
( 1

( 1
 u i
) p i
)

Introduction to Information Retrieval
• All boils down to computing RSV:
RSV d
 log x i

 q i

1 u p i i
( 1

( 1
 u i
) p i
)
 x i

 q i

1 log u p i i
( 1

( 1
 u i
) p i
)
• Equivalently,
RSV d
 x i

 q i

1 c i
; c i
 log u p i i
( 1

( 1
 u i p i
)
)
 log
1
 p i p i
 log
1
 u i u i
So, how do we compute c i
’s from our data ?

Introduction to Information Retrieval
• Estimating RSV coefficients.
• For each term i look at this table of document counts:
Docs x i
=1 x i
=0
Relevant Non-Relevant Total s df i
–
s df i
S
–
s ( N – df i
)
–
( S
– s ) N
– df i
Total S N-S N c i

K ( N , df i
,
• Estimates:
S , s )
 log u p p i i
(

1 s

S u i i
( 1
 p i
)
) u i
 df
N i


S s
 log
( df i
 s ) s
(( N
( S
 s )
 df i
)

( S
 s ))

Introduction to Information Retrieval
• To avoid the possibility of zeroes (e.g., if every or no relevant doc has a particular term) standard procedure is to add ½ to each of the quantities in the center four cells of the table of the previous slide. Accordingly: c
ˆ i

K ( N , df i
, S , s )
 log
( df i
 s
( s


1
2
)
1
2
)
( N
( S

 df s i


1
2
)
S
 s

1
2
)

Introduction to Information Retrieval
 If non-relevant documents are approximated by the whole collection, then u i
(prob. of occurrence in nonrelevant documents for query) is df i
/N and
 log (1– u i
)/u i
= log (N – df i
)/ df i
≈ log N/df i
= IDF!
(assuming df i small compared to N)
 p i
(probability of occurrence in relevant documents) can be estimated in various ways:
 from relevant documents if we 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)

Introduction to Information Retrieval
i
1.
Assume that p i
 p i constant over all x i in query
= 0.5 (even odds) for any given doc; in this case what is
RSV d
 x i

 q i

1 c i given ?
i u i
( 1 p i
) 1 p i u i u i
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
 Use distribution of x i documents containing x i
 p i
= |V i
| / |V| in docs in V. Let V i and u be set of i
, so
 Assume if not retrieved then not relevant
 u i
= (df i
– |V i
|) / (N – |V|)
4.
Go to 2. until converges then return ranking
25

Introduction to Information Retrieval
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: partition
V into relevant VR and non-relevant VNR
3.
Re-estimate p i and u i on the basis of these
 Or can combine new information with original guess (use
Bayesian prior): p i
( k

1 ) 
| VR i
|
|
VR |
 
 p i
(
 k )
 i containing x i
.
is the set of docs in VR
4.
Repeat, generating a succession of approximations to p i
.

Introduction to Information Retrieval
 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

Introduction to Information Retrieval
 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