probabilistic ranking

Ahmet Selman Bozkır

Introduction to conditional, total probability & Bayesian theorem
Historical background of probabilistic information retrieval
Why probabilities in IR?
Document ranking problem
Binary Independence Model

Given some event B with nonzero probability P(B) > 0
We can define conditional prob. as an event A, given B, by
The Probabilty P(A|B) simply reflects the fact that the probability of an event A may depend on a second event B. So if A and B are mutually exclusive, A  B = 

Let’s define three events: 1. A as “draw 47  resistor 2. B as “draw” a resistor with 5% 3. C as “draw” a “100  resistor P(A) = P(47  ) = 44/100 P(B) = P(5%) = 62/100 P(C) = P(100  ) = 32 /100 The joint probabilities are: P(A  B) = P(47   5%) = 28/100 P(A  C) = P(47   100  ) = 0 P(B  C) = P(5%  100  ) = 24/100 I f we use them the cond. prob. : Tolerance Resistance (  )‏ 5% 10% Total 22-  10 14 24 47-  28 26 44 100-  24 8 32 Total: 62 38 100

The probability of P(A) of any event A defined on a sample space S can be expressed in terms of cond. probabilities. Suppose we are given N mutually exclusive events B n ,n = 1,2…. N whose union equals S as ilustrated in figure
A  B n A B1 B3 B2 Bn

The definition of conditional probability applies to any two events. In particular ,let B n be one of the events defined above in the subsection on total probability. İf P(A)≠O,or, alternatively,

if P(B n )≠0, one form of Bayes’ theorem is obtained by equating these two expressions:
Another form derives from a substitution of P(A) as given:

The first attempts to develop a probabilistic theory of retrieval were made over 30 years ago [Maron and Kuhns 1960; Miller 1971], and since then there has been a steady development of the approach. There are already several operational IR systems based upon probabilistic or semiprobabilistic models.

One major obstacle in probabilistic or semiprobabilistic IR models is finding methods for estimating the probabilities used to evaluate the probability of relevance that are both theoretically sound and computationally efficient.

The first models to be based upon such assumptions were the “binary independence indexing model” and the “binary independence retrieval model

One area of recent research investigates the use of an explicit network representation of dependencies. The networks are processed by means of Bayesian inference or belief theory , using evidential reasoning techniques such as those described by Pearl 1988. This approach is an extension of the earliest probabilistic models, taking into account the conditional dependencies present in a real environment.

User Information Need Documents Document Representation Query Representation 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

Classical probabilistic retrieval model
Probability ranking principle, etc.
(Naïve) Bayesian Text Categorization
Bayesian networks for text retrieval
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.

In probabilistic information retrieval, the goal is the estimation of the probability of relevance P(R l q k , d m ) that a document d m will be judged relevant by a user with request q k . In order to estimate this probability, a large number of probabilistic models have been developed.
Typically, such a model is based on representations of queries and documents (e.g., as sets of terms); in addition to this, probabilistic assumptions about the distribution of elements of these representations within relevant and nonrelevant documents are required.
By collecting relevance feedback data from a few documents, the model then can be applied in order to estimate the probability of relevance for the remaining documents in the collection.

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)‏

For events a and b:
Bayes’ Rule
Odds:
Prior P osterior

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/R

Bayes’ Optimal Decision Rule
x is relevant iff p ( R | x ) > p ( NR | x )‏
PRP in action: Rank all documents by p ( R | x )‏

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

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

Estimate how terms contribute to relevance
How tf, df, and length influence your judgments about do things like document relevance?
One answer is the Okapi formulae (S. Robertson)‏
Combine to find document relevance probability
Order documents by decreasing probability

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

Traditionally used in conjunction with PRP
“ Binary” = Boolean : documents are represented as binary incidence vectors of terms (cf. lecture 1):
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!)‏

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:

Constant for a given query Needs estimation
Using Independence Assumption:
So :

Since x i is either 0 or 1:
Then... This can be changed (e.g., in relevance feedback)‏
Let
Assume, for all terms not occurring in the query ( q i =0 )‏

All matching terms Non-matching query terms All matching terms All query terms

Constant for each query Only quantity to be estimated for rankings
Retrieval Status Value:

Estimating RSV coefficients.
For each term i look at this table of document counts:
Estimates:
For now, assume no zero terms.

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)‏

Assume that p i constant over all x i in query
p i = 0.5 (even odds) for any given doc
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!)‏
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|)‏
Go to 2. until converges then return ranking

Guess a preliminary probabilistic description of R and use it to retrieve a first set of documents V, as above.
Interact with the user to refine the description: learn some definite members of R and NR
Reestimate p i and r i on the basis of these
Or can combine new information with original guess (use Bayesian prior):
Repeat, thus generating a succession of approximations to R .
κ is prior weight

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

In general, index terms aren’t independent
Dependencies can be complex
van Rijsbergen (1979) proposed model of simple tree dependencies
Exactly Friedman and Goldszmidt’s Tree Augmented Naive Bayes (AAAI 13, 1996)‏
Each term dependent on one other
In 1970s, estimation problems held back success of this model

What is a Bayesian network?
A directed acyclic graph
Nodes
Events or Variables
Assume values.
For our purposes, all Boolean
Links
model direct dependencies between nodes

a b c
Bayesian networks model causal relations between events
Inference in Bayesian Nets:
Given probability distributions
for roots and conditional
probabilities can compute
apriori probability of any instance
Fixing assumptions (e.g., b
was observed) will cause
recomputation of probabilities
For more information see: R.G. Cowell, A.P. Dawid, S.L. Lauritzen, and D.J. Spiegelhalter. 1999. Probabilistic Networks and Expert Systems . Springer Verlag. J. Pearl. 1988. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan-Kaufman. p(c|ab) for all values for a,b,c p(a)‏ p(b)‏ Conditional dependence

Gloom (g)‏ Finals (f)‏ No Sleep (n)‏ Triple Latte (t)‏ Project Due (d)‏

Goal
Given a user’s information need (evidence), find probability a doc satisfies need
Retrieval model
Model docs in a document network
Model information need in a query network

I - goal node Document Network Query Network Large, but Compute once for each document collection Small, compute once for every query d1 d n d2 t1 t2 t n r1 r2 r3 r k d i - documents t i - document representations r i - “concepts” I q2 q1 c m c2 c1 c i - query concepts q i - high-level concepts

Construct Document Network (once !)‏
For each query
Construct best Query Network
Attach it to Document Network
Find subset of d i ’s which maximizes the probability value of node I (best subset).
Retrieve these d i ’s as the answer to query.

d 1 d 2 r 1 r 3 c 1 c 3 q 1 q 2 i r 2 c 2 Document Network Query Network Documents Terms/Concepts Concepts Query operators ( AND/OR/NOT )‏ Information need

Prior doc probability P ( d ) = 1 /n
P ( r | d )‏
within-document term frequency
tf  idf - based
P ( c | r )‏
1-to-1
thesaurus
P ( q | c ) : canonical forms of query operators
Always use things like AND and NOT – never store a full CPT*
*conditional probability table

Hamlet Macbeth reason double reason two OR NOT User query trouble trouble Document Network Query Network

Prior probs don’t have to be 1/n .
“ User information need” doesn’t have to be a query - can be words typed, in docs read, any combination …
Phrases, inter-document links
Link matrices can be modified over time.
User feedback.
The promise of “personalization”

Document network built at indexing time
Query network built/scored at query time
Representation:
Link matrices from docs to any single term are like the postings entry for that term
Canonical link matrices are efficient to store and compute
Attach evidence only at roots of network
Can do single pass from roots to leaves

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probabilistic ranking

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