Information Retrieval Chap. 02: Modeling - Part 2 Slides from the text book author, modified by L N Cassel September 2003.

Information Retrieval Chap. 02: Modeling - Part 2 Slides from the text book author, modified by L N Cassel September 2003

Probabilistic Model n Objective: to capture the IR problem using a probabilistic framework n Given a user query, there is an ideal answer set n Querying as specification of the properties of this ideal answer set (clustering) n But, what are these properties? n Guess at the beginning what they could be (i.e., guess initial description of ideal answer set) n Improve by iteration

Probabilistic Model n An initial set of documents is retrieved somehow n User inspects these docs looking for the relevant ones (in truth, only top 10-20 need to be inspected) n IR system uses this information to refine description of ideal answer set n By repeating this process, it is expected that the description of the ideal answer set will improve n Have always in mind the need to guess at the very beginning the description of the ideal answer set n Description of ideal answer set is modeled in probabilistic terms

Probabilistic Ranking Principle n Given a user query q and a document dj, the probabilistic model tries to estimate the probability that the user will find the document dj interesting (i.e., relevant). u The model assumes that this probability of relevance depends on the query and the document representations only. u Ideal answer set is referred to as R and should maximize the probability of relevance. Documents in the set R are predicted to be relevant. n But, u how to compute probabilities? u what is the sample space?

The Ranking n Probabilistic ranking computed as: sim(q,dj) = P(dj relevant-to q) / P(dj non-relevant-to q) This is the odds of the document dj being relevant Taking the odds minimizes the probability of an erroneous judgement n Definition: wij  {0,1} P(R | dj) : probability that given document is relevant P(  R | dj) : probability document is not relevant

The Ranking n sim(dj,q) = P(R | dj) / P(  R | dj) = [P( dj | R) * P(R)] [P( dj |  R) * P(  R)] ~ P( dj | R) P( dj |  R) n P( dj | R) : probability of randomly selecting the document dj from the set R of relevant documents n P(R): probability that a document selected at random from the whole set of documents is relevant. P(R) and P(  R) are the same.

Improving the Initial Ranking n sim(dj,q)~ ~  wiq * wij * (log P(ki | R) + log P(ki |  R) ) P(  ki | R) P(  ki |  R) n Let u V : set of docs initially retrieved u Vi : subset of docs retrieved that contain ki n Reevaluate estimates: u P(ki | R) = Vi V u P(ki |  R) = ni - Vi N - V n Repeat recursively

Pluses and Minuses n Advantages: u Docs ranked in decreasing order of probability of relevance n Disadvantages: u need to guess initial estimates for P(ki | R) u method does not take into account tf and idf factors

Brief Comparison of Classic Models n Boolean model does not provide for partial matches and is considered to be the weakest classic model n Salton and Buckley did a series of experiments that indicate that, in general, the vector model outperforms the probabilistic model with general collections n This seems also to be the view of the research community

Set Theoretic Models n The Boolean model imposes a binary criterion for deciding relevance n The question of how to extend the Boolean model to accomodate partial matching and a ranking has attracted considerable attention in the past n We discuss now two set theoretic models for this: u Fuzzy Set Model u Extended Boolean Model (We will not discuss because of time limitations)

Fuzzy Set Model n Queries and docs represented by sets of index terms: matching is approximate from the start n This vagueness can be modeled using a fuzzy framework, as follows: u with each term is associated a fuzzy set u each doc has a degree of membership in this fuzzy set n This interpretation provides the foundation for many models for IR based on fuzzy theory n In here, we discuss the model proposed by Ogawa, Morita, and Kobayashi (1991)

Fuzzy Set Theory n Framework for representing classes whose boundaries are not well defined n Key idea is to introduce the notion of a degree of membership associated with the elements of a set n This degree of membership varies from 0 to 1 and allows modeling the notion of marginal membership n Thus, membership is now a gradual notion, contrary to the crispy notion enforced by classic Boolean logic

Fuzzy Set Theory n Definition u A fuzzy subset A of U is characterized by a membership function  (A,u) : U  [0,1] which associates with each element u of U a number  (u) in the interval [0,1] n Definition u Let A and B be two fuzzy subsets of U. Also, let ¬A be the complement of A. Then, F  (¬A,u) = 1 -  (A,u) F  (A  B,u) = max(  (A,u),  (B,u)) F  (A  B,u) = min(  (A,u),  (B,u))

Fuzzy Information Retrieval n Fuzzy sets are modeled based on a thesaurus n This thesaurus is built as follows: u Let be a term-term correlation matrix u Let c(i,l) be a normalized correlation factor for (ki,kl): c(i,l) = n(i,l) ni + nl - n(i,l) ni: number of docs which contain ki nl: number of docs which contain kl n(i,l): number of docs which contain both ki and kl n We now have the notion of proximity among index terms. c

Fuzzy Information Retrieval n The correlation factor c(i,l) can be used to define fuzzy set membership for a document dj as follows:  (i,j) = 1 -  (1 - c(i,l)) ki  dj  (i,j) : membership of doc dj in fuzzy subset associated with ki n The above expression computes an algebraic sum over all terms in the doc dj (shown here as complement of negated algebraic product) n A doc dj belongs to the fuzzy set for ki, if its own terms are associated with ki

Fuzzy Information Retrieval n  (i,j) = 1 -  (1 - c(i,l)) ki  dj  (i,j) : membership of doc dj in fuzzy subset associated with ki n If doc dj contains a term kl which is closely related to ki, we have u c(i,l) ~ 1 (correlation between terms i, I is high) u  (i,j) ~ 1 (membership of term i in document j is high) u index ki is a good fuzzy index for document j.

Fuzzy IR: An Example n q = ka  (kb   kc) n = (1,1,1) + (1,1,0) + (1,0,0) = cc1 + cc2 + cc3 n  (q,dj) =  (cc1+cc2+cc3,j) = 1 - (1 -  (a,j)  (b,j)  (c,j)) * (1 -  (a,j)  (b,j) (1-  (c,j))) * (1 -  (a,j) (1-  (b,j)) (1-  (c,j))) cc1 cc3 cc2 KaKb Kc qdnf (1,1,1) (1,1,0) (1,0,0) Exercise - put some numbers in the areas and calculate the actual value of  (q,dj)

Fuzzy Information Retrieval n Fuzzy IR models have been discussed mainly in the literature associated with fuzzy theory n Experiments with standard test collections are not available n Difficult to compare at this time

Alternative Probabilistic Models n Probability Theory u Semantically clear u Computationally clumsy n Why Bayesian Networks? u Clear formalism to combine evidences u Modularize the world (dependencies) u Bayesian Network Models for IR F Inference Network (Turtle & Croft, 1991) F Belief Network (Ribeiro-Neto & Muntz, 1996)

Bayesian Inference Basic Axioms: F 0 < P(A) < 1 ; F P(sure)=1; F P(A V B)=P(A)+P(B) if A and B are mutually exclusive

Bayesian Inference Other formulations F P(A)=P(A  B)+P(A  ¬B) F P(A)=   i P(A  B i ), where B i,  i is a set of exhaustive and mutually exclusive events F P(A) + P(¬A) = 1 F P(A|K) belief in A given the knowledge K F if P(A|B)=P(A), we say:A and B are independent F if P(A|B  C)= P(A|C), we say: A and B are conditionally independent, given C F P(A  B)=P(A|B)P(B) F P(A)=   i P(A | B i )P(B i )

Bayesian Inference Bayes’ Rule : the heart of Bayesian techniques P(H|e) = P(e|H)P(H) / P(e) Where, H : a hypothesis and e is an evidence P(H) : prior probability P(H|e) : posterior probability P(e|H) : probability of e if H is true P(e) : a normalizing constant, then we write: P(H|e) ~ P(e|H)P(H)

Bayesian Networks Definition: Bayesian networks are directed acyclic graphs (DAGS) in which the nodes represent random variables, the arcs portray causal relationships between these variables, and the strengths of these causal influences are expressed by conditional probabilities.

Bayes - resource n Look at http://members.aol.com/johnp71/bayes.html

Bayesian Networks y i : parent nodes (in this case, root nodes) x : child node y i cause x Y the set of parents of x The influence of Y on x can be quantified by any function F(x,Y) such that   x F(x,Y) = 1 0 < F(x,Y) < 1 For example, F(x,Y)=P(x|Y) y1y2 yt x …

Bayesian Networks Given the dependencies declared in a Bayesian Network, the expression for the joint probability can be computed as a product of local conditional probabilities, for example, P(x 1, x 2, x 3, x 4, x 5 )= P(x 1 ) P(x 2 | x 1 ) P(x 3 | x 1 ) P(x 4 | x 2, x 3 ) P(x 5 | x 3 ). P(x 1 ) : prior probability of the root node x1 x2x3 x4 x5

Bayesian Networks In a Bayesian network each variable x is conditionally independent of all its non-descendants, given its parents. For example: P(x 4, x 5 | x 2, x 3 )= P(x 4 | x 2, x 3 ) P( x 5 | x 3 ) x1 x2 x3 x4x5

Inference Network Model n Epistemological view of the IR problem n Random variables associated with documents, index terms and queries n A random variable associated with a document d j represents the event of observing that document

Inference Network Model Nodes documents (d j ) index terms (k i ) queries (q, q 1, and q 2 ) user information need (I) Edges from d j to its index term nodes k i indicate that the observation of d j increase the belief in the variables k i.

Inference Network Model d j has index terms k 2, k i, and k t q has index terms k 1, k 2, and k i q 1 and q 2 model boolean formulation q 1 =((k 1  k 2 ) v k i ); I = (q v q 1 )

Inference Network Model Definitions: k 1, d j,, and q random variables. k=(k 1, k 2,...,k t ) a t-dimensional vector k i,  i  {0, 1}, then k has 2 t possible states d j,  j  {0, 1};  q  {0, 1} The rank of a document d j is computed as P(q  d j ) q and d j, are short representations for q=1 and d j =1 (d j stands for a state where d j = 1 and  l  j  d l =0, because we observe one document at a time)

Inference Network Model P(q  d j )=   k P(q  d j | k) P(k) =   k P(q  d j  k) =   k P(q | d j  k) P(d j  k) =   k P(q | k) P(k | d j ) P( d j ) P(¬(q  d j )) = 1 - P(q  d j )

Inference Network Model As the instantiation of d j makes all index term nodes mutually independent P(k | d j ) can be a product,then P(q  d j )=   k [ P(q | k) (   i|g i (k)=1 P(k i | d j ) ) (   i|g i (k)=0 P(¬k i | d j ) ) P( d j ) ] remember that: g i (k)= 1 if k i =1 in the vector k 0 otherwise

Inference Network Model The prior probability P(d j ) reflects the probability associated to the event of observing a given document d j F Uniformly for N documents P(d j ) = 1/N P(¬d j ) = 1 - 1/N F Based on norm of the vector d j P(d j )= 1/|d j | P(¬d j ) = 1 - 1/|d j |

Inference Network Model For the Boolean Model P(d j ) = 1/N P(k i | d j ) = 1 if g i (d j )=1 0 otherwise P(¬k i | d j ) = 1 - P(k i | d j )  only nodes associated with the index terms of the document d j are activated

Inference Network Model For the Boolean Model 1 if  q cc | (q cc  q dnf )  (  k i, g i (k)= g i (q cc ) P(q | k) = 0 otherwise P(¬q | k) = 1 - P(q | k)  one of the conjunctive components of the query must be matched by the active index terms in k

Inference Network Model For a tf-idf ranking strategy P(d j )= 1 / |d j | P(¬d j ) = 1 - 1 / |d j |  prior probability reflects the importance of document normalization

Inference Network Model For a tf-idf ranking strategy P(k i | d j ) = f i,j P(¬k i | d j )= 1- f i,j  the relevance of the a index term k i is determined by its normalized term-frequency factor f i,j = freq i,j / max freq l,j

Inference Network Model For a tf-idf ranking strategy Define a vector k i given by k i = k | ((g i (k)=1)  (  j  i g j (k)=0))  in the state k i only the node k i is active and all the others are inactive

Inference Network Model For a tf-idf ranking strategy idf i if k = k i  g i (q)=1 P(q | k) = 0 if k  k i v g i (q)=0 P(¬q | k) = 1 - P(q | k)  we can sum up the individual contributions of each index term by its normalized idf

Inference Network Model For a tf-idf ranking strategy Applying the previous probabilities we have P(q  d j ) = C j (1/|d j |)   i [ f i,j idf i (1/(1- f i,j )) ]  C j vary from document to document  the ranking is distinct of the one provided by the vector model

Inference Network Model Combining evidential source Let I = q v q 1 P(I  d j )=   k P(I | k) P(k | d j ) P( d j ) =   k [1 - P(¬q|k)P(¬q 1 | k)] P(k| d j ) P( d j )  it might yield a retrieval performance which surpasses the retrieval performance of the query nodes in isolation (Turtle & Croft)

Belief Network Model n As the Inference Network Model u Epistemological view of the IR problem u Random variables associated with documents, index terms and queries n Contrary to the Inference Network Model u Clearly defined sample space u Set-theoretic view u Different network topology

Belief Network Model The Probability Space Define: K={k 1, k 2,...,k t } the sample space (a concept space) u  K a subset of K (a concept) k i an index term (an elementary concept) k=(k 1, k 2,...,k t ) a vector associated to each u such that g i (k)=1  k i  u k i a binary random variable associated with the index term k i, (k i = 1  g i (k)=1  k i  u)

Belief Network Model A Set-Theoretic View Define: a document d j and query q as concepts in K a generic concept c in K a probability distribution P over K, as P(c)=   u P(c|u) P(u) P(u)=(1/2) t P(c) is the degree of coverage of the space K by c

Belief Network Model Network topology query side document side

Belief Network Model Assumption P(d j |q) is adopted as the rank of the document d j with respect to the query q. It reflects the degree of coverage provided to the concept d j by the concept q.

Belief Network Model The rank of d j P(d j |q) = P(d j  q) / P(q) ~ P(d j  q) ~   u P(d j  q | u) P(u) ~   u P(d j | u) P(q | u) P(u) ~   k P(d j | k) P(q | k) P(k)

Belief Network Model For the vector model Define Define a vector k i given by k i = k | ((g i (k)=1)  (  j  i g j (k)=0))  in the state k i only the node k i is active and all the others are inactive

Bayesian Network Models Comparison F Inference Network Model is the first and well known F Belief Network adopts a set-theoretic view F Belief Network adopts a clearly define sample space F Belief Network provides a separation between query and document portions F Belief Network is able to reproduce any ranking produced by the Inference Network while the converse is not true (for example: the ranking of the standard vector model)

Bayesian Network Models Computational costs F Inference Network Model one document node at a time then is linear on number of documents F Belief Network only the states that activate each query term are considered F The networks do not impose additional costs because the networks do not include cycles.

Bayesian Network Models Impact The major strength is net combination of distinct evidential sources to support the rank of a given document.

Information Retrieval Chap. 02: Modeling - Part 2 Slides from the text book author, modified by L N Cassel September 2003.

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Information Retrieval Chap. 02: Modeling - Part 2 Slides from the text book author, modified by L N Cassel September 2003.

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