Chapter 2 Modeling

1. Chapter 2 Modeling Modern Information Retrieval by R. Baeza-Yates and B. Ribeiro-Neto

2. Introduction • Traditional information retrieval systems usually adopt index terms to index and retrieve documents. • An index term is a keyword (or group of related words) which has some meaning of its own (usually a noun). • Advantages: • Simple • The semantic of the documents and of the user information need can be naturally expressed through sets of index terms.

3. IR Models • Ranking algorithms are at the core of information retrieval systems (predicting which documents are relevant and which are not).

4. A taxonomy of information retrieval models Classic Models Set Theoretic U S E R T A S K Boolean Vector Probabilistic Fuzzy Extended Boolean Retrieval: Ad hoc Filtering Algebraic Structured Models Generalized Vector Lat. Semantic Index Neural Networks Non-overlapping lists Proximal Nodes Browsing Probabilistic Browsing Inference Network Belief Network Flat Structured Guided Hypertext

5. Index Terms Full Text Full Text+ Structure Retrieval Classic Set Theoretic Algebraic Probabilistic Classic Set Theoretic Algebraic Probabilistic Structured Browsing Flat Flat Hypertext Structure Guided Hypertext Figure 2.2 Retrieval models most frequently associated with distinct combinations of a document logical view and a user task.

6. Retrieval : Ad hoc and Filtering • Ad hoc (Search): The documents in the collection remain relatively static while new queries are submitted to the system. • Routing (Filtering): The queries remain relatively static while new documents come into the system

7. A formal characterization of IR models • D : A set composed of logical views (or representation) for the documents in the collection. • Q : A set composed of logical views (or representation) for the user information needs (queries). • F : A framework for modeling document representations, queries, and their relationships. • R(qi, dj) : A ranking function which defines an ordering among the documents with regard to the query.

8. Define • ki : A generic index term • K : The set of all index terms {k1,…,kt} • wi,j : A weight associated with index term ki of a document dj • gi: A function returns the weight associated with ki in any t-dimensoinal vector( gi(dj)=wi,j )

9. Classic IR Model • Basic concepts : Each document is described by a set of representative keywords called index terms. • Assign a numerical weights to distinct relevance between index terms.

10. Boolean model • Binary decision criterion • Data retrieval model • Advantage • clean formalism, simplicity • Disadvantage • It is not simple to translate an information need into a Boolean expression. • exact matching may lead to retrieval of too few or too many documents

11. Example: • Can be represented as a disjunction of conjunction vectors (in DNF). • Q= qa(qbqc)=(1,1,1)  (1,1,0)  (1,0,0) • Formal definition • For the Boolean model, the index term weight are all binary. • A query is a conventional Boolean expression, which can be transformed to a disjunctive normal form if (qcc )(ki, wi,j=gi(qcc))

12. Vector model • Assign non-binary weights to index terms in queries and in documents. => TFxIDF • Compute the similarity between documents and query. => Sim(Dj, Q) • More precise than Boolean model.

13. The IR problem  A clustering problem • We think of the documents as a collection C of objects and think of the user query as a specification of a set A of objects. • Intra-cluster: • What are the features which better describe the objects in the set A? • Inter-cluster: • What are the features which better distinguish the objects in the set A?

14. Idea for TFxIDF • TF: inter-clustering similarity is quantified by measuring the raw frequency of a term ki inside a document dj, such term frequency is usually referred to as the tf factor and provides one measure of how well that term describes the document contents. • IDF : inter-clustering similarity is quantified by measuring the inverse of the frequency of a term ki among the documents in the collection.This frequency is often referred to as the inverse document frequency.

15. Vector Model (1/4) • Index terms are assigned positive and non-binary weights. • The index terms in the query are also weighted. • Term weights are used to compute the degree of similarity between documents and the user query. Then, retrieved documents are sorted in decreasing order.

16. Vector Model (2/4) • Degree of similarity

17. Vector Model (3/4) • Definition • normalized frequency • inverse document frequency • term-weighting schemes • query-term weights

18. Vector Model (4/4) • Advantages • its term-weighting scheme improves retrieval performance • its partial matching strategy allows retrieval of documents that approximate the query conditions • its cosine ranking formula sorts the documents according to their degree of similarity to the query • Disadvantage • The assumption of mutual independence between index terms

19. Orthogonal v1: (1,0) (1,0) v2: (1,1) (0,1) v3: (0,1) (-1,1) Cos(v1,v2)=1/2 Cos(v2,v3)=1/2 Cos(v1,v3)=0 Cos(v1,v2)=0 Cos(v2,v3)=1/2 Cos(v1,v3)=-1/2 v2 v3 v1

20. Probabilistic Model (1/6) • Introduced by Roberston and Sparck Jones, 1976 • Also called binary independence retrieval (BIR) model • Idea: Given a user query q, and the ideal answer set of the relevant documents, the problem is to specify the properties for this set. • i.e. the probabilistic model tries to estimate the probability that the user will find the document dj relevant with ratio P(dj relevant to q)/P(dj nonrelevant to q)

21. Probabilistic Model (2/6) • Definition • All index term weights are all binary i.e., wi,j {0,1} • Let R be the set of documents know to be relevant to query q • Let be the complement of R • Let be the probability that the document dj is relevant to the query q • Let be the probability that the document dj is nonelevant to query q

22. Probabilistic Model (3/6) • The similarity sim(dj,q) of the document dj to the query q is defined as the ratio • Using Bayes’ rule, • P(R) stands for the probability that a document randomly selected from the entire collection is relevant • stands for the probability of randomly selecting the document dj from the set R of relevant documents

23. Probabilistic Model (4/6) • Assuming independence of index terms and given q=(d1, d2, …, dt),

24. Probabilistic Model (5/6) • Pr(ki |R) stands for the probability that the index term ki is present in a document randomly selected from the set R • stands for the probability that the index term ki is not present in a document randomly selected from the set R

25. Probabilistic Model (6/6)

26. Estimation of Term Relevance In the very beginning: Next, the ranking can be improved as follows: For small values for V Let V be a subset of the documents initially retrieved

27. Alternative Set Theoretic Models Fuzzy Set Model Extended Boolean Model

28. Fuzzy Theory • A fuzzy subset A of a universe U is characterized by a membership function uA: U{0,1} which associates with each element uU a number uA • Let A and B be two fuzzy subsets of U,

29. Fuzzy Information Retrieval • Using a term-term correlation matrix • Define a fuzzy set associated to each index term ki. • If a term klis strongly related to ki, that is ci,l ~1, then ui(dj)~1 • If a term klis loosely related to ki, that is ci,l ~0, then ui(dj)~0

30. Example • Disjunctive Normal Form

31. Algebraic Sum and Product • The degree of membership in a disjunctive fuzzy set is computed using an algebraic sum, instead of max function. • The degree of membership in a conjunctive fuzzy set is computed using an algebraic product, instead of min function. • More smooth than max and min functions.

32. Alternative Algebraic Models Generalized Vector Space Model Latent Semantic Model

33. Latent Semantic Indexing (1/5) • Let A be a term-document association matrix with m rows and n columns. • Latent semantic indexing decomposes A using singular value decompositions. • U (mm) is the matrix of eigenvectors derived from the term-to-term correlation matrix (AAT) • V (nn) is the matrix of eigenvectors derived from the the document-to-document matrix (ATA) •  is an mn diagonal matrix of singular values, where rmin(t,N) is the rank of A.

34. Latent Semantic Indexing (2/5) • Consider now only the s largest singular values of S, and their corresponding columns in U and V. (The remaining singular values of  are deleted). • The resultant matrix As (rank s) is closest to the original matrix A in the least square sense. • s<r is the dimensionality of a reduced concept space.

35. Latent Semantic Indexing (3/5) • The selection of s attempts to balance two opposing effects: • s should be large enough to allow fitting all the structure in the real data • s should be small enough to allow filtering out all the non-relevant representational details • Us={u1, u2, …, us} are the s principle components of column space (document space) Rm • Vs={v1, v2, …, vs} are the s principle components of row space (term space) Rn

36. Latent Semantic Indexing (4/5) • Consider the relationship between any two documents • is the projected vector for document di (RmRs) • is the projected vector for term vector ti (RnRs)

37. Latent Semantic Indexing (5/5) • To rank documents with regard to a given user query, we model the query as a pseudo-document in the matrix A (original). • Assume the query is modeled as the document with number k. • Then the kth row in the matrix provides the ranks of all documents with respect to this query.

38. Speedup • The matrix vector multiplication requires a total of Nt scalar multiplications. • While requires only (n+m)s scalar multiplications.

39. Alternative Probabilistic Model Bayesian Networks Inference Network Model Belief Network Model

40. Bayesian Network • Let xi be a node in a Bayesian network G and xi be the set of parent nodes of xi. • The influence of xi on xi can be specified by any set of functions that satisfy: • P(x1,x2,x3,x4,x5)=P(x1)P(x2|x1)P(x3|x1)P(x4|x2,x3)P(x5|x3)

41. Belief Network Model (1/6) • The probability space The set K={k1, k2, …, kt} is the universe. To each subset u is associated a vector such that gi()=1  kiu. • Random variables • To each index term kiis associated a binary random variable.

42. Belief Network Model (2/6) • Concept space • A document dj is represented as a concept composed of the terms used to index dj. • A user query q is also represented as a concept composed of the terms used to index q. • Both user query and document are modeled as subsets of index terms. • Probability distribution P over K

43. Belief Network Model (3/6) • A query is modeled as a network node • This variable is set to 1 whenever q completely covers the concept space K • P(q) computes the degree of coverage of the space K by q • A document dj is modeled as a network node • This random variable is 1 to indicate that dj completely covers the concept space K • P(dj) computes the degree of coverage of the space K by dj

44. Belief Network Model (4/6)

45. Belief Network Model (5/6) • Assumption • P(dj |q) is adopted as the rank of the document dj with respect to the query q.

46. Belief Network Model (6/6) • Specify the conditional probabilities as follows: • Thus, the belief network model can be tuned to subsume the vector model.

47. Comparison • Belief network model • Belief network model is based on set-theoretic view • Belief network model provides a separation between the document and the query • Belief network model is able to reproduce any ranking strategy generated by the inference network model • Inference network model