Ahmet Selman Bozkr Probabilistic Information Retrieval Models Systems

Ahmet Selman Bozkır Probabilistic Information Retrieval Models & Systems

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

Conditioanal Probability � � 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 =

Conditioanal Probability Tolerance Resistance ( ) 5% 10% Total 22 - 10 14 24 47 - 28 26 44 100 - 24 8 32 Total: 62 38 100 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. : 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

Total Probability � 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 Bn , n = 1, 2…. N whose union equals S as ilustrated in figure A Bn B 1 B 3 B 2 A Bn

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

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

Historical Background of PIR � � 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 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.

Why probabilities in IR? User Information Need Query Representation Understanding of user need is uncertain How to match? Document s 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?

Probabilistic IR topics � 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.

Introduction � In probabilistic information retrieval, the goal is the estimation of the probability of relevance P(R l qk, dm) that a document dm will be judged relevant by a user with request qk. 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.

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)

Recall a few probability basics For events a and b: Bayes’ Rule Prior Posterior Odds:

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 p(x|R), p(x|NR) - probability that if a relevant (non-relevant) document is retrieved, it is x.

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

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

Probabilistic Retrieval Strategy �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

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 (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!)

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 • Using Independence Assumption: • So : Needs estimation

Binary Independence Model • Since xi is either 0 or 1: • Let • Assume, for all terms not occurring in the query (qi=0) Then. . . This can be changed (e. g. , in relevance feedback)

Binary Independence Model All matching terms Non-matching query terms All query terms

Binary Independence Model Constant for each query • Retrieval Status Value: Only quantity to be estimated for rankings

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

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. 2. 3. Assume that pi constant over all xi in query pi = 0. 5 (even odds) for any given doc V is fixed size set of highest ranked documents on this model (note: now a bit like tf. idf!) Determine guess of relevant document set: We need to improve our guesses for pi and ri, so 4. Use distribution of xi in docs in V. Let Vi be set of documents containing xi ▪ pi = |Vi| / |V| Assume if not retrieved then not relevant ▪ ri = (ni – |Vi|) / (N – |V|) Go to 2. until converges then return ranking

Probabilistic Relevance Feedback 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 pi and ri on the basis of these 1. 4. Or can combine new information with original guess (use Bayesian prior): 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 Exactly Friedman and Goldszmidt’s Tree Augmented Naive Bayes (AAAI 13, 1996) Each term dependent on one other In 1970 s, estimation problems held back success of this model

Bayesian Networks for Text Retrieval (Turtle and Croft 1990) 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

Bayesian Networks • Bayesian networks model causal relations between events a b p(a) c p(c|ab) for all values for a, b, c p(b) Conditional dependence • 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.

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

Model for Text Retrieval �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

Bayesian Nets for IR: Idea Document Network di -documents d 1 d 2 ti. Large, - document but representations t 1 t 2 ri. Compute - “concepts” once for each document collection r 1 r 2 r 3 c 1 c 2 q 1 dn tn rk ci - query concepts cm Small, compute once for every query qi - high-level concepts q 2 Query Network I I - goal node

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

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

Link matrices and probabilities 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

Example: “reason trouble –two” Hamlet Macbeth reason trouble OR double two NOT User query Document Network Query Network

Extensions �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”

Computational details �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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