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1 CS 430 / INFO 430 Information Retrieval Lecture 12 Probabilistic Information Retrieval

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2 Course Administration Discussion Class 6 This is a long paper. The discussion class will be on Sections 1 and 2 (up to page 16). Assignment 2 Remember to check the FAQ for hints and answers to questions that people have raised.

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3 Course Administration Midterm Examination See the Examinations page on the Web site On Wednesday October 11, 7:30-9:00 p.m., Phillips 203, instead of the discussion class. The topics to be are examined are all lectures and discussion class readings before the midterm break. See the Web site for a sample paper. Laptops may be used to store course materials and your notes, but for no other purposes. Hand calculators are allowed. No communication. No other electronic devices.

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4 Three Approaches to Information Retrieval Many authors divide the classical methods of information retrieval into three categories: Boolean (based on set theory) Vector space (based on linear algebra) Probabilistic (based on Bayesian statistics) In practice, the latter two have considerable overlap.

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5 Probability revision: independent random variables and conditional probability Let a, b be two events, with probability P(a) and P(b). Independent events The events a and b are independent if and only if: P(a b) = P(b) P(a) Conditional probability P(a | b) is the probability of a given b, also called the conditional probability of a given b. Conditional independence The events a 1,..., a n are conditionally independent if and only if: P(a i | a j ) = P(a i ) for all i and j

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6 Probability: Conditional probability P(a) = x + y P(b) = w + x P(a | b) = x / (w + x) P(a | b) P(b) = P(a b) = P(b | a) P(a) a b w z y x a b where a is the event not a

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7 Example: independent random variables and conditional probability Independent a and b are the results of throwing two dice P(a=5 | b=3) = P(a=5) = 1 / 6 Not independent a and b are the results of throwing two dice t is the sum of the two dice t = a + b P(t=8 | a=2) = 1 / 6 P(t=8 | a=1) = 0

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8 Probability Theory -- Bayes Theorem Notation Let a, b be two events. P(a | b) is the probability of a given b Bayes Theorem P(a | b) = Derivation P(a | b) P(b) = P(a b) = P(b | a) P(a) P(b | a) P(a) P(b)P(b)

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9 Probability Theory -- Bayes Theorem Terminology used with Bayes Theorem P(a | b) = P(a) is called the prior probability of a P(a | b) is called the posterior probability of a given b P(b | a) P(a) P(b)P(b)

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10 Example of Bayes Theorem Example a Weight over 200 lb. b Height over 6 ft. Over 200 lb Over 6 ft w z y x P(a | b) = x / (w+x) = x / P(b) P(b | a) = x / (x+y) = x / P(a) x is P(a b)

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11 Probability Ranking Principle "If a reference retrieval system’s response to each request is a ranking of the documents in the collections in order of decreasing probability of usefulness to the user who submitted the request, where the probabilities are estimated as accurately a possible on the basis of whatever data is made available to the system for this purpose, then the overall effectiveness of the system to its users will be the best that is obtainable on the basis of that data." W.S. Cooper

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12 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

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13 Probabilistic Principle Basic concept: The probability that a document is relevant to a query is assumed to depend on the terms in the query and the terms used to index the document, only. Given a user query q, the ideal answer set, R, is the set of all relevant documents. Given a user query q and a document d in the collection, the probabilistic model estimates the probability that the user will find d relevant, i.e., that d is a member of R.

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14 Probabilistic Principle Initial probabilities: Given a query q and a document d the model needs an estimate of the probability that the user finds d relevant. i.e., P(R | d). Similarity measure: S(d, q), the similarity of d to q, is the ratio: probability that d is relevant to q probability that d is not relevant to q This measure runs from near zero, if the probability is small that the document is relevant, to large as the probability of relevance approaches one. In practice it is often convenient to use S' = log(S).

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15 Probabilistic Principle S (d, q) = = by Bayes Theorem = x k where k is constant P(R | d) P(d | R) P(R) P(d | R) P(d | R) is the probability of randomly selecting d from R.

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16 Binary Independence Retrieval Model (BIR) Let x = (x 1, x 2,... x n ) be the term incidence vector for d. x i = 1 if term i is in the document and 0 otherwise. We estimate P(d | R) by P(x | R) If the index terms are independent P(x | R) = P(x 1 R) P(x 2 R)... P(x n R) = P(x 1 | R) P(x 2 | R)... P(x n | R) = ∏ P(x i | R) {This is known as the Naive Bayes probabilistic model.}

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17 Binary Independence Retrieval Model (BIR) ∏ P(x i | R) Since the x i are either 0 or 1, this can we written: P(x i = 1 | R) P(x i = 0 | R) x i = 1 P(x i = 1 | R) x i = 0 P(x i = 0 | R) S = k ∏ S(d, q) = k

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18 Binary Independence Retrieval Model (BIR) For terms that appear in the query let p i = P(x i = 1 | R) r i = P(x i = 1 | R) For terms that do not appear in the query assume p i = r i p i 1 - p i x i = q i = 1 r i x i = 0, q i = 1 1 - r i p i (1 - r i ) 1 - p i x i = q i = 1 r i (1 - p i ) q i = 1 1 - r i S = k = k ∏ constant for a given query terms with q i = 0 are p i /r i equal to 1

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19 Binary Independence Retrieval Model (BIR) Taking logs and ignoring factors that are constant for a given query, we have: p i (1 - r i ) (1 - p i ) r i where the summation is taken over those terms that appear in both the query and the document. = ∑ log { } S' = log(S)

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20 Relationship to Term Vector Space Model Suppose that, in the term vector space, document d is represented by a vector that has component in dimension i of: p i (1 - r i ) (1 - p i ) r i and the query q is represented by a vector with value 1 in each dimension that corresponds to a term in the vector. Then the Binary Independence Retrieval similarity, S' is the inner product of these two vectors. Thus this approach can be considered as a probabilistic way of determining term weights in the vector space model { } log

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21 Practical Application The probabilistic model is an alternative to the term vector space model. The Binary Independence Retrieval similarity measure is used instead of the cosine similarity measure to rank all documents against the query q. Techniques such as stoplists and stemming can be used with either model. Variations to the model result in slightly different expressions for the similarity measure.

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22 Practical Application Early uses of probabilistic information retrieval were based on relevance feedback R is a set of documents that are guessed to be relevant and R the complement of R. 1. Guess a preliminary probabilistic description of R and use it to retrieve a first set of documents. 2. Interact with the user to refine the description of R (relevance feedback). 3.Repeat, thus generating a succession of approximations to R.

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23 Initial Estimates of P(x i | R) Initial guess, with no information to work from: p i = P(x i | R) = c r i = P(x i | R) = n i / N where: c is an arbitrary constant, e.g., 0.5 n i is the number of documents that contain x i N is the total number of documents in the collection

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24 Initial Similarity Estimates p i (1 - r i ) (1 - p i ) r i = ∑ log{(N - n i )/n i } where the summation is taken over those terms that appear in both the query and the document. S' (d, q) = ∑ log { } With these assumptions:

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25 Improving the Estimates of P(x i | R) Human feedback -- relevance feedback Automatically (a) Run query q using initial values. Consider the t top ranked documents. Let s i be the number of these documents that contain the term x i. (b) The new estimates are: p i = P(x i | R) = s i / t r i = P(x i | R) = (n i - s i ) / (N - t)

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26 Discussion of Probabilistic Model Advantages Based on firm theoretical basis Disadvantages Initial definition of R has to be guessed. Weights ignore term frequency Assumes independent index terms (as does vector model)

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