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Language Models Naama Kraus (Modified by Amit Gross) Slides are based on Introduction to Information Retrieval Book by Manning, Raghavan and Schütze

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IR approaches Boolean retrieval – Boolean constrains of term occurrences in documents – no ranking Vector space model – Queries and vectors are represented as vectors in a high dimensional space – Notions of similarity (cosine similarity) implying ranking Probabilistic model – Rank documents by the probability P(R|d,q) – Estimate P(R|d,q) using relevance feedback technique Language Models – today’s class

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Intuition Users who try to think of a good query, think of words that are likely to appear in relevant documents Language model approach: A document is a good match to a query, if the document model is likely to generate the query – If document contains query words often

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Illustration Language Model document query

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Traditional language model Finite automata Generative model I wish I wish I wish I wish I wish …… The language of the automaton: the full set of strings that it can generate

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Probabilistic language model Each node has a probability distribution over generating different terms A language model is a function that puts a probability measure over strings drawn from some vocabulary The model is called Finite State Transducer

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Language model example s the0.2 a0.1 frog0.01 toad0.01 said0.03 likes0.02 that0.04 ….. STOP0.2 state emission probabilities (partial) unigram language model P(frog said that toad likes frog) = 0.01 x 0.03 x 0.04 x 0.01 x 0.02 x 0.01 (We ignore continue/stop probabilities assuming they are fixed for all queries) Probability that some text (e.g. a query) was generated by the model:

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Query likelihood sfrogsaidthattoadlikesthatdog M M q = frog likes toad P(q | M1) = 0.01 x 0.02 x 0.01 P(q | M2) = x 0.04 x P(q|M1) > P(q|M2) => M1 is more likely to generate query q

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Types of language models How do we build probabilities over sequence of terms? P(t1 t2 t3 t4) = P(t1) x P(t2|t1) x P(t3|t1 t2) x P(t4|t1 t2 t3) Unigram language model – most simplest ; no conditioning context P(t1 t2 t3 t4) = P(t1) x P(t2) x P(t3) x P(t4) Bigram language model – condition on previous term P(t1 t2 t3 t4) = P(t1) x P(t2|t1) x P(t3|t2) x P(t4|t3) Trigram language model … Unigram model is the most common in IR Often sufficient to judge the topic of a document Data sparseness issues when using richer models Simple and efficient implementation

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The query likelihood model Goal: rank documents by P(d|q) – The probability that a user querying q, had the document d in mind Bayes Rule: P(d|q) = P(q|d)P(d)/P(q) P(q) – same for all documents ignored P(d) – often treated as uniform across documents ignored – Could be non uniform prior based on criteria like authority, length, genre, newness … Rank by P(q|d)

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The query likelihood model (2) P(q|d) - the probability that a query q was generated by a language model derived from document d – The probability that a query would be observed as a random sample from the respective document model Algorithm: 1.Infer a Language Model Md for each document d 2.Estimate P(q|Md) 3.Rank the documents according to these probabilities

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Illustration d1 Md1 query d2 Md2 d3 Md3 P(q|Md1) P(q|Md2) P(q|Md3) E.g., P(q|Md3) > P(q|Md1) > P(q|Md2) d3 is first, d1 is second, d2 is third

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Estimating P(q|Md) Use Maximum Likelihood Estimation - MLE Assume a unigram language model (terms occur independently) unigramMLE Length of document Term frequency

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Sparse data problem Documents are sparse – Some words don’t appear in the document – In particular, some of the query terms P(q|d) = 0 ; zero probability problem – Conjunctive semantics Occurring words are poorly estimated – A single documents is small training set – Occurring words are over estimated Their occurrence was partly by chance

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Solution: smoothing Smooth probabilities in Language Models – overcome zero probabilities – give some probability mass to unseen words The probability of a non occurring term should be close to its probability to occur in the collection P(t|Mc) = cf(t)/T cf(t) = #occurrences of term t in the collection T – length of the collection = sum of all document lengths

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Smoothing methods Linear Interpolation Bayesian smoothing Summary, with linear interpolation In practice, log in taken from both sides of the equation to avoid multiplying many small numbers

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Exercise Given a collection of two documents D1, D2 D1: Xyzzy reports a profit but revenue is down D2: Quorus narrows quarter loss but revenue decreases further A user submitted the query: “revenue down” Rank D1 and D2 - Use an MLE unigram model and a linear interpolation smoothing with lambda parameter 0.5

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Extended LM approaches query model document Document model query likelihood document likelihood model comparison Query likelihood P(q|d) – the probability of document LM to generate query we’ve seen in previous slides … Document likelihood P(d|q) – the probability of query LM to generate document in the next slides … Model comparison R(d;q) – compare between document and query models in the next slides … P(t|query) P(t|document)

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Document likelihood model P(d|q) – the probability of query LM to generate document Problem: queries are short bad model estimation [Zhai and Lafferty 2001] – Expand the query with terms taken from relevant documents in the usual way and hence update the language mode

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KL divergence Kullback–Leibler (KL) divergence An asymmetric divergence measure from information theory Measures the difference between two probability distributions P, Q Typically Q is an estimation of P Properties Non negative Equals 0 iff P equals Q May have an infinite value Asymmetric, thus not a metric Jensen–Shannon (JS) divergence Based on KL divergence (D) Always finite 0 <= JSD <= 1 Symmetric

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Model comparison Make LM from both query and document Measure `how different` these LMs from each other Use KL divergence Rank by KLD - the closer to 0 the higher is the rank

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Language models - summary Probabilistic model – mathematically precise Intuitive, simple concept Achieves very good retrieval results – Still, no evidence that it exceeds the traditional vector space model Relation to the Vector Space Model – Both use term frequency – Smoothing with collection generation probability is a little like idf Terms rare in the general collection but common in some documents will have a greater influence on the document’s ranking – Probabilistic vs. geometric – Mathematical mode vs. heuristic model

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