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Using query likelihood language models in IR
Language modeling is a quite general formal approach to IR, with many
variant realizations. The original and basic method for using language
models in IR is the query likelihood model . In it, we
construct from
each document in the collection a language model . Our goal is
to rank documents by , where the probability of a document is
interpreted as the likelihood that it is relevant to the query. Using
Bayes rule (as introduced in probirsec), we have:

(98) 
is the same for all documents, and so can be ignored.
The prior probability of a document is often treated as uniform
across all and so it can also be
ignored, but we could implement a genuine prior which could include
criteria like authority, length, genre, newness, and number of previous
people who have read the document. But, given these simplifications, we
return results ranked by simply , the probability of the query
under the language model derived from .
The Language Modeling approach thus attempts to model the query generation
process: Documents are ranked by the probability that a query would be
observed as a random sample from the respective document model.
The most common way to do this is using the multinomial unigram language
model, which is equivalent to a multinomial Naive Bayes
model (page 13.3 ), where the
documents are the classes, each treated in the estimation as a separate
``language''. Under this model, we have that:

(99) 
where, again
is the multinomial coefficient for the query ,
which we will henceforth ignore, since it is a constant for a
particular query.
For retrieval based on a language model (henceforth LM ), we
treat the generation of queries as a random process.
The approach is to
 Infer a LM for each document.
 Estimate , the probability of generating the query
according to each of these document models.
 Rank the documents according to these probabilities.
The intuition of the basic model is that the user has a prototype document
in mind, and
generates a query based on words that appear in this document.
Often, users
have a reasonable idea of terms that are likely to occur in documents of
interest and they will choose query terms that distinguish these
documents from others in the collection.^{}Collection statistics are an integral part of the language model, rather
than being used heuristically as in many other approaches.
Next: Estimating the query generation
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