By Adam Berger
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Additional info for Statistical machine learning for information retrieval
Example text
By scaling these I(w, d) values appropriately, one can construct an artificial cumulative distribution function I˜ over words in each document. Drawing m ∼ φ(· | d) random samples from the document according to this distribution results in a query q = q1 , . . , qm . Several such queries were generated for each document.
We’ve already seen an example: x − 1 is an auxiliary function for log x in the sense that x − 1 ≥ log x for all x. This observation might prove useful if we’re trying to establish that some function f (x) lies on or above log x: if we can show f (x) lies on or above x − 1, then we’re done, since x − 1 itself lies above log x. (Incidentally, it’s also true that log x is an auxiliary function for x − 1, albeit in the other direction). We’ll be making use of a particular type of auxiliary function: one that bounds the change in log-likelihood between two models.
14) 62 Document ranking This formula shows that the probability of the query is given as a product of terms. Yet the query term translations are not independent, due to the process of filtering out the generated list to remove duplicates. 14) will henceforth be denoted as Model 1 . Model 1 has an interpretation in terms of binomial random variables. Suppose that a word w does not belong to the query with probability β w = e−λ(d)p(w | d) . Then Model 1 amounts to flipping independent βw -biased coins to determine which set of words comprise the query [36].
