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In this paper, Brown et al. proposed the usage of dictionary entries in addition to the machine translation probabilities that we're used to in the Math of Statistical Machine Translation.

He describes:

We formalize an intuitive account of lexicographic behavior as follows. We imagine that a lexicographer, when constructing an entry for the English word or phrase e, first chooses a random size s, and then selects at random a sample of s instances of the use of e, each with its French translation. We imagine, further, that he includes in his entry for e a list consisting of all of tile translations that occur at least once in his random sample. The probability that he will, in this way, obtain tile list fi, ..., f,,, is

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That much is understandable, and the Pr(f|e) in the formula is what we get from the normal machine translation probabilities. and then reweighted by the binomial recursion of finding the probability of the lexicographer choosing the sample sentence given the english word/phrase, i.e. Pr(s|e).

Then he went on to simply the formula and estimated a Poisson distribution instead of a binomial one and that will simplify the computation to:

enter image description here

Everything still looks okay but there's an additional variable exp^lambda(e) and that he explains is the mean of the Possion distribution of selecting the sample examples given the English source word.

So the question is how did he estimate that exp^lambda(e) from the corpus?

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