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I have a text corpus, from which I have computed the unigram probabilities of every word.

Now, let $p_{i}$ & $p_{j}$ be the unigram probabilities of $i$ and $j$ respectively, what will be the probability with which a pair of words $i$ and $j$ will co-occur(within a maximum gap of 1 word) when independence between $i$ and $j$ is assumed?

EDIT: Here $p_{i}$ is the probability of any given word being $i$.

By co-occurrence probability, I mean the chance that word $i$ & $j$ co-occur in a random sequence of three words drawn from the corpus. By co-occurring I mean that $i$ and $j$ occurring anywhere in this 3 word random sequence within a maximum word distance of 1. (For example, $i$*$j$ and $ij$ are both valid co-occurrences of $i$ & $j$ but $i**j$ is not a valid co-occurrence since the distance between $i$ & $j$ is more than 1.)

And finally, $ijij$ counts as 3 co-occurrences.

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  • $\begingroup$ I think we need a bit more information. What precisely is $p_i$: is it the probability of any given word being $i$, or the probability of $i$ occurring in a document? What do you mean by co-occurring: is it $i$ and $j$ occurring at a specific place, or $i$ and $j$ occurring anywhere in a document? Finally, would something like $ijij$ count as 1, 2 or 3 co-occurrences? $\endgroup$ – Simon Byrne Jul 20 '11 at 11:57
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By the Principle of Inclusion-Exclusion, the probability of no co-occurrences of i and j in three words equals the probability that i fails to appear plus the probability that j fails to appear minus the probability that both i and j fail to appear. Subtracting this from 1 will yield the answer. The independence assumption lets us compute all three probabilities by multiplying the relevant "unigram" probabilities, whence the probability of a co-occurrence of i and j equals

$$1 - (1-p_i)^3 - (1-p_j)^3 + (1 - p_i - p_j)^3$$

$$= 3 p_i p_j (2 - p_i - p_j).$$

For example, when $p_i=p_j=1/3$, this formula gives $4/9$. The 12 possible patterns of co-occurrence can be symbolized as iij, iji, ijj, ij*, i*j, jii, jij, ji*, jji, j*i, *ij, *ji, each with equal probability of $(1/3)^3$. The chance therefore is $12/3^3=4/9$, agreeing with the formula.

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