We start by observing that, given the definition of the Dirichlet distribution, it follows that Considering the different sets of indices in isolation, let i=j 1 α 2 i , and ψ 2 = km 2 i=k 1 α 2 i .
we rewrite (7) for the two conditions: 1. w has one meaning: 2. w has two meanings: with some algebraic manipulations, it can be shown that M (1) > M (2) if the following holds: when (8) holds, the average cosine distance between two samples of a certain word w gets larger as w acquires more meanings.