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I have the cardinalities of sets $N_i, \forall i \in 1,2..n$, and the cardinalities $|N_i \cap T|, |N_i|, |T|, |N_i \cup T|, \forall i$, are known.

Here, the set $T$ and sets $N_i$'s are all subsets of a mother :) set M and $|M|$ is also known.

Using these quantities, I would like to compute the normalized mutual information $NMI(N_i,T)$, for each set $N_i$.

Do I have all the necessary info, to compute the $NMI(.)$ using the cardinalities-and what are the assumptions that go with it? In specific, the probabilistic assumptions required over these sets, and the normalization of the mutual information is a grey area to me.

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