Let be a Boolean vector and are Boolean variables. Assume that there is a joint distribution over and we'd like to find a joint distribution over such that:
1. The marginal of on , equals .
2. are independent of under , i.e., .
3. is maximized,
where denotes the mutual information. For now I don't even know what is a nontrivial upper bound of given that ? Furthermore, is it possible we can know the optimal distribution that achieves the upper bound?
My conjecture is that the upper bound of should have something to do with the correlation (coupling?) between and , so ideally it should contain something related to that.