Given a finite set $S$ (WLOG, we assume $S = [n]$ for some $n \in \mathbb N$), we consider the space of all the distributions on $S$, $f \colon 2^S \to [0, 1]$ with $\sum_{S' \subseteq S} f(S' ) = 1$.
Background: Naive parameterization needs $2^n$ parameters, i.e., all the $f(S')$ values, which would be intractable for large $n$ (e.g., $2^{100} \approx 1.27 \times 10^{30}$)
Question: Can we have parameterization with much fewer parameters?
Independent case: An immediate idea I have is to assume the independency between all the events $i \in f(S')$, i.e., we can only use $n$ parameters $p_1, p_2, \ldots, p_n$ and let $f(S') = \prod_{i \in S'} p_i \prod_{j \notin S'} 1 - p_j$. This definitely cannot model all the possible distributions but it does model a subset of distributions.
Can we have other options with reasonably many parameters, e.g., $O(n^2)$?