Assume we have $N$ random variables $x_1, x_2, \dots, x_N$. Is there a way to express the joint entropy $H(x_1, x_2, \dots, x_N)$ in terms of single-variable or pairwise measures such as the pairwise mutual information $I(x_i; x_j)$ or the entropy $H(x_i)$ of each variable? In other words, are there functions $g_{ij}(\cdot,\cdot)$ such that
$$ H(x_1, x_2, \dots, x_N) = \sum_{i\in \{1, \dots, N\}}\sum_{j\in \{1, \dots, N\}} g_{ij}(x_i, x_j). $$
This is not possible. One can have three random variables $X_1,X_2,X_3$ that are pairwise independent but not mutually independent (e.g., Example of Pairwise Independent but not Jointly Independent Random Variables?). The pairwise data therefore do not provide enough to deduce whether $X_1$, $X_2$, and $X_3$ are mutually independent. However, the joint entropy equals $H(X_1) + H(X_2) + H(X_3)$ if and only if $X_1$, $X_2$, and $X_3$ are mutually independent.
