# Can the joint entropy be expressed in terms of pairwise measures?

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.