Compression of an entropy sum
= Compression of an entropy sum
A <union-intersection compression> can only decrease the sum of <information entropies> $\sum_{A\in\mathcal A}H(X_A)$ for a finite-valued <random vector> $X$. Each elementary step follows from <entropy submodularity>; iteration proves the assertion with <multiset> multiplicities. Uniform coordinate multiplicity gives <Shearer inequality> by compressing to repeated full sets and empty sets.