# Shearer's inequality

$H[(X_1,\dots,X_d)] \leq \frac{1}{r}\sum_{i=1}^n H[(X_j)_{j\in S_i}]$
where $(X_{j})_{j\in S_{i}}$ is the Cartesian product of random variables $X_{j}$ with indices j in $S_{i}$ (so the dimension of this vector is equal to the size of $S_{i}$).