Shearer's inequality

In probability theory, Shearer's inequality states that if X₁, ..., X_d are random variables and S₁, ..., S_n are subsets of {1, 2, ..., d} such that every integer between 1 and d lies in exactly r of these subsets, then

$H[(X_1,\dots,X_d)] \leq \frac{1}{r}\sum_{i=1}^n H[(X_j)_{j\in S_i}]$

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 $S_{i}$ ).