Shearer's inequality

In information theory, Shearer's inequality, named after James Shearer, 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 at least 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 $S_{i}$).