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

where ( X j ) j ∈ 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 ).