Shearer's inequality
Shearer's inequality or also Shearer's lemma, in mathematics, is an inequality in information theory relating the entropy of a set of variables to the entropies of a collection of subsets. It is named for mathematician James B. Shearer.
Concretely, it states that if X1, ..., Xd are random variables and S1, ..., Sn are subsets of such that every integer between 1 and d lies in at least r of these subsets, then
where is entropy and is the Cartesian product of random variables with indices j in.
The inequality generalizes the subadditivity property of entropy, which can be recovered by taking for.
Combinatorial version
Let be a family of subsets of with each included in at least members of. Let be another set of subsets of. Thenwhere the set of possible intersections of elements of with.