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 {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 $$H$$ is entropy and $$ (X_{j})_{j\in S_{i}}$$ is the Cartesian product of random variables $$X_{j}$$ with indices j in $$S_{i}$$.

Combinatorial version
Let $$\mathcal{F}$$ be a family of subsets of [n] (possibly with repeats) with each $$i\in [n]$$ included in at least $$t$$ members of $$\mathcal{F}$$. Let $$\mathcal{A}$$ be another set of subsets of $$\mathcal F$$. Then


 * $$ \mathcal |\mathcal{A}|\leq \prod_{F\in \mathcal{F}}|\operatorname{trace}_{F}(\mathcal{A})|^{1/t}$$

where $$ \operatorname{trace}_{F}(\mathcal{A})=\{A\cap F:A\in\mathcal{A}\}$$ the set of possible intersections of elements of $$ \mathcal{A}$$ with $$ F$$.