// Workers AI · dad joke modeWhy did Shearer's inequality go to therapy? It had uneven issues.

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

The inequality generalizes the subadditivity property of entropy, which can be recovered by taking $S_{i}=\{i\}$ for $i\in \{1,\ldots ,n\}$.

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 $[n]$. 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 $F$.

See also

• Lovász local lemma