Shearer's inequality

Let $X=(X_1,\dots ,X_n)$ be a discrete random vector, and let $\mathcal{S}$ be a collection of subsets of $[n]$ in which every index occurs at least $r$ times. ThenApply the chain rule for information entropy to each projection $X_S$ and use conditioning reduces entropy to compare every term with the corresponding conditional entropy in the full chain rule.

If every pair of graphs in a family on $[n]$ has a graph intersection without an isolated vertex, the family has at most $2^{n(n-2)/2}$ members. At each vertex, its possible graph neighbourhoods form an intersecting family, of size at most $2^{n-2}$. Every edge occurs in two such neighbourhood projections. Shearer inequality bounds twice the full information entropy by the sum of their information entropies.