 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.

Shearer's inequality is a result in information theory related to the concept of conditional independence. It provides a way to bound the joint information of a collection of random variables in terms of the information of subsets of those variables.