Converse theorem

Redirect to:

• Shearer's inequality

If $X_1,\mathrm{...}{X}_d$ are random variables and $S_1,\mathrm{...},S_n$ are subsets of $\{1,2,\mathrm{...},d\}$ such that every integer between 1 and d lie in exactly r of these subsets, then

$H[(X_1,\mathrm{...},X_d)]\le \frac{1}{r}\sum _{i=1}^{n}H[(X_j{)}_{j\in S_i}]$

where $(X_j{)}_{j\in S_i}$ is the Cartesian product of random variables $X_j$ with indexes j in $S_i$ (so the dimension of this vector is equal to the size of $S_i$).