Shearer's inequality is an inequality in information theory relating the entropy a set of variables to the entropies of a collection of subsets. It is named for mathematician James 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
\( {\displaystyle H[(X_{1},\dots ,X_{d})]\leq {\frac {1}{r}}\sum _{i=1}^{n}H[(X_{j})_{j\in S_{i}}]} \)
where H is entropy and \( (X_{j})_{j\in S_{i}} \) is the Cartesian product of random variables \( X_{{j}} \) with indices j in \( S_{i} \). [1]
References
Chung, F.R.K.; Graham, R.L.; Frankl, P.; Shearer, J.B. (1986). "Some Intersection Theorems for Ordered Sets and Graphs". J. Comb. Theory A. 43: 23–37.
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