Top Qs
Timeline
Chat
Perspective

Shearer's inequality

From Wikipedia, the free encyclopedia

Remove ads

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, ..., 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

where is entropy and is the Cartesian product of random variables with indices j in .[1]

The inequality generalizes the subadditivity property of entropy, which can be recovered by taking for .[2]

Remove ads

Combinatorial version

Let be a family of subsets of [n] (possibly with repeats) with each included in at least members of . Let be another set of subsets of . Then

where the set of possible intersections of elements of with .[2]

Remove ads

See also

References

Loading related searches...

Wikiwand - on

Seamless Wikipedia browsing. On steroids.

Remove ads