From Wikipedia, the free encyclopedia
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
X
1
, ...,
X
d
are
random variables
and
S
1
, ...,
S
n
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]
Combinatorial version
[
edit
]
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]
See also
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]
References
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]