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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
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
H
[
(
X
1
,
…
,
X
d
)
]
≤
1
r
∑
i
=
1
n
H
[
(
X
j
)
j
∈
S
i
]
{\displaystyle H[(X_{1},\dots ,X_{d})]\leq {\frac {1}{r}}\sum _{i=1}^{n}H[(X_{j})_{j\in S_{i}}]}
where
H
{\displaystyle H}
is entropy and
(
X
j
)
j
∈
S
i
{\displaystyle (X_{j})_{j\in S_{i}}}
is the Cartesian product of random variables
X
j
{\displaystyle X_{j}}
with indices j in
S
i
{\displaystyle S_{i}}
.[1]
Combinatorial version [ edit ]
Let
F
{\displaystyle {\mathcal {F}}}
be a family of subsets of [n ] (possibly with repeats) with each
i
∈
[
n
]
{\displaystyle i\in [n ]}
included in at least
t
{\displaystyle t}
members of
F
{\displaystyle {\mathcal {F}}}
. Let
A
{\displaystyle {\mathcal {A}}}
be another set of subsets of
F
{\displaystyle {\mathcal {F}}}
. Then
|
A
|
≤
∏
F
∈
F
|
trace
F
(
A
)
|
1
/
t
{\displaystyle {\mathcal {|}}{\mathcal {A}}|\leq \prod _{F\in {\mathcal {F}}}|\operatorname {trace} _{F}({\mathcal {A}})|^{1/t}}
where
trace
F
(
A
)
=
{
A
∩
F
:
A
∈
A
}
{\displaystyle \operatorname {trace} _{F}({\mathcal {A}})=\{A\cap F:A\in {\mathcal {A}}\}}
the set of possible intersections of elements of
A
{\displaystyle {\mathcal {A}}}
with
F
{\displaystyle F}
.[2]
See also [ edit ]
References [ edit ]
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Shearer%27s_inequality&oldid=1232243724 "
C a t e g o r i e s :
● I n f o r m a t i o n t h e o r y
● I n e q u a l i t i e s
H i d d e n c a t e g o r i e s :
● W i k i p e d i a a r t i c l e s t h a t a r e t o o t e c h n i c a l f r o m D e c e m b e r 2 0 2 1
● A l l a r t i c l e s t h a t a r e t o o t e c h n i c a l
● T h i s p a g e w a s l a s t e d i t e d o n 2 J u l y 2 0 2 4 , a t 1 8 : 3 8 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
● P r i v a c y p o l i c y
● A b o u t W i k i p e d i a
● D i s c l a i m e r s
● C o n t a c t W i k i p e d i a
● C o d e o f C o n d u c t
● D e v e l o p e r s
● S t a t i s t i c s
● C o o k i e s t a t e m e n t
● M o b i l e v i e w