Journal of Inequalities in Pure and
Applied Mathematics
ON SOME APPLICATIONS OF THE AG INEQUALITY IN INFORMATION THEORY
volume 2, issue 1, article 11,
2001.
BERTRAM MOND AND JOSIP E. PEČARIĆ
School of Mathematics And Statistics
La Trobe University, Bundoora,
Victoria, 3083 AUSTRALIA.
EMail: B.Mond@latrobe.edu.au
URL: http://www.latrobe.edu.au/www/mathstats/Staff/mond.html
Department of Mathematics
Faculty of Textile Technology
University of Zagreb, CROATIA.
EMail: pecaric@mahazu.hazu.hr
URL: http://mahazu.hazu.hr/DepMPCS/indexJP.html
Received 3 November, 2000;
accepted 11 January 2001.
Communicated by: A. Lupaş
Abstract
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School of Communications and Informatics,Victoria University of Technology
ISSN (electronic): 1443-5756
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Abstract
Recently, S.S. Dragomir used the concavity property of the log mapping and
the weighted arithmetic mean-geometric mean inequality to develop new inequalities that were then applied to Information Theory. Here we extend these
inequalities and their applications.
2000 Mathematics Subject Classification: 26D15.
Key words: Arithmetic-Geometric Mean, Kullback-Leibler Distances, Shannon’s Entropy.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
An Inequality of I.A. Abou-Tair and W.T. Sulaiman . . . . . . . . 4
3
On Some Inequalities of S.S. Dragomir . . . . . . . . . . . . . . . . . . 6
4
Some Inequalities for Distance Functions . . . . . . . . . . . . . . . . 12
5
Applications for Shannon’s Entropy . . . . . . . . . . . . . . . . . . . . . 15
6
Applications for Mutual Information . . . . . . . . . . . . . . . . . . . . 17
References
On Some Applications of the AG
Inequality in Information Theory
Bertram Mond and
Josip E. Pečarić
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1.
Introduction
One of the most important inequalities is the arithmetic-geometric means inequality:
P
Let ai , pi , i = 1, . . . , n be positive numbers, Pn = ni=1 pi . Then
(1.1)
n
Y
i=1
p /P
ai i n
n
1 X
≤
p i ai ,
Pn i=1
with equality iff a1 = · · · = an .
It is well-known that using (1.1) we can prove the following generalization
of another well-known inequality, that is Hölder’s inequality:
PmLet pij , qi (i = 1, . . . , m; j = 1, . . . , n) be positive numbers with Qm =
i=1 qi . Then
(1.2)
m
n Y
m
n
X
Y
X
qi
Q
m
(pij )
≤
pij
j=1 i=1
i=1
!
qi
Qm
On Some Applications of the AG
Inequality in Information Theory
Bertram Mond and
Josip E. Pečarić
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.
j=1
In this note, we show that using (1.1) we can improve some recent results which
have applications in information theory.
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2.
An Inequality of I.A. Abou-Tair and W.T. Sulaiman
The main result from [1] is:
Let pij , qi (i = 1, . . . , m; j = 1, . . . , n) be positive numbers. Then
n Y
m
m
n
X
qi
1 XX
(pij ) Qm ≤
pij qi .
Q
m
j=1 i=1
i=1 j=1
(2.1)
Moreover, set in (1.1), n = m, pi = qi , ai =
m
n
Y
X
(2.2)
i=1
!
pij
j=1
qi
Qm
Pn
j=1
pij . We now have
!
m
n
1 X X
≤
pij qi .
Qm i=1 j=1
m
n
n Y
m
Y
X
X
qi
(pij ) Qm ≤
pij
j=1 i=1
Bertram Mond and
Josip E. Pečarić
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Now (1.2) and (2.2) give
(2.3)
On Some Applications of the AG
Inequality in Information Theory
i=1
j=1
! Qqi
m
m
n
1 XX
≤
pij qi .
Qm i=1 j=1
which is an interpolation of (2.1). Moreover, the generalized Hölder inequality
was obtained in [1] as a consequence of (2.1). This is not surprising since (2.1),
for n = 1, becomes
m
m
Y
qi
1 X
(pi1 ) Qm ≤
pi1 qi
Q
m
i=1
i=1
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which is, in fact, the A-G inequality (1.1) (set m = n, pi1 = ai and qi = pi ).
Theorem 3.1 in [1] is the well-known Shannon inequality:
Pn
Pn
Given
i=1 ai = a,
i=1 bi = b. Then
a ln
a
b
≤
n
X
i=1
ai
; ai , bi > 0.
ai ln
bi
It was obtained from (2.1) through the special case
a
(2.4)
n ai
Y
bi
i=1
ai
b
≤ .
a
Let us note that (2.4) is again a direct consequence of the A-G inequality. Indeed, in (1.1), setting ai → bi /ai , pi → ai , i =
(2.4).
Pm1, . . . , n wePhave
m
Theorem 3.2 from [1] is Rényi’s inequality. Given
i=1 ai = a,
i=1 bi = b,
then for α > 0, α 6= 1,
m
X 1
1
(aα b1−α − a) ≤
aαi b1−α
− ai ; ai , bi ≥ 0.
i
α−1
α−1
i=1
In fact, in the proof given in [1], it was proved that Hölder’s inequality is a consequence of (2.1). As we have noted, Hölder’s inequality is also a consequence
of the A-G inequality.
On Some Applications of the AG
Inequality in Information Theory
Bertram Mond and
Josip E. Pečarić
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3.
On Some Inequalities of S.S. Dragomir
The following theorems were proved in [2]:
Theorem 3.1.
P Let ai ∈ (0, 1) and bi > 0(i = 1, . . . , n). If pi > 0 (i = 1, . . . , n)
is such that ni=1 pi = 1, then
#
" n
#
" n
n
X a2 X
X ai a i
i
(3.1)
exp
pi −
pi ai ≥ exp
pi
−1
bi
bi
i=1
i=1
i=1
n ai pi
Y
ai
≥
bi
i=1
"
a i #
n
X
pi
≥ exp 1 −
pi
ai
" n i=1
#
n
X
X
≥ exp
p i ai −
p i bi
i=1
Theorem 3.2. Let ai ∈ (0, 1) (i =P1, . . . , n) and bj > 0 (j = 1, . . . , m). If
pi > 0 (i P
= 1, . . . , n) is such that ni=1 pi = 1 and qj > 0 (j = 1, . . . , m) is
such that m
j=1 qj = 1, then we have the inequality
(3.2)
exp
n
X
i=1
pi a2i
m
X
qj
j=1
bj
−
n
X
i=1
p i ai
"
≥ exp
n X
m
X
i=1
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Josip E. Pečarić
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i=1
with equality iff ai = bi for all i ∈ {1, . . . , n}.
!
On Some Applications of the AG
Inequality in Information Theory
#
a i
ai
p i qj
−1
bj
j=1
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n
Q
≥
aai i pi
i=1
m
Q
j=1
q
bj j
"
Pni=1 pi ai
a i #
bj
≥ exp 1 −
p i qj
ai
i=1 j=1
!
n
m
X
X
≥ exp
p i ai −
qj b j .
i=1
n X
m
X
j=1
The equality holds in (3.2) iff a1 = · · · = an = b1 = · · · = bm .
On Some Applications of the AG
Inequality in Information Theory
Bertram Mond and
Josip E. Pečarić
First we give an improvement of the second and third inequality in (3.1).
Theorem
3.3. Let ai , bi and pi (i = 1, . . . , n) be positive real numbers with
Pn
i=1 pi = 1. Then
−1 X
a i
a i
n
ai
ai
(3.3)
exp pi
−1
≥
pi
bi
bi
i=1
n pi ai
Y
ai
≥
bi
i=1
" n
#−1
X b i a i
≥
pi
ai
i=1
"
a i #
n
X
bi
≥ exp 1 −
pi
,
a
i
i=1
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with equality iff ai = bi , i = 1, . . . , n.
Proof. The first inequality in (3.3) is a simple consequence of the following
well-known elementary inequality
(3.4)
ex−1 ≥ x, for all x ∈ R
with equality iff x = 1. The second inequality is a simple consequence of the
A-G inequality that is, in (1.1), set ai → (ai /bi )ai , i = 1, . . . , n. The third
inequality is again a consequence of (1.1). Namely, for ai → (bi /ai )ai , i =
1, . . . , n, (1.1) becomes
a i
n ai pi
n
Y
X
bi
bi
≤
pi
ai
ai
i=1
i=1
which is equivalent to the third inequality. The last inequality is again a consequence of (3.4).
Theorem 3.4.
ai ∈ (0, 1) and bi > 0 (i = 1, . . . , n). If pi > 0, i = 1, . . . , n
PLet
n
is such that
i=1 pi = 1, then
#
" n
#
" n
n
X ai a i
X a2 X
i
(3.5) exp
−
pi ai ≥ exp
pi
−1
pi
b
b
i
i
i=1
i=1
i=1
a i
n
X
ai
≥
pi
bi
i=1
n
pa
Y
ai i i
≥
bi
i=1
On Some Applications of the AG
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Josip E. Pečarić
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ai #−1
bi
≥
pi
ai
i=1
"
a i #
n
X
bi
≥ exp 1 −
pi
ai
" n i=1
#
n
X
X
≥ exp
p i ai −
pi bi
"
n
X
i=1
i=1
On Some Applications of the AG
Inequality in Information Theory
with equality iff ai = bi for all i = 1, . . . , n.
Bertram Mond and
Josip E. Pečarić
Proof. The theorem follows from Theorems 3.1 and 3.3.
Theorem 3.5. Let ai , pi (i = 1, . . . , n); bj , qj (j = 1, . . . , m) be positive
numbers
with
Pn
Pm
i=1 pi =
j=1 qj = 1. Then
"
(3.6)
exp
n X
m
X
i=1
#
a i
ai
−1
p i qj
bj
j=1
≥
n X
m
X
i=1
a i
ai
p i qj
bj
j=1
n
Q
≥
q bj j
Pn
i=1
≥
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i=1
m
Q
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m
X
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"
≥ exp 1 −
ai #−1
bj
.
p i qj
a
i
j=1
n X
m
X
i=1
Equality in (3.6) holds iff a1 = · · · = an = b1 = · · · = bm .
Proof. The first and the last inequalities are simple consequences of (3.4). The
second is also a simple consequence of the A-G inequality. Namely, we have
n
Q
i=1
m
Q
q
bj j
aai i pi
Pni=1 pi ai =
n Y
m ai pi qj
Y
ai
i=1 j=1
bj
≤
n X
m
X
i=1
a i
ai
p i qj
,
bj
j=1
j=1
which is the second inequality in (3.6). By the A-G inequality, we have
a i
n Y
m ai pi qj
n X
m
Y
X
bj
bj
≤
p i qj
ai
ai
i=1 j=1
i=1 j=1
which gives the third inequality in (3.6).
Theorem 3.6. Let the assumptions of Theorem 3.2 be satisfied. Then
#
" n
X
m n
X
X
qj
2
(3.7)
−
p i ai
exp
p i ai
bj
i=1
j=1
i=1
" n m
#
a i
XX
ai
−1
≥ exp
p i qj
bj
i=1 j=1
On Some Applications of the AG
Inequality in Information Theory
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Josip E. Pečarić
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≥
n X
m
X
i=1
a i
ai
p i qj
bj
j=1
n
Q
≥ Q
m
aai i pi
i=1
q
bj j
Pni=1 pi ai
j=1
ai #−1
bj
≥
p i qj
ai
i=1 j=1
"
a i #
n X
m
X
bj
≥ exp 1 −
p i qj
ai
i=1 j=1
" n
#
m
X
X
≥ exp
p i ai −
qj b j .
"
n X
m
X
i=1
j=1
Equality holds in (3.7) iff a1 = · · · = an = b1 = · · · = bm .
Proof. The theorem is a simple consequence of Theorems 3.2 and 3.5.
On Some Applications of the AG
Inequality in Information Theory
Bertram Mond and
Josip E. Pečarić
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4.
Some Inequalities for Distance Functions
In 1951, Kullback and Leibler introduced the following distance function in
Information Theory (see [4] or [5])
KL(p, q) :=
(4.1)
n
X
pi log
i=1
pi
,
qi
provided that p, q ∈ Rn++ := {x = (x1 , . . . , xn )∈ Rn , xi > 0, i = 1, . . . , n}.
Another useful distance function is the χ2 -distance given by (see [5])
Dχ2 (p, q) :=
(4.2)
n
X
p2 − q 2
i
i=1
i
qi
,
where p, q ∈ Rn++ . S.S. Dragomir [2] introduced the following two new distance
functions
n pi
X
pi
(4.3)
P2 (p, q) :=
−1
qi
i=1
On Some Applications of the AG
Inequality in Information Theory
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and
n pi
X
qi
P1 (p, q) :=
−
+1 ,
p
i
i=1
(4.4)
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Rn++ .
provided p, q ∈
The following inequality connecting all the above four
distance functions holds.
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Theorem 4.1. Let p, q ∈ Rn++ with pi ∈ (0, 1). Then we have the inequality:
Dχ2 (p, q) + Qn − Pn ≥ P2 (p, q)
1
≥ n ln
P2 (p, q) + 1
n
≥ KL(p, q)
1
≥ −n ln −
P1 (p, q) + 1
n
≥ P1 (p, q)
≥ Pn − Qn ,
Pn
Pn
where Pn =
p
=
1,
Q
=
i
n
i=1
i=1 qi . Equality holds in (4.5) iff pi =
qi (i = 1, . . . , n).
(4.5)
Proof. Set in (3.5), pi = 1/n, ai = pi , bi = qi (i = 1, . . . , n) and take
logarithms. After multiplication by n, we get (4.5).
Corollary 4.2. Let p, q be probability distributions. Then we have
(4.6)
Dχ2 (p, q) ≥ P2 (p, q)
1
≥ n ln
P2 (p, q) + 1
n
≥ KL(p, q)
1
≥ −n ln 1 −
P1 (p, q)
n
≥ P1 (p, q) ≥ 0.
On Some Applications of the AG
Inequality in Information Theory
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Equality holds in (4.6) iff p = q.
Remark 4.1. Inequalities (4.5) and (4.6) are improvements of related results in
[2].
On Some Applications of the AG
Inequality in Information Theory
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5.
Applications for Shannon’s Entropy
The entropy of a random variable is a measure of the uncertainty of the random
variable, it is a measure of the amount of information required on the average to
describe the random variable. Let p(x), x ∈ χ be a probability mass function.
Define the Shannon’s entropy f of a random variable X having the probability
distribution p by
(5.1)
H(X) :=
X
p(x) log
x∈χ
1
.
p(x)
In the above definition
we use the convention (based on continuity argu
0
ments) that 0 log q = 0 and p log p0 = ∞. Now assume that |χ| (card(χ) =
1
|χ|) is finite and let u(x) = |χ|
be the uniform probability mass function in χ.
It is well known that [5, p. 27]
X
p(x)
(5.2)
KL(p, q) =
p(x) log
= log |χ| − H(X).
q(x)
x∈χ
The following result is important in Information Theory [5, p. 27]:
Theorem 5.1. Let X, p and χ be as above. Then
(5.3)
H(X) ≤ log |χ|,
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with equality if and only if X has a uniform distribution over χ.
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In what follows, by the use of Corollary 4.2, we are able to point out the
following estimate for the difference log |χ| − H(X), that is, we shall give the
following improvement of Theorem 9 from [2]:
Theorem 5.2. Let X, p and χ be as above. Then
X
|χ|E(X) − 1 ≥
(5.4)
|χ|p(x) [p(x)]p(x) − 1
x∈χ
(
≥ |χ| ln
1 X p(x)
[|χ| [p(x)]p(x)
|χ| x∈χ
)
On Some Applications of the AG
Inequality in Information Theory
Bertram Mond and
Josip E. Pečarić
≥ ln |χ| − H(X)
(
)
1 X −p(x)
≥ −|x| ln
|χ|
[p(x)]−p(x)
|χ| x∈χ
X
≥
|χ|−p(x) [p(x)]−p(x) − 1 ≥ 0,
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x∈χ
where E(X) is the informational energy of X, i.e., E(X) :=
1
equality holds in (5.4) iff p(x) = |χ|
for all x ∈ χ.
P
x∈χ
Proof. The proof is obvious by Corollary 4.2 by choosing u(x) =
p2 (x). The
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6.
Applications for Mutual Information
We consider mutual information, which is a measure of the amount of information that one random variable contains about another random variable. It is the
reduction of uncertainty of one random variable due to the knowledge of the
other [6, p. 18].
To be more precise, consider two random variables X and Y with a joint
probability mass function r(x, y) and marginal probability mass functions p(x)
and q(y), x ∈ X , y ∈ Y. The mutual information is the relative entropy
between the joint distribution and the product distribution, that is,
X
r(x, y)
I(X; Y ) =
r(x, y) log
= D(r, pq).
p(x)q(y)
x∈χ,y∈Y
The following result is well known [6, p. 27].
Theorem 6.1. (Non-negativity of mutual information). For any two random
variables X, Y
(6.1)
I(X, Y ) ≥ 0,
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with equality iff X and Y are independent.
In what follows, by the use of Corollary 4.2, we are able to point out the following estimate for the mutual information, that is, the following improvement
of Theorem 11 of [2]:
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Theorem 6.2. Let X and Y be as above. Then we have the inequality
#
"
X X r2 (x, y)
X X r(x, y) r(x,y)
−1≥
−1
p(x)q(y)
p(x)q(y)
x∈χ y∈Y
x∈χ y∈Y
"
r(x,y) #
X
X
r(x, y)
1
≥ |χ| |Y| ln
|χ||Y| x∈χ y∈Y p(x)q(y)
≥ I(X, Y )
(
X X p(x)q(y) r(x,y)
1
|χ||Y| x∈χ y∈Y
r(x, y)
"
#
r(x,y)
XX
r(x, y)
≥
1−
≥ 0.
p(x)q(y)
x∈χ y∈Y
≥ −|χ| |Y| ln
The equality holds in all inequalities iff X and Y are independent.
On Some Applications of the AG
Inequality in Information Theory
Bertram Mond and
Josip E. Pečarić
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References
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On Some Applications of the AG
Inequality in Information Theory
Bertram Mond and
Josip E. Pečarić
Title Page
Contents
[6] T.M. COVER AND J.A.THOMAS, Elements of Information Theory, John
Wiley and Sons, Inc., 1991.
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