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In mathematics, the '''log sum inequality''' is an [[inequality (mathematics)|inequality]] which is useful for proving several theorems in [[information theory]].


==Statement==
Let <math>a_1,\ldots,a_n</math> and <math>b_1,\ldots,b_n</math> be nonnegative numbers.  Denote the sum of all <math>a_i\;</math>s by <math>a</math> and the sum of all <math>b_i\;</math>s by <math>b</math>.  The log sum inequality states that


:<math>\sum_{i=1}^n a_i\log\frac{a_i}{b_i}\geq a\log\frac{a}{b},</math>
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with equality if and only if <math>\frac{a_i}{b_i}</math> are equal for all <math>i</math>.
 
==Proof==
 
Notice that after setting <math>f(x)=x\log x</math> we have
 
:<math>
\begin{align}
\sum_{i=1}^n a_i\log\frac{a_i}{b_i} & {} = \sum_{i=1}^n b_i f\left(\frac{a_i}{b_i}\right)
= b\sum_{i=1}^n \frac{b_i}{b} f\left(\frac{a_i}{b_i}\right) \\
& {} \geq b f\left(\sum_{i=1}^n \frac{b_i}{b}\frac{a_i}{b_i}\right) = b f\left(\frac{1}{b}\sum_{i=1}^n a_i\right)
= b f\left(\frac{a}{b}\right) \\
& {} = a\log\frac{a}{b},
\end{align}
</math>
where the inequality follows from [[Jensen's inequality]] since <math>\frac{b_i}{b}\geq 0</math>, <math>\sum_i\frac{b_i}{b}= 1</math>, and <math>f</math> is convex.
 
==Applications==
 
The log sum inequality can be used to prove several inequalities in information theory such as [[Gibbs' inequality]] or the convexity of [[KL-divergence|Kullback-Leibler divergence]].
 
For example to prove Gibbs' inequality it is enough to substitute <math>p_i\;</math>s for <math>a_i\;</math>s, and <math>q_i\;</math>s for <math>b_i\;</math>s to get
 
:<math>D_{\mathrm{KL}}(P\|Q) \equiv \sum_{i=1}^n p_i \log_2 \frac{p_i}{q_i} \geq 1\log\frac{1}{1} = 0.</math>
 
==Generalizations==
 
The inequality remains valid for <math>n=\infty</math> provided that <math>a<\infty</math> and <math>b<\infty</math>.
Generalizations to convex functions other than the logarithm is given in Csiszár, 2004.
 
==References==
* T.S. Han, K. Kobayashi, ''Mathematics of information and coding.'' American Mathematical Society, 2001. ISBN 0-8218-0534-7.
* Information Theory course materials, Utah State University [http://ocw.usu.edu/Electrical_and_Computer_Engineering/Information_Theory/lecture3.pdf]. Retrieved on 2009-06-14.
* {{cite journal
|first1=I. |last1=Csiszár |authorlink1=Imre Csiszár
|first2=P. |last2=Shields
|year=2004
|title=Information Theory and Statistics: A Tutorial
|journal=Foundations and Trends in Communications and Information Theory
|volume=1 |issue=4 |pages=417&ndash;528
|doi=10.1561/0100000004
|url=http://www.renyi.hu/~csiszar/Publications/Information_Theory_and_Statistics:_A_Tutorial.pdf |accessdate=2009-06-14
}}
 
[[Category:Inequalities]]
[[Category:Information theory]]
[[Category:Articles containing proofs]]

Latest revision as of 23:34, 3 September 2014


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