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In mathematics, the log sum inequality is an inequality which is useful for proving several theorems in information theory.

Statement

Let $a_1,\dots ,a_n$ and $b_1,\dots ,b_n$ be nonnegative numbers. Denote the sum of all $a_i$s by $a$ and the sum of all $b_i$s by $b$. The log sum inequality states that

$\sum _{i=1}^n{a}_i\log \frac{a_i}{b_i}\ge a\log \frac{a}{b},$

with equality if and only if $\frac{a_i}{b_i}$ are equal for all $i$.

Proof

Notice that after setting $f(x)=x\log x$ we have

$\begin{array}{rl}\sum _{i=1}^n{a}_i\log \frac{a_i}{b_i}& =\sum _{i=1}^n{b}_{i}f\left(\frac{a_i}{b_i}\right)=b\sum _{i=1}^n\frac{b_i}{b}f\left(\frac{a_i}{b_i}\right)\\ & \ge bf\left(\sum _{i=1}^n\frac{b_i}{b}\frac{a_i}{b_i}\right)=bf\left(\frac{1}{b}\sum _{i=1}^n{a}_i\right)=bf\left(\frac{a}{b}\right)\\ & =a\log \frac{a}{b},\end{array}$

where the inequality follows from Jensen's inequality since $\frac{b_i}{b}\ge 0$, $\sum _i\frac{b_i}{b}=1$, and $f$ 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 Kullback-Leibler divergence.

For example to prove Gibbs' inequality it is enough to substitute $p_i$s for $a_i$s, and $q_i$s for $b_i$s to get

$D_{\mathrm{K}\mathrm{L}}(P\Vert Q)\equiv \sum _{i=1}^n{p}_i{\log }_2\frac{p_i}{q_i}\ge 1\log \frac{1}{1}=0.$

Generalizations

The inequality remains valid for $n=\mathrm{\infty }$ provided that $a<\mathrm{\infty }$ and $b<\mathrm{\infty }$. 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 [1]. Retrieved on 2009-06-14.
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