Log sum inequality

The log sum inequality is used for proving theorems in information theory.

Statement

Let $a_{1},\ldots ,a_{n}$ and $b_{1},\ldots ,b_{n}$ 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

$\sum _{i=1}^{n}a_{i}\log {\frac {a_{i}}{b_{i}}}\geq a\log {\frac {a}{b}},$

with equality if and only if ${\frac {a_{i}}{b_{i}}}$ are equal for all $i$, in other words $a_{i}=cb_{i}$ for all $i$.

(Take $a_{i}\log {\frac {a_{i}}{b_{i}}}$ to be $0$ if $a_{i}=0$ and $\infty$ if $a_{i}>0,b_{i}=0$. These are the limiting values obtained as the relevant number tends to $0$.)

Proof

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

${\begin{aligned}\sum _{i=1}^{n}a_{i}\log {\frac {a_{i}}{b_{i}}}&{}=\sum _{i=1}^{n}b_{i}f\left({\frac {a_{i}}{b_{i}}}\right)=b\sum _{i=1}^{n}{\frac {b_{i}}{b}}f\left({\frac {a_{i}}{b_{i}}}\right)\\&{}\geq bf\left(\sum _{i=1}^{n}{\frac {b_{i}}{b}}{\frac {a_{i}}{b_{i}}}\right)=bf\left({\frac {1}{b}}\sum _{i=1}^{n}a_{i}\right)=bf\left({\frac {a}{b}}\right)\\&{}=a\log {\frac {a}{b}},\end{aligned}}$

where the inequality follows from Jensen's inequality since ${\frac {b_{i}}{b}}\geq 0$, $\sum _{i=1}^{n}{\frac {b_{i}}{b}}=1$, and $f$ is convex.

Generalizations

The inequality remains valid for $n=\infty$ provided that $a<\infty$ and $b<\infty$. The proof above holds for any function $g$ such that $f(x)=xg(x)$ is convex, such as all continuous non-decreasing functions. Generalizations to non-decreasing functions other than the logarithm is given in Csiszár, 2004.

Applications

The log sum inequality can be used to prove inequalities in information theory. Gibbs' inequality states that the Kullback-Leibler divergence is non-negative, and equal to zero precisely if its arguments are equal. One proof uses the log sum inequality.

Proof
Let $P=(p_{i})_{i\in \mathbb {N} }$ and $Q=(q_{i})_{i\in \mathbb {N} }$ be pmfs. In the log sum inequality, substitute $n=\infty$, $a_{i}=p_{i}$ and $b_{i}=q_{i}$ to get $\mathbb {D} _{\mathrm {KL} }(P\|Q)\equiv \sum _{i}p_{i}\log _{2}{\frac {p_{i}}{q_{i}}}\geq 1\log {\frac {1}{1}}=0$ with equality if and only if $p_{i}=q_{i}$ for all i (as both $P$ and $Q$ sum to 1).

The inequality can also prove convexity of Kullback-Leibler divergence.

References

• Thomas M. Cover; Joy A. Thomas (1991). Elements of Information Theory. Hoboken, New Jersey: Wiley. ISBN 978-0-471-24195-9.
• Csiszár, I.; Shields, P. (2004). "Information Theory and Statistics: A Tutorial" (PDF). Foundations and Trends in Communications and Information Theory. 1 (4): 417–528. doi:10.1561/0100000004. Retrieved 2009-06-14.
• 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.
• MacKay, David J.C. (2003). Information Theory, Inference, and Learning Algorithms. Cambridge University Press. ISBN 0-521-64298-1.