Log sum inequality

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

Statement

Let and be nonnegative numbers. Denote the sum of all s by and the sum of all s by . The log sum inequality states that

with equality if and only if are equal for all , in other words for all .[1]

(Take to be if and if . These are the limiting values obtained as the relevant number tends to .)[1]

Proof

Notice that after setting we have

where the inequality follows from Jensen's inequality since , , and is convex.[1]

Generalizations

The inequality remains valid for provided that and . The proof above holds for any function such that is convex, such as all continuous non-decreasing functions. Generalizations to non-decreasing functions other than the logarithm is given in Csiszár, 2004.

Another generalization is due to Dannan, Neff and Thiel, who showed that if and are positive real numbers with and , and , then . [2]

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.[3] One proof uses the log sum inequality.

The inequality can also prove convexity of Kullback-Leibler divergence.[4]

Notes

  1. Cover & Thomas (1991), p. 29.
  2. F. M. Dannan, P. Neff, C. Thiel (2016). "On the sum of squared logarithms inequality and related inequalities" (PDF). Journal of Mathematical Inequalities. 10 (1): 1–17. doi:10.7153/jmi-10-01. S2CID 23953925. Retrieved 12 January 2023.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  3. MacKay (2003), p. 34.
  4. Cover & Thomas (1991), p. 30.

References

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