## The Annals of Applied Probability

### Dictator functions maximize mutual information

#### Abstract

Let $(\boldsymbol{\mathsf{X}},\boldsymbol{\mathsf{Y}})$ denote $n$ independent, identically distributed copies of two arbitrarily correlated Rademacher random variables $(\mathsf{X},\mathsf{Y})$. We prove that the inequality $\mathrm{I}(f(\boldsymbol{\mathsf{X}});g(\boldsymbol{\mathsf{Y}}))\le \mathrm{I}(\mathsf{X};\mathsf{Y})$ holds for any two Boolean functions: $f,g\colon \{-1,1\}^{n}\to \{-1,1\}$ [$\mathrm{I}(\cdot ;\cdot)$ denotes mutual information]. We further show that equality in general is achieved only by the dictator functions $f(\boldsymbol{x})=\pm g(\boldsymbol{x})=\pm x_{i}$, $i\in \{1,2,\ldots,n\}$.

#### Article information

Source
Ann. Appl. Probab., Volume 28, Number 5 (2018), 3094-3101.

Dates
Revised: January 2018
First available in Project Euclid: 28 August 2018

https://projecteuclid.org/euclid.aoap/1535443243

Digital Object Identifier
doi:10.1214/18-AAP1384

Mathematical Reviews number (MathSciNet)
MR3847982

Zentralblatt MATH identifier
06974774

#### Citation

Pichler, Georg; Piantanida, Pablo; Matz, Gerald. Dictator functions maximize mutual information. Ann. Appl. Probab. 28 (2018), no. 5, 3094--3101. doi:10.1214/18-AAP1384. https://projecteuclid.org/euclid.aoap/1535443243

#### References

• [1] Anantharam, V., Gohari, A. A., Kamath, S. and Nair, C. (2013). On hypercontractivity and the mutual information between Boolean functions. In Proc. 51st Annual Allerton Conference on Communication, Control, and Computing 13–19.
• [2] Courtade, T. A. and Kumar, G. R. (2014). Which Boolean functions maximize mutual information on noisy inputs? IEEE Trans. Inform. Theory 60 4515–4525.
• [3] Cover, T. M. and Thomas, J. A. (2006). Elements of Information Theory, 2nd ed. Wiley, Hoboken, NJ.
• [4] Klotz, J. G., Kracht, D., Bossert, M. and Schober, S. (2014). Canalizing Boolean functions maximize mutual information. IEEE Trans. Inform. Theory 60 2139–2147.
• [5] Kumar, G. R. and Courtade, T. A. (2013). Which Boolean functions are most informative? In Proc. IEEE Int. Symp. on Inform. Theory 226–230. DOI:10.1109/ISIT.2013.6620221.
• [6] O’Donnell, R. (2014). Analysis of Boolean Functions. Cambridge Univ. Press, New York.
• [7] Pichler, G. (2017). Clustering by mutual information. Ph.D. thesis, Vienna Univ. Technology.
• [8] Roberts, A. W. and Varberg, D. E. (1973). Convex Functions. Pure and Applied Mathematics 57. Academic Press, New York.