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Journal of Theoretical Probability

, Volume 29, Issue 1, pp 48–62 | Cite as

The Kearns–Saul Inequality for Bernoulli and Poisson-Binomial Distributions

  • Eckhard Schlemm
Article

Abstract

We give a direct rigorous proof of the Kearns–Saul inequality, which bounds the Laplace transform of a generalised Bernoulli random variable. We extend the arguments to generalised Poisson-binomial distributions and characterise the set of parameters such that an analogous inequality holds for the sum of two generalised Bernoulli random variables.

Keywords

Bernoulli distribution Kearns–Saul inequality Laplace transform Poisson-binomial distribution 

Mathematics Subject Classification (2010)

60E10 

References

  1. 1.
    Berend, D., Kontorovich, A.: On the concentration of the missing mass. Electron. Commun. Probab. 18(3), 1–7 (2013)MathSciNetGoogle Scholar
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    Chen, S.X., Liu, J.S.: Statistical applications of the Poisson-binomial and conditional Bernoulli distributions. Stat. Sin. 7(4), 875–892 (1997)MATHGoogle Scholar
  3. 3.
    Kearns, M., Saul, L.: Large deviation methods for approximate probabilistic inference. In: Proceedings of the Fourteenth conference on Uncertainty in artificial intelligence, pp. 311–319. Morgan Kaufmann Publishers Inc., Los Altos (1998)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Wolfson CollegeUniversity of CambridgeCambridgeUK

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