An optimal uniform concentration inequality for discrete entropies on finite alphabets in the high-dimensional setting

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Abstract

We prove an exponential decay concentration inequality to bound the tail probability of the difference between the log-likelihood of discrete random variables on a finite alphabet and the negative entropy. The concentration bound we derive holds uniformly over all parameter values. The new result improves the convergence rate in an earlier result of Zhao (2020), from (K2 log K)/n = o(1) to (log K)2 /n = o(1), where n is the sample size and K is the size of the alphabet. We further prove that the rate (log K)2 /n = o(1) is optimal. The result is extended to misspecified log-likelihoods for grouped random variables. We give applications of the new result in information theory.

Original languageEnglish (US)
Pages (from-to)1892-1911
Number of pages20
JournalBernoulli
Volume28
Issue number3
DOIs
StatePublished - Aug 2022
Externally publishedYes

Keywords

  • Concentration inequality
  • entropy
  • log-likelihood
  • non-convex optimization
  • source coding theorem
  • typical set

ASJC Scopus subject areas

  • Statistics and Probability

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