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A new entropy power inequality for integer-valued random variables

Saeid Haghighatshoar, Emmanuel Abbé, Emre Telatar

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Abstract

The entropy power inequality (EPI) provides lower bounds on the differential entropy of the sum of two independent real-valued random variables in terms of the individual entropies. Versions of the EPI for discrete random variables have been obtained for special families of distributions with the differential entropy replaced by the discrete entropy, but no universal inequality is known (beyond trivial ones). More recently, the sumset theory for the entropy function yields a sharp inequality H(X + X') - H(X) ≥ 1/2 - o(l) when X,X' are i.i.d. with high entropy. This paper provides the inequality H(X + X') - H(X) ≥ g(H(X)), where X, X' are arbitrary i.i.d. integer-valued random variables and where g is a universal strictly positive function on R+satisfying g(0) = 0. Extensions to non identically distributed random variables and to conditional entropies are also obtained.

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What this paper is about

The entropy power inequality (EPI) provides lower bounds on the differential entropy of the sum of two independent real-valued random variables in terms of the individual entropies. Versions of the EPI for discrete random variables have been obtained for special families of distributions with the differential entropy replaced by the discrete entropy, but no universal inequality is known (beyond trivial ones). More recently, the sumset theory for the entropy function yields a sharp inequality H(X + X') - H(X) ≥ 1/2 - o(l) when X,X' are i.i.d. with high entropy. This paper provides the inequality H(X + X') - H(X) ≥ g(H(X)), where X, X' are arbitrary i.i.d. integer-valued random variables and where g is a universal strictly positive function on R+satisfying g(0) = 0. Extensions to non identically distributed random variables and to conditional entropies are also obtained.

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Available abstract

The entropy power inequality (EPI) provides lower bounds on the differential entropy of the sum of two independent real-valued random variables in terms of the individual entropies. Versions of the EPI for discrete random variables have been obtained for special families of distributions with the differential entropy replaced by the discrete entropy, but no universal inequality is known (beyond trivial ones). More recently, the sumset theory for the entropy function yields a sharp inequality H(X + X') - H(X) ≥ 1/2 - o(l) when X,X' are i.i.d. with high entropy. This paper provides the inequality H(X + X') - H(X) ≥ g(H(X)), where X, X' are arbitrary i.i.d. integer-valued random variables and where g is a universal strictly positive function on R+satisfying g(0) = 0. Extensions to non identically distributed random variables and to conditional entropies are also obtained.

Key concepts: Entropy power inequality, Mathematics, Differential entropy, Random variable, Independent and identically distributed random variables, Entropy (arrow of time), Conditional entropy, Discrete mathematics

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