A new entropy power inequality for integer-valued random variables
A new entropy power inequality for integer-valued random variables
复制标题
整数值随机变量的新熵幂不等式
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
E. Telatar
中科院分区：
文献类型：
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作者：
Saeid Haghighatshoar;E. Abbe;E. Telatar
熵幂不等式 (EPI) 提供了两个独立实值随机变量之和的微分熵的下界（以各个熵表示）。离散随机变量的 EPI 版本已经针对特殊的分布族获得，其中微分熵被离散熵取代，但不存在普遍的不等式（除了平凡的不等式）。最近，当 X,X' 独立同分布时，熵函数的求和集理论产生了尖锐的不等式 H(X + X') - H(X) ≥ 1/2 - o(l)。具有高熵。本文提供了不等式 H(X + X') - H(X) ≥ g(H(X))，其中 X、X' 是任意独立同分布。整数值随机变量，其中 g 是 R+ 上满足 g(0) = 0 的通用严格正函数。还获得了对非同分布随机变量和条件熵的扩展。
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.