A proof of the Shepp-Olkin entropy concavity conjecture
A proof of the Shepp-Olkin entropy concavity conjecture
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DOI：
10.3150/16-bej860
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发表时间：
2017-11-01
期刊：
影响因子：
1.5
通讯作者：
Johnson, Oliver
中科院分区：
文献类型：
--
作者：
Hillion, Erwan;Johnson, Oliver
我们证明了谢普-奥尔金猜想，即独立的伯努利随机变量之和的熵在单个随机变量的参数中是凹的。我们的证明改进了同一作者之前提出的一个论点，解决了单调情况下的猜想(其中所有参数都同时增加)。事实上，通过仔细分析概率质量函数导数的凹性，我们证明了单调情况是最坏的情况。我们推广了谢普和奥尔金的猜想，考虑了Renyi和Tsallis熵。
We prove the Shepp-Olkin conjecture, which states that the entropy of the sum of independent Bernoulli random variables is concave in the parameters of the individual random variables. Our proof refines an argument previously presented by the same authors, which resolved the conjecture in the monotonic case (where all the parameters are simultaneously increasing). In fact, we show that the monotonic case is the worst case, using a careful analysis of concavity properties of the derivatives of the probability mass function. We propose a generalization of Shepp and Olkin's original conjecture, to consider Renyi and Tsallis entropies.