Sharp Bounds on the Entropy of the Poisson Law and Related Quantities

Details Sharp Bounds on the Entropy of the Poisson Law and Related Quantities Sharp Bounds on the Entropy of the Poisson Law and Related Quantities

2010-01-17

arxiv.org/abs/1001.2897v1

IEEE Trans. Inform. Theory 56 (2010) 2299 - 2306

Computer Science

Information Theory

To appear, IEEE Trans. Inform. Theory

Scientific paper

10.1109/TIT.2010.2044057

One of the difficulties in calculating the capacity of certain Poisson channels is that H(lambda), the entropy of the Poisson distribution with mean lambda, is not available in a simple form. In this work we derive upper and lower bounds for H(lambda) that are asymptotically tight and easy to compute. The derivation of such bounds involves only simple probabilistic and analytic tools. This complements the asymptotic expansions of Knessl (1998), Jacquet and Szpankowski (1999), and Flajolet (1999). The same method yields tight bounds on the relative entropy D(n, p) between a binomial and a Poisson, thus refining the work of Harremoes and Ruzankin (2004). Bounds on the entropy of the binomial also follow easily.

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