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The Shannon entropy is a widely used summary statistic, for example, networktraffic measurement, anomaly detection, neural computations, spike trains, etc.This study focuses on estimating Shannon entropy of data streams. It is knownthat Shannon entropy can be approximated by Reenyi entropy or Tsallis entropy,which are both functions of the p-th frequency moments and approach Shannonentropy as p->1.  Compressed Counting (CC) is a new method for approximating the p-th frequencymoments of data streams. Our contributions include:  1) We prove that Renyi entropy is (much) better than Tsallis entropy forapproximating Shannon entropy.  2) We propose the optimal quantile estimator for CC, which considerablyimproves the previous estimators.  3) Our experiments demonstrate that CC is indeed highly effectiveapproximating the moments and entropies. We also demonstrate the crucialimportance of utilizing the variance-bias trade-off.

Estimating Entropy of Data Streams Using Compressed Counting