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coincidences and estimation of entropies of random variables with large cardinalities巧合,估计随机变量的熵大的基数.pdf

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coincidences and estimation of entropies of random variables with large cardinalities巧合,估计随机变量的熵大的基数

Entropy 2011, 13, 2013-2023; doi:10.3390/ OPEN ACCESS entropy ISSN 1099-4300 /journal/entropy Article Coincidences and Estimation of Entropies of Random Variables with Large Cardinalities Ilya Nemenman Departments of Physics and Biology and Computational and Life Sciences Initiative, Emory University, 400 Dowman Dr., Atlanta, GA 30322, USA; E-Mail: ilya.nemenman@ Received: 1 November 2011; in revised form: 8 December 2011 / Accepted: 15 December 2011 / Published: 19 December 2011 Abstract: We perform an asymptotic analysis of the NSB estimator of entropy of a discrete random variable. The analysis illuminates the dependence of the estimates on the number of coincidences in the sample and shows that the estimator has a well deﬁned limit for a large cardinality of the studied variable. This allows estimation of entropy with no a priori assumptions about the cardinality. Software implementation of the algorithm is available. Keywords: entropy estimation; coincidences; bias-variance tradeoff; model selection Classiﬁcation: MSC 94A17, 62F12, 62F15 1. Introduction Estimation of functions of a discrete random variable with an unknown probability distribution is one of the simplest problems in statistics. However, the simplicity vanishes in an extremely undersampled regime, where , the cardinality or the alphabet size of the variable, is much larger than , the number of the samples. In this case, the average number of samples per possible outcome, or bin, is less than one, and the relative uncertainty about the underlying probability dis