On Measure Theoretic definitions of Generalized Information Measures and Maximum Entropy Prescriptions

Details On Measure Theoretic definitions of Generalized Information Measures and Maximum Entropy Prescriptions On Measure Theoretic definitions of Generalized Information Measures and Maximum Entropy Prescriptions

2006-01-18

arxiv.org/abs/cs/0601080v1

Computer Science

Information Theory

Scientific paper

Though Shannon entropy of a probability measure $P$, defined as $- \int_{X} \frac{\ud P}{\ud \mu} \ln \frac{\ud P}{\ud\mu} \ud \mu$ on a measure space $(X, \mathfrak{M},\mu)$, does not qualify itself as an information measure (it is not a natural extension of the discrete case), maximum entropy (ME) prescriptions in the measure-theoretic case are consistent with that of discrete case. In this paper, we study the measure-theoretic definitions of generalized information measures and discuss the ME prescriptions. We present two results in this regard: (i) we prove that, as in the case of classical relative-entropy, the measure-theoretic definitions of generalized relative-entropies, R\'{e}nyi and Tsallis, are natural extensions of their respective discrete cases, (ii) we show that, ME prescriptions of measure-theoretic Tsallis entropy are consistent with the discrete case.

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