A Preferred Definition of Conditional Rényi Entropy

The Renyi entropy is a generalization of Shannon entropy to a one-parameter family of entropies. Tsallis entropy too is a generalization of Shannon entropy. The measure for Tsallis entropy is non-logarithmic. After the introduction of Shannon entropy , the conditional Shannon entropy was derived and its properties became known. Also, for Tsallis entropy, the conditional entropy was introduced and its properties were shown. But no specific definition has been given for the conditional Renyi entropy. Several authors have used some definitions of the conditional Renyi entropy, to find their properties and relations among them, but there is no general agreement on any specific definition In this paper, we focus on the definitions of the conditional Renyi entropy, and select one of them on the basis of a relation between Renyi and Tsallis entropies, and show that the chain rule holds generally for the case of conditional Renyi entropy. Then, using this definition, we show some of the properties of conditional Renyi entropy. Finally, we show the relations among Renyi, Shannon and Tsallis entropies.