The EPI Of Extensive Entropy

This paper is divided into eight parts, the six parts in front of it are the main body of the article.The first part introduces the origins and development of entropy.The second part gives several commonentropies(Shannon entropy, Tsallis entropy, Renyi entropy, etc.) and the definition of theirproperties.The third part briefly describes the relationships between entropies, including Shannonentropy and Tsallis entropy, Shannon entropy and Renyi entropy Tsallis entropy and Renyi entropy.Theforth part is the main purpose of this paper:the entropy power inequality of Shannon entropy andextended to the EPI of extensive entropy (Renyi entropy, Tsallis entropy, the general form of entropy)and prove them,finally spread the form of entropy power inequality.The fifth part makes good use ofmaximum likelihood estimation and minimum mean square error estimation method to estimate theextensive entropy. The sixth part is the summary and prospection of this paper. The seventh and eighthpart is the acknowledgements and references statement.The forth part of this paper will use the nature of entropy, the nature of mutual information, aswell as the maximum entropy principle to prove entropy power inequality of Shannon entropy; will usethe nature of Renyi entropy and monotonicity to prove the entropy power inequality of Renyi entropy;because Tsallis entropy is specified, this paper will only discuss its entropy inequality in exceptionalcircumstances.. This paper gives three forms of extended of entropy power inequality.The first form ismainly based on classical entropy power inequality so that it can be extended to the n random variables.The second form makes good use of the ideas of arrangement to promote it.The third extended formjust applied to the proof of the EPI of Shannon entropy.The fifth part of the paper uses maximum likelihood estimation、minimum mean square errorestimation method and least error method. to gives three conditions,and get three conclusions,and getthe estimation of extensive entropy.