A generalized class of certainty and information measures (Q759728)

The objective of the paper is to introduce a generalized probabilistic theory of discrete information measures. Within this framework, information measures are developed with the help of certainty measures. The authors give a definition of a generalized measure of average certainty. In the characterization theorem, the property of multiplicativity (for independent stochastic experiments) plays an important role. Other properties of the certainty measure are also studied. As a special case, the quadratic certainty measure (information energy) is obtained: \(\sum^{n}_{i=1}p^ 2_ i\), for \(p_ i\geq 0\), \(i=1,...,n,\) \(\sum^{n}_{i=1}p_ i=1.\) Another section of the paper deals with the relation between certainty and information, and discusses the various methods in order to obtain measures of information based on the certainty measure. The characterization theorem for the information measures H(X) makes use of the following axiom: for stochastically independent experiments X, Y it holds that \(H(XY)=H(X)+H(Y)+cH(X)H(Y),\) \(c\in {\mathbb{R}}\). This axiom (pseudoadditivity) is a direct consequence of the multiplicativity property of the certainty measure. Properties of the general classes of information measures are also studied. As special cases, there may be obtained most of the presently known information measures.