Characterizations of sum form information measures on open domains. (English) Zbl 0880.39019
This is a survey of information measures depending on \(k\) complete \(n\)-ary probability distributions (the probabilities being either nonnegative or positive), which are sums of terms depending upon only the first, only the second, …, only the \(n\)-th \(k\)-component probability vectors. In most results reported, just measurability or no regularity at all is supposed regarding the functions generating these dependencies. Additivity requires that the (measure of) information obtained from two \(k\)-tuples of probability distributions equals the sum of informations obtained from the individual probability distributions (one \(m\)-ary, the other \(n\)-ary). Generalizations of this property and symmetries are also considered.
The authors, who also did much of the original work, report that most problems of determining all such information measures have been solved. Some open problems lead to the functional equation \(f(pq) + f(p(1-q)) + f((1-p)q) +f((1-p)(1-q)) = g(p)h(q)\), \(p,q\in]0,1[^{k}\). An 81-item list of references closes the paper.
The authors, who also did much of the original work, report that most problems of determining all such information measures have been solved. Some open problems lead to the functional equation \(f(pq) + f(p(1-q)) + f((1-p)q) +f((1-p)(1-q)) = g(p)h(q)\), \(p,q\in]0,1[^{k}\). An 81-item list of references closes the paper.
Reviewer: J.Aczél (Waterloo/Ontario)
MSC:
| 39B22 | Functional equations for real functions |
| 26A24 | Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems |
| 28A20 | Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence |
| 39B62 | Functional inequalities, including subadditivity, convexity, etc. |
| 94A17 | Measures of information, entropy |
Keywords:
sum form information measures; open domains; probability distributions; functional equation; Information improvement; Deviation; inaccuracy; characterization; Generating function; Differentiable; Derivation; Additive; Multiplicative type; Logarithmic functionsReferences:
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