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Recursive inset entropies of multiplicative type on open domains

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Summary

Let ℬ be a ring of sets, and letI be thek-dimensional open unit interval. The functional equation

$$\begin{gathered} \varphi (E \cup F,G;p) + \mu (1 - p)\varphi \left( {E,F;\frac{q}{{1 - p}}} \right) \hfill \\ = \varphi (E \cup G,F;q) + \mu (1 - q)\varphi \left( {E,G;\frac{p}{{1 - q}}} \right), \hfill \\ \end{gathered} $$

for all disjoint triplesE, F, G of nonvoid sets in ℬ and all pairsp, q inI withp + q ∈ I, is solved for ϕ and multiplicative μ. This problem was posed by Aczél in Aequationes Math.26 (1984), 255–260. Our solution to this problem leads to an axiomatic characterization of measures of inset informationI n (E 1,⋯,E n ;\(\bar p\) 1,⋯,\(\bar p\) n) which have the representations

$$\begin{gathered} f(E_1 ) + \sum\limits_2^n {g(E_i ) - f\left( {\bigcup\limits_1^n {E_i } } \right) + l(\bar p_1 ) + \sum\limits_1^n {\lambda (E_i ,\bar p_i )} } (for \mu = 1) \hfill \\ f(E_1 )\mu (\bar p_1 ) + \sum\limits_2^n {g(E_i )\mu (\bar p_i ) - f\left( {\bigcup\limits_1^n {E_i } } \right) + \sum\limits_1^n {l(\bar p_1 )\mu (\bar p_i )} } (for \mu additive) \hfill \\ f( \hfill \\ \end{gathered} $$

Herel is logarithmic, λ is additive in the events and logarithmic in the probabilities, andf andg are arbitrary functions.

A key step in the process is to solve the equation

$$J(E \cup F,G) + J(E,F) = J(E \cup G,F) + J(E,G)$$

for disjoint triplesE, F, G of nonempty sets in ℬ. A new construction was developed to handle the case where ℬ happens to be an algebra.

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References

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Authors and Affiliations

  1. Center for Information Theory, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada

    B. R. Ebanks, PL. Kannappan & C. T. Ng

  2. Department of Mathematics, University of Louisville, 40292, Louisville, KY, USA

    B. R. Ebanks, PL. Kannappan & C. T. Ng

Authors
  1. B. R. Ebanks
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  2. PL. Kannappan
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  3. C. T. Ng
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Ebanks, B.R., Kannappan, P. & Ng, C.T. Recursive inset entropies of multiplicative type on open domains. Aeq. Math. 36, 268–293 (1988). https://doi.org/10.1007/BF01836096

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