Structure of possibilistic information metrics and distances: properties.(English)Zbl 0703.94002

Summary: Systematic analysis of identities and inequalities satisfied by possibilistic information distances is conducted. The analysis is based on their representations as discrete sums and on certain inequalities for rearrangements of sequences. These identities and inequalities express several properties that are usually deemed characteristic of information distances and measures. In the companion paper those properties are used to obtain several axiomatic characterizations of possibility distances.
The basic distance is $$g(p,q)$$ defined for the distributions $$p=(p_ 1,\dots,p_ n)$$ and $$q=(q_ 1,\dots,q_ n)$$ such that $$p_ i\leq q_ i$$, $$i=1,\dots,n$$. They serve to define a metric $$G(p,q)=g(p,p\vee q)+g(q,p\vee q)$$ and a distance $$H(p,q)=g(p\wedge q,p)+g(p\wedge q,q)$$. All these distances are, in turn based on the $$U$$-uncertainty information function.

MSC:

 94A17 Measures of information, entropy 94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory)
Full Text:

References:

 [1] Aczel J., On Measures of Information and their Characterization. (1975) · Zbl 0308.02045 [2] Dubois D., Possibility Theory. (1988) [3] DOI: 10.1080/03081078208960799 · Zbl 0497.94008 [4] DOI: 10.1080/03081078308960805 · Zbl 0498.94006 [5] DOI: 10.1016/0165-0114(87)90090-X · Zbl 0632.94039 [6] Mathai A., Basic Concepts in Information Theory and Statistics. (1975) · Zbl 0346.94014 [7] Ramer A., Int. J. General Systems 14 (1989) [8] Ramer A., University of Oklahoma Technical Report OU-TR-PPI-89-I [9] DOI: 10.1016/0165-0114(87)90091-1 · Zbl 0637.94027 [10] DOI: 10.1016/S0020-7373(80)80029-0 · Zbl 0435.90006 [11] DOI: 10.1016/0165-0114(78)90029-5 · Zbl 0377.04002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.