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Some Results on Set-Valued Possibilistic Distributions
Kramosil, Ivan
When proposing and processing uncertainty decision making algorithms of various kinds and purposes we meet more and more often probability distributions ascribing to random events non-numerical uncertainty degrees. The reason is that we have to process systems of uncertainties for which the classical conditions like sigma-additivity or linear ordering of values are too restrictive to define sufficiently closely the nature of uncertainty we would like to specify and process. For the case of non-numerical uncertainty degrees at least the two criteria may be considered. First systems with rather complicated, but sophisticated and nontrivially formally analyzable uncertainty degrees. E.g., uncertainties supported by some algebras or partially ordered structures. Contrary, we may consider more easy non-numerical, but on the intuitive level interpretable relations. Well-known examples of such structures are set-valued possibilistic measures. Some perhaps interesting particular results in this direction will be introduced and analyzed in the contribution.
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Doplňování fragmentů posibilistických distribucí s hodnotami v úplném svazu podle principu maximální hodnoty entropie
Kramosil, Ivan
Investigated are Boolean-valued possibilistic distributions taking their values in the power-set of all sets of positive integers. However, some of these possibility degrees may be known only fragmentally in the sense that for the characteristic sequence (identifier) of the set-value in question not all members of this sequence are known. A simple possibilistic entropy function is defined and completions of fragments of possibility degrees with respect to the classical (optimistic or global, in a sense) principle of maximal entropy as well as with respect to some weakened (local or pessimistic, in a sense) versions of this principle are introduced and analyzed.
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Posibilistické entropické funkce s hodnotami ve svazu
Kramosil, Ivan
Lattice-valued entropy functions defined by a lattice-valued possibilistic distribution Pi on a space Omega are defined as the expected value (in the sense of Sugeno integral) of the complement of the value Pi(omega) with omega ranging over the space Omega. The analysis is done, in parallel, for two alternative interpretations of the notion of complement in the complete lattice in question. Supposing that this complete lattice is completely distributive in the defined sense, the entropy value defined by possibilistically independent (noninteractive, in other terms) products of finite sequences of lattice-valued possibilistic distributions are proved to be defined by the supremum value of the entropies defined by particular distributions.
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