Statistical Estimations of Lattice-Valued Possibilistic Distributions

Abstract

The most often applied non-numerical uncertainty degrees are those taking their values in complete lattices, but also their weakened versions may be of interest. In what follows, we introduce and analyze possibilistic distributions and measures taking values in finite upper-valued possibilistic lattices, so that only for finite sets of such values their supremum is defined. For infinite sets of values of the finite lattice in question we apply the idea of the so called Monte-Carlo method: sample at random and under certain conditions a large enough finite subset of the infinite set in question, and take the supremum over this finite sample set as a "good enough" estimation of the undefined supremum of the infinite set. A number of more or less easy to prove assertions demonstrate the conditions when and in which sense the quality of the results obtained by replacing non-existing or non-accessible supremum values by their random estimations tend to the optimum results supposing that the probabilistic qualities of the statistical estimations increases as demanded by Monte-Carlo methods.

Several results on set-valued possibilistic distributions

“…Inspired by Lemma 2.4 and Lemma 4.1, we propose in [7,8,9] some modifications of the space of values in which the mapping π : Ω → T takes its values in such a way that π(ω 0 ) = 1 T is valid for only one ω 0 ∈ Ω. In [7], the mapping π, defined on Ω, takes its values in a complete chained lattice; let us recall, for the reader's convenience, the way leading to this notion.…”