Journal:Informatica
Volume 4, Issues 1-2 (1993), pp. 21–44
Abstract
As a rule, a measure is a mapping from a σ-field of sets into the set of reals, or more generally, into some Banach space. A concept of set-valued measure (SV-measure) is introduced in the paper being a specific mapping from a σ-field of sets into a power set of a set. Properties of SV-measures are analyzed and illustrated on examples. Close relationship between SV-measures and a new nonstandard approach in artificial intelligence (AI) is explained. Then, the construction of factorization of the measures is mentioned, a special class of σ-quasiatomic SV-measures is defined and corresponding characterization theorem is proved. This class involves SV-measures ranging in a countable set which were used in modelling uncertainty in AI. It enables to answer one question arising in connection with this application.
Journal:Informatica
Volume 3, Issue 4 (1992), pp. 497–523
Abstract
It is well known that many practical optimization problems with random elements lead from the mathematical point of view to deterministic optimization problems depending on the random elements through probability laws only. Further, it is also well known that these probability laws are known very seldom. Consequently, statistical estimates of the unknown probability measure, if they exist, must be employed to obtain some estimates of the optimal value and the optimal solution, at least.
If the theoretical distribution function is completely unknown then an empirical distribution usually substitutes it [2, 3, 9, 17, 31]. The great attention has been already paid to the studying of statistical properties of such arised empirical estimates, in the literature. We can remember here the works [4, 5, 6, 10, 13, 16, 32], for example. The aim of this paper is to discuss the convergence rate. For this we shall employed the assertions of the papers [10, 11, 13].