945 research outputs found
The (sub/super)additivity assertion of Choquet
The assertion in question comes from the short final section in the Theory of Capacities of Choquet 1953/54, in connection with his prototype of the subsequent Choquet integral. The problem was whether and when this formation is additive. Choquet had the much more abstract idea that all functionals in a certain wide class must be subadditive, and the counterpart with superadditive. His treatment of this point was kind of an outline, and his proof limited to a rather narrow special case. Thus the adequate context and scope of the assertion remained open even up to now. In this paper we present a counterexample which shows that the initial context has to be modified, and then in new context a comprehensive theorem which fulfils all needs turned up so far
Measure and integration : an attempt at unified systematization
The Fundamentals for Set Functions. The Outer and Inner Extension Theorems. Consequences and Applications. The Fundamentals for Functionals. Comparison with the Traditional Daniell-Stone and Bourbaki Procedures
Modeling attitudes toward uncertainty through the use of the Sugeno integral
The aim of the paper is to present under uncertainty, and in an ordinal framework, an axiomatic treatment of the Sugeno integral in terms of preferences which parallels some earlier derivations devoted to the Choquet integral. Some emphasis is given to the characterization of uncertainty aversion.Sugeno integral; uncertainty aversion; preference relations; ordinal information
How regular can maxitive measures be?
We examine domain-valued maxitive measures defined on the Borel subsets of a
topological space. Several characterizations of regularity of maxitive measures
are proved, depending on the structure of the topological space. Since every
regular maxitive measure is completely maxitive, this yields sufficient
conditions for the existence of a cardinal density. We also show that every
outer-continuous maxitive measure can be decomposed as the supremum of a
regular maxitive measure and a maxitive measure that vanishes on compact
subsets under appropriate conditions.Comment: 24 page
Envelopes of conditional probabilities extending a strategy and a prior probability
Any strategy and prior probability together are a coherent conditional
probability that can be extended, generally not in a unique way, to a full
conditional probability. The corresponding class of extensions is studied and a
closed form expression for its envelopes is provided. Then a topological
characterization of the subclasses of extensions satisfying the further
properties of full disintegrability and full strong conglomerability is given
and their envelopes are studied.Comment: 2
A statistical inference method for the stochastic reachability analysis.
The main contribution of this paper is the characterization of reachability problem associated to stochastic hybrid systems in terms of imprecise probabilities. This provides the connection between reachability problem and Bayesian statistics. Using generalised Bayesian statistical inference, a new concept of conditional reach set probabilities is defined. Then possible algorithms to compute the reach set probabilities are derived
Finitely additive extensions of distribution functions and moment sequences: The coherent lower prevision approach
We study the information that a distribution function provides about the finitely additive probability measure inducing it. We show that in general there is an infinite number of finitely additive probabilities associated with the same distribution function. Secondly, we investigate the relationship between a distribution function and its given sequence of moments. We provide formulae for the sets of distribution functions, and finitely additive probabilities, associated with some moment sequence, and determine under which conditions the moments determine the distribution function uniquely. We show that all these problems can be addressed efficiently using the theory of coherent lower previsions
Measure and integral : new foundations after one hundred years
The present article aims to describe the main ideas and developments in the theory of measure and integral in the course and at the end of the first century of its existence
Learning weights in the generalized OWA operators
This paper discusses identification of parameters of generalized ordered weighted averaging (GOWA) operators from empirical data. Similarly to ordinary OWA operators, GOWA are characterized by a vector of weights, as well as the power to which the arguments are raised. We develop optimization techniques which allow one to fit such operators to the observed data. We also generalize these methods for functional defined GOWA and generalized Choquet integral based aggregation operators.<br /
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