56 research outputs found
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
A unifying approach to integration for bounded positive charges
This paper deals with n-monotone functionals, which constitute a generalisation of n-monotone set functions. Using the notion of exactness of a functional, we introduce a new notion of lower and upper integral which subsumes as particular cases most of the approaches to integration in the literature. As a consequence, we can characterise which types of integrals can be used to calculate the natural extension (the lower envelope of all linear extensions) of a positive bounded charge
Jensen's and Cantelli's Inequalities with Imprecise Previsions
We investigate how basic probability inequalities can be extended to an
imprecise framework, where (precise) probabilities and expectations are
replaced by imprecise probabilities and lower/upper previsions. We focus on
inequalities giving information on a single bounded random variable ,
considering either convex/concave functions of (Jensen's inequalities) or
one-sided bounds such as or (Markov's and Cantelli's
inequalities). As for the consistency of the relevant imprecise uncertainty
measures, our analysis considers coherence as well as weaker requirements,
notably -coherence, which proves to be often sufficient. Jensen-like
inequalities are introduced, as well as a generalisation of a recent
improvement to Jensen's inequality. Some of their applications are proposed:
extensions of Lyapunov's inequality and inferential problems. After discussing
upper and lower Markov's inequalities, Cantelli-like inequalities are proven
with different degrees of consistency for the related lower/upper previsions.
In the case of coherent imprecise previsions, the corresponding Cantelli's
inequalities make use of Walley's lower and upper variances, generally ensuring
better bounds.Comment: Published in Fuzzy Sets and Systems -
https://dx.doi.org/10.1016/j.fss.2022.06.02
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
The Moment Problem for Finitely Additive Probabilities
We study the moment problem for finitely additive probabilities and show that the information provided by the moments is equivalent to the one given by the associated lower and upper distribution functions
The Hausdorff moment problem under finite additivity
We investigate to what extent finitely additive probability measures on the unit interval are determined by their moment sequence. We do this by studying the lower envelope of all finitely additive probability measures with a given moment sequence. Our investigation leads to several elegant expressions for this lower envelope, and it allows us to conclude that the information provided by the moments is equivalent to the one given by the associated lower and upper distribution functions
Two laws of large numbers for sublinear expectations
In this paper, we consider the sublinear expectation on bounded random vari-
ables. With the notion of uncorrelatedness for random variables under the
sublinear expectation, a weak law of large numbers is obtained. With the no-
tion of independence for random variable sequences and regular property for
sublinear expectations, we get a strong one.Comment: 11 pages;5 reference
Decision Making on Oil Extraction under Z-information
AbstractIn modern conditions, the refining process is complicated and ambiguous, requiring a precise knowledge of all the internal and external factors. However, in many cases, it is impossible to get complete information. Therefore, the process of oil production takes place in conditions of uncertainty accompanying the various situations. A partial absence of beliefs and fuzziness are some of the aspects of uncertainty. In this paper we consider a somewhat different framework for representing our knowledge. Zadeh suggested a Z-number notion, based on a reliability of the given information. In this study we apply Z- information to decision making on oil extraction problem and suggest the framework for decision making on a base of Z-numbers. The method associates with the construction of a non-additive measure as a lower prevision and uses this capacity in Choquet integral for constructing a utility function
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
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