23 research outputs found
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
Extreme lower probabilities
We consider lower probabilities on finite possibility spaces as models for the uncertainty about the state. These generalizations of classical probabilities can have some interesting properties; for example: k-monotonicity, avoiding sure loss, coherence, permutation invariance. The sets formed by all the lower probabilities satisfying zero or more of these properties are convex. We show how the extreme points and rays of these sets ─ the extreme lower probabilities ─ can be calculated and we give an illustration of our results
Extreme lower probabilities
We consider lower probabilities on finite possibility spaces as models for the uncertainty about the state. These generalizations of classical probabilities can have some interesting properties; for example: k-monotonicity, avoiding sure loss, coherence, permutation invariance. The sets formed by all the lower probabilities satisfying zero or more of these properties are convex. We show how the extreme points and rays of these sets ─ the extreme lower probabilities ─ can be calculated and we give an illustration of our results
Symmetry of models versus models of symmetry
A model for a subject's beliefs about a phenomenon may exhibit symmetry, in
the sense that it is invariant under certain transformations. On the other
hand, such a belief model may be intended to represent that the subject
believes or knows that the phenomenon under study exhibits symmetry. We defend
the view that these are fundamentally different things, even though the
difference cannot be captured by Bayesian belief models. In fact, the failure
to distinguish between both situations leads to Laplace's so-called Principle
of Insufficient Reason, which has been criticised extensively in the
literature.
We show that there are belief models (imprecise probability models, coherent
lower previsions) that generalise and include the Bayesian belief models, but
where this fundamental difference can be captured. This leads to two notions of
symmetry for such belief models: weak invariance (representing symmetry of
beliefs) and strong invariance (modelling beliefs of symmetry). We discuss
various mathematical as well as more philosophical aspects of these notions. We
also discuss a few examples to show the relevance of our findings both to
probabilistic modelling and to statistical inference, and to the notion of
exchangeability in particular.Comment: 61 page
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
Unifying Practical Uncertainty Representations: II. Clouds
There exist many simple tools for jointly capturing variability and
incomplete information by means of uncertainty representations. Among them are
random sets, possibility distributions, probability intervals, and the more
recent Ferson's p-boxes and Neumaier's clouds, both defined by pairs of
possibility distributions. In the companion paper, we have extensively studied
a generalized form of p-box and situated it with respect to other models . This
paper focuses on the links between clouds and other representations.
Generalized p-boxes are shown to be clouds with comonotonic distributions. In
general, clouds cannot always be represented by random sets, in fact not even
by 2-monotone (convex) capacities.Comment: 30 pages, 7 figures, Pre-print of journal paper to be published in
International Journal of Approximate Reasoning (with expanded section
concerning clouds and probability intervals
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
Relating Imprecise Representations of imprecise Probabilities
International audienceThere exist many practical representations of probability families that make them easier to handle. Among them are random sets, possibility distributions, probability intervals, Ferson's p-boxes and Neumaier's clouds. Both for theoretical and practical considerations, it is important to know whether one representation has the same expressive power than other ones, or can be approximated by other ones. In this paper, we mainly study the relationships between the two latter representations and the three other ones
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