3,075 research outputs found
Outer Approximations of Coherent Lower Probabilities Using Belief Functions
We investigate the problem of outer approximating a coherent lower probability with a more tractable model. In particular, in this work we focus on the outer approximations made by belief functions. We show that they can be obtained by solving a linear programming problem. In addition, we consider the subfamily of necessity measures, and show that in that case we can determine all the undominated outer approximations in a simple manner
Completely monotone outer approximations of lower probabilities on finite possibility spaces
Drawing inferences from general lower probabilities on finite possibility spaces usually involves solving linear programming problems. For some applications this may be too computationally demanding. Some special classes of lower probabilities allow for using computationally less demanding techniques. One such class is formed by the completely monotone lower probabilities, for which inferences can be drawn efficiently once their Möbius transform has been calculated. One option is therefore to draw approximate inferences by using a completely monotone approximation to a general lower probability; this must be an outer approximation to avoid drawing inferences that are not implied by the approximated lower probability. In this paper, we discuss existing and new algorithms for performing this approximation, discuss their relative strengths and weaknesses, and illustrate how each one works and performs
Addressing ambiguity in randomized reinsurance stop-loss treaties using belief functions
The aim of the paper is to model ambiguity in a randomized reinsurance stop-loss treaty. For this, we consider the lower envelope of the set of bivariate joint probability distributions having a precise discrete marginal and an ambiguous Bernoulli marginal. Under an independence assumption, since the lower envelope fails 2-monotonicity, inner/outer Dempster-Shafer approximations are considered, so as to select the optimal retention level by maximizing the lower expected insurer's annual profit under reinsurance. We show that the inner approximation is not suitable in the reinsurance problem, while the outer approximation preserves the given marginal information, weakens the independence assumption, and does not introduce spurious information in the retention level selection problem. Finally, we provide a characterization of the optimal retention level
UNIFYING PRACTICAL UNCERTAINTY REPRESENTATIONS: I. GENERALIZED P-BOXES
Pre-print of final version.International audienceThere exist several simple representations of uncertainty that are easier to handle than more general ones. Among them are random sets, possibility distributions, probability intervals, and more recently Ferson's p-boxes and Neumaier's clouds. Both for theoretical and practical considerations, it is very useful to know whether one representation is equivalent to or can be approximated by other ones. In this paper, we define a generalized form of usual p-boxes. These generalized p-boxes have interesting connections with other previously known representations. In particular, we show that they are equivalent to pairs of possibility distributions, and that they are special kinds of random sets. They are also the missing link between p-boxes and clouds, which are the topic of the second part of this study
Valid and efficient imprecise-probabilistic inference with partial priors, II. General framework
Bayesian inference requires specification of a single, precise prior
distribution, whereas frequentist inference only accommodates a vacuous prior.
Since virtually every real-world application falls somewhere in between these
two extremes, a new approach is needed. This series of papers develops a new
framework that provides valid and efficient statistical inference, prediction,
etc., while accommodating partial prior information and imprecisely-specified
models more generally. This paper fleshes out a general inferential model
construction that not only yields tests, confidence intervals, etc.~with
desirable error rate control guarantees, but also facilitates valid
probabilistic reasoning with de~Finetti-style no-sure-loss guarantees. The key
technical novelty here is a so-called outer consonant approximation of a
general imprecise probability which returns a data- and partial prior-dependent
possibility measure to be used for inference and prediction. Despite some
potentially unfamiliar imprecise-probabilistic concepts in the development, the
result is an intuitive, likelihood-driven framework that will, as expected,
agree with the familiar Bayesian and frequentist solutions in the respective
extreme cases. More importantly, the proposed framework accommodates partial
prior information where available and, therefore, leads to new solutions that
were previously out of reach for both Bayesians and frequentists. Details are
presented here for a wide range of examples, with more practical details to
come in later installments.Comment: Follow-up to arXiv:2203.06703. Feedback welcome at
https://researchers.one/articles/22.11.0000
Special Cases
International audienceThis chapter reviews special cases of lower previsions, that are instrumental in practical applications. We emphasize their various advantages and drawbacks, as well as the kind of problems in which they can be the most useful
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