5,275 research outputs found
A unified view of some representations of imprecise probabilities
International audienceSeveral methods for the practical representation of imprecise probabilities exist such as Ferson's p-boxes, possibility distributions, Neumaier's clouds, and random sets . In this paper some relationships existing between the four kinds of representations are discussed. A cloud as well as a p-box can be modelled as a pair of possibility distributions. We show that a generalized form of p-box is a special kind of belief function and also a special kind of cloud
Probability theory and its models
This paper argues for the status of formal probability theory as a
mathematical, rather than a scientific, theory. David Freedman and Philip
Stark's concept of model based probabilities is examined and is used as a
bridge between the formal theory and applications.Comment: Published in at http://dx.doi.org/10.1214/193940307000000347 the IMS
Collections (http://www.imstat.org/publications/imscollections.htm) by the
Institute of Mathematical Statistics (http://www.imstat.org
Other uncertainty theories based on capacities
International audienceThe two main uncertainty representations in the literature that tolerate imprecision are possibility distributions and random disjunctive sets. This chapter devotes special attention to the theories that have emerged from them. The first part of the chapter discusses epistemic logic and derives the need for capturing imprecision in information representations. It bridges the gap between uncertainty theories and epistemic logic showing that imprecise probabilities subsume modalities of possibility and necessity as much as probability. The second part presents possibility and evidence theories, their origins, assumptions and semantics, discusses the connections between them and the general framework of imprecise probability. Finally, chapter points out the remaining discrepancies between the different theories regarding various basic notions, such as conditioning, independence or information fusion and the existing bridges between them
Clouds, p-boxes, fuzzy sets, and other uncertainty representations in higher dimensions
Uncertainty modeling in real-life applications comprises some serious problems such as the curse of dimensionality and a lack of sufficient amount of statistical data. In this paper we give a survey of methods for uncertainty handling and elaborate the latest progress towards real-life applications with respect to the problems that come with it. We compare different methods and highlight their relationships. We introduce intuitively the concept of potential clouds, our latest approach which successfully copes with both higher dimensions and
incomplete information
Imprecise Label Learning: A Unified Framework for Learning with Various Imprecise Label Configurations
Learning with reduced labeling standards, such as noisy label, partial label,
and multiple label candidates, which we generically refer to as
\textit{imprecise} labels, is a commonplace challenge in machine learning
tasks. Previous methods tend to propose specific designs for every emerging
imprecise label configuration, which is usually unsustainable when multiple
configurations of imprecision coexist. In this paper, we introduce imprecise
label learning (ILL), a framework for the unification of learning with various
imprecise label configurations. ILL leverages expectation-maximization (EM) for
modeling the imprecise label information, treating the precise labels as latent
variables.Instead of approximating the correct labels for training, it
considers the entire distribution of all possible labeling entailed by the
imprecise information. We demonstrate that ILL can seamlessly adapt to partial
label learning, semi-supervised learning, noisy label learning, and, more
importantly, a mixture of these settings. Notably, ILL surpasses the existing
specified techniques for handling imprecise labels, marking the first unified
framework with robust and effective performance across various challenging
settings. We hope our work will inspire further research on this topic,
unleashing the full potential of ILL in wider scenarios where precise labels
are expensive and complicated to obtain.Comment: 29 pages, 3 figures, 16 tables, preprin
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
Why Credences Cannot be Imprecise
Beliefs formed under uncertainty come in different grades, which are called credences or degrees of belief. The most common way of measuring the strength of credences is by ascribing probabilities to them. What kind of probabilities may be used remains an open question and divides the researchers in two camps: the sharpers who claim that credences can be measured by the standard single-valued precise probabilities. The non-sharpers, on the other hand, claim that credences are imprecise and can only be measured by imprecise probabilities. The latter view has recently gained in popularity. According to non-sharpers, credences must be imprecise when the evidence is essentially imprecise (ambiguous, vague, conflicting or scarce).
This view is, however, misleading. Imprecise credences can lead to irrational behaviour and do not make much sense after a closer examination. I provide a coherence-based principle which enables me to demonstrate that there is no need for imprecise credences. This principle is then applied to three special cases, which are prima facie best explained by use of imprecise credences: the jellyfish guy case, Ellsberg paradox and the Sleeping Beauty problem.
The jellyfish guy case deals with a strange situation, where the evidence is very ambiguous. Ellsberg Paradox demonstrates a problem that occurs when comparing precise and imprecise credences. The Sleeping Beauty problem demonstrates that imprecise credences are not useless, but rather misguided. They should be understood as sets of possible precise credences, of which only one can be selected at a given time
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