Results 1  10
of
23
Towards a unified theory of imprecise probability
 Int. J. Approx. Reasoning
, 2000
"... Belief functions, possibility measures and Choquet capacities of order 2, which are special kinds of coherent upper or lower probability, are amongst the most popular mathematical models for uncertainty and partial ignorance. I give examples to show that these models are not sufficiently general to ..."
Abstract

Cited by 40 (0 self)
 Add to MetaCart
Belief functions, possibility measures and Choquet capacities of order 2, which are special kinds of coherent upper or lower probability, are amongst the most popular mathematical models for uncertainty and partial ignorance. I give examples to show that these models are not sufficiently general to represent some common types of uncertainty. Coherent lower previsions and sets of probability measures are considerably more general but they may not be sufficiently informative for some purposes. I discuss two other models for uncertainty, involving sets of desirable gambles and partial preference orderings. These are more informative and more general than the previous models, and they may provide a suitable mathematical setting for a unified theory of imprecise probability.
New Semantics For Quantitative Possibility Theory
 2ND INTERNATIONAL SYMPOSIUM ON IMPRECISE PROBABILITIES AND THEIR APPLICATIONS, ITHACA, NEW YORK
, 2001
"... New semantics for numerical values given to possibility measures are provided. For epistemic possibilities, the new approach is based on the semantics of the transferable belief model, itself based on betting odds interpreted in a less drastic way than what subjective probabilities presupposes. It i ..."
Abstract

Cited by 31 (3 self)
 Add to MetaCart
New semantics for numerical values given to possibility measures are provided. For epistemic possibilities, the new approach is based on the semantics of the transferable belief model, itself based on betting odds interpreted in a less drastic way than what subjective probabilities presupposes. It is shown that the least informative among the belief structures that are compatible with prescribed betting rates is nested, i.e. corresponds to a possibility measure. It is also proved that the idempotent conjunctive combination of two possibility measures corresponds to the hypercautious conjunctive combination of the belief functions induced by the possibility measures. This view di#ers from the subjective semantics first proposed by Giles and relying on upper and lower probability induced by nonexchangeable bets. For objective possibility degrees, the semantics is based on the most informative possibilistic approximation of a probability measure derived from a histogram. The motivation for this semantics is its capability to extend a wellknown kind of confidence intervals around the mode of a distribution to a fuzzy confidence interval. We show how the idempotent disjunctive combination of possibility functions is related to the convex mixture of probability distributions.
A Random Set Description of a Possibility Measure and Its Natural Extension
 IEEE Transactions on Systems, Man and Cybernetics
, 1997
"...  The relationship is studied between possibility and necessity measures dened on arbitrary spaces, the theory of imprecise probabilities, and elementary random set theory. It is shown how special random sets can be used to generate normal possibility and necessity measures, as well as their natural ..."
Abstract

Cited by 17 (7 self)
 Add to MetaCart
 The relationship is studied between possibility and necessity measures dened on arbitrary spaces, the theory of imprecise probabilities, and elementary random set theory. It is shown how special random sets can be used to generate normal possibility and necessity measures, as well as their natural extensions. This leads to interesting alternative formulas for the calculation of these natural extensions. KeywordsUpper probability, upper prevision, coherence, natural extension, possibility measure, random sets. I. Introduction P OSSIBILITY measures were introduced by Zadeh [1] in 1978. In his view, these supremum preserving set functions are a mathematical representation of the information conveyed by typical armative statements in natural language. For recent discussions of this interpretation within the behavioural framework of the theory of imprecise probabilities, we refer to [2], [3], [4]. Supremum preserving set functions can also be found in the literature under a number o...
Joint propagation of probability and possibility in risk analysis: Towards a formal framework
 INTERNATIONAL JOURNAL OF APPROXIMATE REASONING
, 2007
"... ..."
A behavioural model for vague probability assessments
, 2003
"... I present an hierarchical uncertainty model that is able to represent vague probability assessments, and to make inferences based on them. This model can be given an interpretation in terms of the behaviour of a modeller in the face of uncertainty, and is based on Walley’s theory of imprecise proba ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
I present an hierarchical uncertainty model that is able to represent vague probability assessments, and to make inferences based on them. This model can be given an interpretation in terms of the behaviour of a modeller in the face of uncertainty, and is based on Walley’s theory of imprecise probabilities. It is formally closely related to Zadeh’s fuzzy probabilities, but it has a different interpretation, and a different calculus. Through rationality (coherence) arguments, the hierarchical model is shown to lead to an imprecise firstorder uncertainty model that can be used in decision making, and as a prior in statistical reasoning.
Precision–imprecision equivalence in a broad class of imprecise hierarchical uncertainty models
 Journal of Statistical Planning and Inference
, 2000
"... ABSTRACT. Hierarchical models are rather common in uncertainty theory. They arise when there is a ‘correct ’ or ‘ideal ’ (socalled firstorder) uncertainty model about a phenomenon of interest, but the modeler is uncertain about what it is. The modeler’s uncertainty is then called secondorder unce ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
ABSTRACT. Hierarchical models are rather common in uncertainty theory. They arise when there is a ‘correct ’ or ‘ideal ’ (socalled firstorder) uncertainty model about a phenomenon of interest, but the modeler is uncertain about what it is. The modeler’s uncertainty is then called secondorder uncertainty. For most of the hierarchical models in the literature, both the first and the secondorder models are precise, i.e., they are based on classical probabilities. In the present paper, I propose a specific hierarchical model that is imprecise at the second level, which means that at this level, lower probabilities are used. No restrictions are imposed on the underlying firstorder model: that is allowed to be either precise or imprecise. I argue that this type of hierarchical model generalizes and includes a number of existing uncertainty models, such as imprecise probabilities, Bayesian models, and fuzzy probabilities. The main result of the paper is what I call Precision–Imprecision Equivalence: the implications of the model for decision making and statistical reasoning are the same, whether the underlying firstorder model is assumed to be precise or imprecise. 1.
Reconciling Frequentist Properties With The Likelihood Principle
, 1998
"... This paper is devoted primarily to a presentation of some main features of these developments, which seem to have intrinsic as well as historical interest. These developments include an apparently decisive negative outcome. It has seemed to some (including this writer) that any adequate concept of s ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
This paper is devoted primarily to a presentation of some main features of these developments, which seem to have intrinsic as well as historical interest. These developments include an apparently decisive negative outcome. It has seemed to some (including this writer) that any adequate concept of statistical evidence must meet at least certain minimum versions of both of the criteria just indicated. But the difficulties of developing such a concept have become increasingly apparent, and it now seems rather clear that no such adequate concept of statistical evidence can exist.
Imprecise reliability: An introductory overview
"... The main aim of this paper is to define what imprecise reliability is, and discuss a variety of problems that can be solved by means of a framework of imprecise probabilities. From this point of view, various branches of reliability analysis are considered, including analysis of monotone systems, re ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
The main aim of this paper is to define what imprecise reliability is, and discuss a variety of problems that can be solved by means of a framework of imprecise probabilities. From this point of view, various branches of reliability analysis are considered, including analysis of monotone systems, repairable systems, multistate systems, structural reliability, software reliability, human reliability, fault tree analysis. Various types of initial information used in imprecise reliability are considered. Some open problems are briefly discussed in the concluding section.
Possibilistic Previsions
, 1998
"... The paper deals with a possibilistic imprecise secondorder probability model. It is argued that such models appear naturally in a number of situations. They lead to the introduction of a new type of previsions, called possibilistic previsions, which formally generalise coherent upper and lower prev ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
The paper deals with a possibilistic imprecise secondorder probability model. It is argued that such models appear naturally in a number of situations. They lead to the introduction of a new type of previsions, called possibilistic previsions, which formally generalise coherent upper and lower previsions. The converse problem is also looked at: given a possibilistic prevision, under what conditions can it be generated by a secondorder possibility distribution? This leads to the definition of normality, representability and natural extension of possibilistic previsions. Finally, some attention is paid to the special class of full possibilistic previsions, which can be formally related to Zadeh's fuzzy probabilities. The results have immediate applicability in decision making and statistical reasoning. 1 Introduction Consider an unstable radioactive nucleus. Its probability of decay in a given time interval t is given by 1 \Gamma e \Gammat , where the parameter is the decay rate o...
Practical Implementation of Possibilistic Probability Mass Functions
 In Proceedings of Fifth Workshop on Uncertainty Processing (WUPES 2000) (Jindvrichouv Hradec, Czech Republic
, 2000
"... Probability assessments of events are often linguistic in nature. We model them by means of possibilistic probabilities (a version of Zadeh's fuzzy probabilities with a behavioural interpretation) with a suitable shape for practical implementation (on a computer). Employing the tools of interval ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
Probability assessments of events are often linguistic in nature. We model them by means of possibilistic probabilities (a version of Zadeh's fuzzy probabilities with a behavioural interpretation) with a suitable shape for practical implementation (on a computer). Employing the tools of interval analysis and the theory of imprecise probabilities we argue that the verication of coherence for these possibilistic probabilities, the corrections of noncoherent to coherent possibilistic probabilities and their extension to other events and gambles can be performed by nite and exact algorithms. The model can furthermore be transformed into an imprecise rstorder model, useful for decision making and statistical inference. 1 Introduction Consider a football match in which the three possible outcomes are win (w), draw (d) and loss (l) for the home team. Suppose we have the following probability judgements for a specic match: win is likely to occur, draw and loss both have a chan...