Constructing Probability Boxes and Dempster-Shafer Structures

Ferson, Scott; Kreinovick, Vladik; Ginzburg, Lev; Sentz, Fari

doi:10.2172/809606

Cited by 530 publications

(496 citation statements)

References 164 publications

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“…< x n . A probability box or p-box [4] (F , F ) is a pair of cumulative distribution functions F , F : X → [0, 1] such that F ≤ F . Here we interpret p-boxes as lower and upper bounds of an ill-known cumulative distribution, that induce a credal set…”

Section: Basic Definitionsmentioning

confidence: 99%

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On extreme points of p-boxes and belief functions

Montes

Destercke

2017

Ann Math Artif Intell

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The extreme points of convex probability sets play an important practical role, especially for specific, easier to manipulate sets. Although this problem has been studied for many models (probability intervals, possibility distributions), it remains to be studied for imprecise cumulative distributions (a.k.a. p-boxes). This is what we do in this paper, where we characterize the maximal number of extreme points of a p-box, give a family of p-boxes that attains this number and show an algorithm that allows to compute the extreme points of a given p-box. To achieve all this, we also provide what we think to be a new characterization of extreme points of a belief function. IntroductionImprecise probability theory [10] is a powerful unifying framework for uncertainty treatment, relying on convex sets of probabilities, or credal sets, to model the uncertainty. Formally, they encompass many existing models: belief functions, possibility distributions, probability intervals, . . . . To apply such models, it is important to study their practical aspects, among which is the characterization of their extreme points. Indeed, these extreme points can be used in many settings, such as graphical models or statistical learning.Extreme points of many models have already been studied. For instance, Dempster [3] shows that the maximal number of extreme points of a belief function on a n-element space is n!. It was later [6] proved that the maximal number of extreme points for possibility distributions in a n-element space is 2 n−1 , and in [8] an algorithm to extract them was provided. In [2], authors studied the extreme points of probability intervals. One practical and popular model for which extreme points have not been characterized are p-boxes [4]. They are special kinds of belief functions whose focal elements are ordered intervals [5,9,10], and are quite instrumental in applications such as risk and reliability analysis.In this paper, we investigate extreme point of p-boxes: we demonstrate that their maximal number is the Pell number, and give the family of p-boxes for which this bound is obtained. To do so, we introduce a new way to characterize the extreme points of a belief function. We also provide an algorithm to compute the extreme points of a given p-box. Section 2 introduces the new characterization, while Section 3.2 studies the extreme points of p-boxes. Due to space restrictions, proofs and side results have been removed. Extreme points of belief functionsGiven a space X = {x 1 , . . . , x n }, a probability mass function is a function m : P(X ) → These two functions are conjugate since Bel(A) = 1 − Pl(A c ), and we can focus on one of them. A focal set of the belief function Bel is a set E such that m(E) > 0, and F will denote the set of focal sets. A belief function also induces a credal setBeing convex, the set M(Bel) can be characterized by its extreme points 1 , that we will denote Ext(Bel). It is known [1,3] that there is a correspondence between the extreme points of a belief function and the perm...

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Section: Basic Definitionsmentioning

confidence: 99%

“…In [2], authors studied the extreme points of probability intervals. One practical and popular model for which extreme points have not been characterized are p-boxes [4]. They are special kinds of belief functions whose focal elements are ordered intervals [5,9,10], and are quite instrumental in applications such as risk and reliability analysis.…”

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confidence: 99%

On extreme points of p-boxes and belief functions

Montes

Destercke

2017

Ann Math Artif Intell

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“…The probability box (p-box) approach, see e.g. [9], can be used to define imprecise stochastic numbers by its lower bound l F (x) and upper bound u F (x) cumulative distribution function (cdf). In general, arbitrary stochastic models, including empirical distributions, can be used for the lower and upper bound cdf.…”

Section: Probability Box Approachmentioning

confidence: 99%

Real-Time Fuzzy Analysis of Machine Driven Tunneling

Cao¹,

Freitag²,

Meschke³

2017

Proceedings of the 2nd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECO

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Abstract. Reliability assessment in mechanized tunneling requires to take into account limited information describing the local geology and the corresponding geotechnical parameters. The geotechnical data are often quite limited and generally not available in the form of precise models and parameter values. In this case, epistemic uncertainty should be considered within the reliability assessment. The concept of fuzzy numbers is applied to predict tunneling induced settlements in real-time. An advanced numerical simulation is utilized as forward model for the settlement prediction. However, to achieve real-time capabilities, surrogate models are required. Deterministic surrogate models can be used together with an ↵-cut optimization approach to compute fuzzy data. In this paper, a surrogate modeling strategy is introduced to directly process fuzzy input-output data. Within this approach, the time-consuming optimization procedure is replaced by a surrogate model to obtain the fuzzy settlement field prediction in realtime. The significant reduction in computation time maintaining similar prediction performance leads to potential applications in steering of mechanized tunneling processes.

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“…P-boxes [17] and DempsterShafer structures [36], which are both related to capacities, are used to propagate both probabilistic and non-deterministic information in numerical simulation for instance. Arithmetic rules on P-boxes were defined in e.g.…”

Section: Related Workmentioning

confidence: 99%

“…Remark that there exists another popular model for imprecise probabilities, namely Probability-boxes [17] (P-box in short). We already showed in [5] that finite DSI structures and discrete P-box are equivalent.…”

Section: Dempster-shafer Structuresmentioning

confidence: 99%

Static Analysis of Programs with Imprecise Probabilistic Inputs

Adjé

Bouissou

Goubault-Larrecq

et al. 2014

Verified Software: Theories, Tools, Experiments

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Abstract. Having a precise yet sound abstraction of the inputs of numerical programs is important to analyze their behavior. For many programs, these inputs are probabilistic, but the actual distribution used is only partially known. We present a static analysis framework for reasoning about programs with inputs given as imprecise probabilities: we define a collecting semantics based on the notion of previsions and an abstract semantics based on an extension of Dempster-Shafer structures. We prove the correctness of our approach and show on some realistic examples the kind of invariants we are able to infer.

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Constructing Probability Boxes and Dempster-Shafer Structures

Cited by 530 publications

References 164 publications

On extreme points of p-boxes and belief functions

On extreme points of p-boxes and belief functions

Real-Time Fuzzy Analysis of Machine Driven Tunneling

Static Analysis of Programs with Imprecise Probabilistic Inputs

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