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## Lp consonant approximations of belief functions (2012)

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3132 |
Mathematical theory of evidence
- Shafer
- 1979
(Show Context)
Citation Context ... have to be computed in order to find the overall approximation. Interpretations of the obtained approximations in terms of degrees of belief are proposed. 1 Introduction The theory of evidence (ToE) =-=[1]-=- is a popular approach to uncertainty description. Probabilities are there replaced by belief functions (b.f.s), which assign values between 0 and 1 to subsets of the sample space Θ instead of single ... |

489 | The transferable belief model, in:
- Smets, Kennes
- 2008
(Show Context)
Citation Context ...rdinate axes {XA, ∅ � A � Θ} in RN−2 , a belief function b can be represented by the vector b = ∑ ∅�A�Θ b(A)XA. . If we denote by bA = b ∈ B s.t. mbA(A) = 1, mbA(B) = 0 ∀B ⊆ Θ, B = A the categorical =-=[20]-=- belief function assigning all the mass to a single subset A ⊆ Θ, we can prove that [21, 17] the set of points of RN−2 which correspond to a b.f. or “belief space” B coincides with the convex closure ... |

318 |
Possibility Theory
- Dubois, Prade
- 1988
(Show Context)
Citation Context ...′ ∈ B representing two belief functions b, b ′, such norms read as: ‖b − b ′ ‖1 . = ∑ ∅�B⊆Θ |b(B) − b ′ (B)|; ‖b − b ′ ‖2 . = √ ∑ ∅�B⊆Θ (b(B) − b ′ (B)) 2 ; ‖b − b ′ . ‖∞ = max ∅�B⊆Θ |b(B) − b′ (B)|. =-=(2)-=- As the consonant complex CO is a collection of linear spaces (better, simplices which generate a linear space), solving the consonant approximation problem involves finding a number of partial soluti... |

228 | Combinatorial Theory - Aigner - 1979 |

135 | Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion,” - Chateauneuf, Jaffray - 1989 |

90 | Bayesian and non-Bayesian evidential updating - Kyburg - 1987 |

89 | Modern geometry: Methods and applications - Dubrovin, Novikov, et al. - 1979 |

77 | Computationally efficient approximation of Dempster-Shafer theory,” - Voorbraak - 1989 |

71 | Approximations for efficient computation in the theory of evidence,” - Tessem - 1993 |

62 | Sufficient conditions for convergence of the sum-product algorithm. - Mooij, Kappen - 2007 |

50 |
Consonant approximations of belief functions.
- Dubois, Prade
- 1990
(Show Context)
Citation Context ...ation of the Dempster-Shafer theory to a fuzzy valued measure. The links between transferable belief model and possibility theory have been briefly investigated by Ph. Smets in [12].Dubois and Prade =-=[3]-=-, more specifically, have extensively worked on consonant approximations of belief functions. Their work has been later considered in [4, 5]. In particular, the notion of “outer consonant approximatio... |

47 | A geometric approach to the theory of evidence.
- Cuzzolin
- 2008
(Show Context)
Citation Context ...ojection π[b] of a belief function onto the probability simplex [14], and studied consistent approximations of belief functions induced by classical Lp norms [15, 16] in the space of belief functions =-=[17]-=-. In [18] he has shown that norm minimization can also be used to define families of geometric conditional belief functions. Jousselme et al [19] have recently conducted a very nice survey of the dist... |

47 | Additive representation of non-additive measures and the Choquet integral. - Gilboa, Schmeidler - 1994 |

47 | Belief functions versus probability functions,” in Uncertainty and Intelligent - Smets - 1988 |

47 | On the Plausibility Transformation Method for Translating Belief Function Models to Probability Models,” - Cobb, Shenoy - 2006 |

39 | New semantics for quantitative possibility theory. In: - Dubois, Prade, et al. - 2001 |

37 | Two new bayesian approximations of belief functions based on convex geometry
- Cuzzolin
- 2007
(Show Context)
Citation Context ...f functions by minimizing appropriate distance functions has been explored. The author has indeed introduced the notion of orthogonal projection π[b] of a belief function onto the probability simplex =-=[14]-=-, and studied consistent approximations of belief functions induced by classical Lp norms [15, 16] in the space of belief functions [17]. In [18] he has shown that norm minimization can also be used t... |

32 | Inner and outer approximation of belief structures using a hierarchical clustering approach,” Int - Denoeux - 2001 |

31 | The nature of the unnormalized beliefs encountered in the transferable belief model. In: - Smets - 1992 |

24 | On transformations of belief functions to probabilities,” - Daniel - 2006 |

23 | Unfair coins and necessity measures: Towards a possibilistic interpretation of histograms. - Dubois, Prade - 1983 |

21 | Geometry of Dempster’s rule of combination - Cuzzolin - 2004 |

20 |
A class of fuzzy measures generated from a Dempster-Shafer belief structure,
- Yager
- 1999
(Show Context)
Citation Context ...d by their values on the singletons P os(x), x ∈ Θ, they are less computationally expensive than b.f.s, making the approximation process interesting for many applications. Many authors, such as Yager =-=[7]-=- and Romer [8] amongst others, have studied the connection between fuzzy numbers and Dempster-Shafer theory. Klir et al have published an excellent discussion [9] on the relations among fuzzy and beli... |

19 | Geometric analysis of belief space and conditional subspaces, in
- Cuzzolin, Frezza
- 2001
(Show Context)
Citation Context ...tor b = ∑ ∅�A�Θ b(A)XA. . If we denote by bA = b ∈ B s.t. mbA(A) = 1, mbA(B) = 0 ∀B ⊆ Θ, B = A the categorical [20] belief function assigning all the mass to a single subset A ⊆ Θ, we can prove that =-=[21, 17]-=- the set of points of RN−2 which correspond to a b.f. or “belief space” B coincides with the convex closure of all the categorical belief functions bA: B = Cl(bA, { ∅ � A ⊆ Θ), where Cl denotes the co... |

18 | Distances in evidence theory: Comprehensive survey and generalizations. - Jousselme, Maupin - 2012 |

18 | A Monte-Carlo Algorithm for Dempster-Shafer Belief - Wilson - 1991 |

18 | Message errors in belief propagation - Ihler, Fisher, et al. - 2004 |

17 | On transforming belief function models to probability models”, Working Paper - Cobb, Shenoy |

16 | Extending consonant approximations to capacities. In:
- Baroni
- 2004
(Show Context)
Citation Context ...e to introduce the notion of “outer consonant approximations” [3] of a belief function b, i.e., those co.b.f.s such that ∀A ⊆ Θ co(A) ≤ b(A). Dubois and Prade’s work has been later extended by Baroni =-=[6]-=- to capacities. In [13] the author has provided a comprehensive description of the geometry of the set of outer consonant approximations. In recent times the opportunity of seeking probability or cons... |

16 |
Constructing fuzzy measures in expert systems, fuzzy sets and systems 92
- Klir, J
- 1997
(Show Context)
Citation Context ...ications. Many authors, such as Yager [7] and Romer [8] amongst others, have studied the connection between fuzzy numbers and Dempster-Shafer theory. Klir et al have published an excellent discussion =-=[9]-=- on the relations among fuzzy and belief measures and possibility theory. Heilpern [10] has also presented the theoretical background of fuzzy numbers connected with the possibility and Dempster-Shafe... |

16 |
Sovremennaja geometrija. Metody i prilozenija
- Dubrovin, Novikov, et al.
- 1986
(Show Context)
Citation Context ... associated with singletons1 : P = Cl(bx, x ∈ Θ). The consonant complex. In this framework the geometry of consonant belief functions can be described by resorting to the notion of simplicial complex =-=[22]-=-. A simplicial complex is a collection Σ of simplices of arbitrary dimensions possessing the following properties: 1. if a simplex belongs to Σ, then all its faces of any dimension belong to Σ; 2. the... |

14 | Minimal Information Loss Possibilistic Approximations of Random
- Joslyn, Klir
- 1992
(Show Context)
Citation Context ...en briefly investigated by Ph. Smets in [12].Dubois and Prade [3], more specifically, have extensively worked on consonant approximations of belief functions. Their work has been later considered in =-=[4, 5]-=-. In particular, the notion of “outer consonant approximation” has received considerable attention in the past. Indeed, belief functions admit the following order relation: b ≤ b ′ ≡ ∀A ⊆ Θb(A) ≤ b ′ ... |

14 |
Representation and application of fuzzy numbers, Fuzzy sets and Systems
- Heilpern
(Show Context)
Citation Context ...he connection between fuzzy numbers and Dempster-Shafer theory. Klir et al have published an excellent discussion [9] on the relations among fuzzy and belief measures and possibility theory. Heilpern =-=[10]-=- has also presented the theoretical background of fuzzy numbers connected with the possibility and Dempster-Shafer theories, describing some types of representation of fuzzy numbers and studying the n... |

14 | On possibility/probability transformations, in: Fuzzy Logic: State of the - Dubois, Prade, et al. - 1993 |

13 |
The transferable belief model and possibility theory
- Smets
- 1990
(Show Context)
Citation Context ...e suggested a generalization of the Dempster-Shafer theory to a fuzzy valued measure. The links between transferable belief model and possibility theory have been briefly investigated by Ph. Smets in =-=[12]-=-.Dubois and Prade [3], more specifically, have extensively worked on consonant approximations of belief functions. Their work has been later considered in [4, 5]. In particular, the notion of “outer ... |

13 | The geometry of consonant belief functions: Simplicial complexes of possibility measures. Fuzzy Sets and Systems (under review)
- Cuzzolin
- 2007
(Show Context)
Citation Context ...ion of “outer consonant approximations” [3] of a belief function b, i.e., those co.b.f.s such that ∀A ⊆ Θ co(A) ≤ b(A). Dubois and Prade’s work has been later extended by Baroni [6] to capacities. In =-=[13]-=- the author has provided a comprehensive description of the geometry of the set of outer consonant approximations. In recent times the opportunity of seeking probability or consonant approximations/tr... |

13 | Equivalence between belief theories and naive bayesian fusion for systems with independent evidential data: Part II, the example - Sudano - 2003 |

12 | Possibilistic Normalization of Inconsistent Random
- Joslyn
- 1997
(Show Context)
Citation Context ...en briefly investigated by Ph. Smets in [12].Dubois and Prade [3], more specifically, have extensively worked on consonant approximations of belief functions. Their work has been later considered in =-=[4, 5]-=-. In particular, the notion of “outer consonant approximation” has received considerable attention in the past. Indeed, belief functions admit the following order relation: b ≤ b ′ ≡ ∀A ⊆ Θb(A) ≤ b ′ ... |

12 | Representation and application of fuzzy numbers - Heilpern - 1997 |

12 | new Bayesian approximations of belief functions based on convex geometry - “Two |

11 | A mathematical analysis of information-preserving transformations between probabilistic and possibilistic formulations of Uncertainty”, Intern - Geer, Klir - 1992 |

10 | Extreme points of credal sets generated by 2-alternating capacities. - Miranda, Couso, et al. - 2003 |

9 |
Applicability analysis of fuzzy inference by means of generalized Dempster-Shafer theory
- Roemer, Kandel
- 1995
(Show Context)
Citation Context ...ues on the singletons P os(x), x ∈ Θ, they are less computationally expensive than b.f.s, making the approximation process interesting for many applications. Many authors, such as Yager [7] and Romer =-=[8]-=- amongst others, have studied the connection between fuzzy numbers and Dempster-Shafer theory. Klir et al have published an excellent discussion [9] on the relations among fuzzy and belief measures an... |

8 | Geometric conditioning of belief functions
- Cuzzolin
- 2010
(Show Context)
Citation Context ...π[b] of a belief function onto the probability simplex [14], and studied consistent approximations of belief functions induced by classical Lp norms [15, 16] in the space of belief functions [17]. In =-=[18]-=- he has shown that norm minimization can also be used to define families of geometric conditional belief functions. Jousselme et al [19] have recently conducted a very nice survey of the distance or s... |

8 | A similarity measure between basic belief assignments - Diaz, Rifqi, et al. - 2006 |

8 | On some properties of distances in evidence theory - Jousselme, Maupin |

7 |
Generalization of the dempster-shafer theory: A fuzzy-valued measure
- Caro
(Show Context)
Citation Context ...Dempster-Shafer theories, describing some types of representation of fuzzy numbers and studying the notions of distance and order between fuzzy numbers based on these representations. Caro and Nadjar =-=[11]-=-, instead, have suggested a generalization of the Dempster-Shafer theory to a fuzzy valued measure. The links between transferable belief model and possibility theory have been briefly investigated by... |

7 | Constructing consonant belief functions from sample data using confidence sets of pignistic probabilities - Aregui, Denœux - 2008 |

6 | Consistent approximation of belief functions
- Cuzzolin
- 2009
(Show Context)
Citation Context ...eed introduced the notion of orthogonal projection π[b] of a belief function onto the probability simplex [14], and studied consistent approximations of belief functions induced by classical Lp norms =-=[15, 16]-=- in the space of belief functions [17]. In [18] he has shown that norm minimization can also be used to define families of geometric conditional belief functions. Jousselme et al [19] have recently co... |

6 | A new evidential distance measure based on belief intervals, Scientia Iranica - Khatibi, Montazer - 2010 |

5 | A new method to determine evidence distance - Shi, Cheng, et al. |

5 | Belief functions on real numbers, Int - Smets |

5 | geometric approach to the theory of evidence - “A - 2008 |

5 | The geometry of relative plausibility and belief of singletons,” submitted to the International Journal of Approximate Reasoning - Cuzzolin - 2006 |

4 | A new method to determine evidence discounting coefficient - Jiang, Zhang, et al. |

4 | A Monte-Carlo algorithm for combining Dempster-Shafer belief based on approximate precomputation,” ECSQARU’99 - Moral, Salmeron |

3 | Consonant approximations of belief functions,” Int - Dubois, Prade - 1990 |

2 | Complexes of outer consonant approximations - Cuzzolin - 2009 |

1 | approximations of belief functions,” Int - “Consonant - 1990 |

1 | approximation of belief functions - “Consistent - 2009 |

1 | approximations of belief functions - “Consonant - 1990 |

1 | consistent approximations of belief functions - “Lp |

1 | of outer consonant approximations - “Complexes - 2009 |

1 | Monte-Carlo estimations for belief functions - Kramosil - 1998 |