A sufficient condition for belief function construction from conditional belief functions (original) (raw)
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Idempotent merging of belief functions: Extending the minimum rule of possibility theory
Workshop on the Theory on Belief …, 2010
When merging belief functions, Dempster rule of combination is justified only when information sources can be considered as independent and reliable. When dependencies are ill-known, it is usual to require the combination rule to be idempotent, as it ensures a cautious behaviour in the face of dependent sources. There are different strategies to find such rules for belief functions. The strategy considered here consists in relying on idempotent rules used in a more specific frameworks and to study its extension to belief functions. We study two possible extensions of the minimum rule of possibility theory to belief functions. We first investigate under which conditions it can be extended to general contour functions.We then further investigate the combination rule that maximises the expected cardinality of the resulting random set.
Cautious conjunctive merging of belief functions
Symbolic and Quantitative …, 2007
When merging belief functions, Dempster rule of combination is justified only when information sources can be considered as independent. When this is not the case, one must find out a cautious merging rule that adds a minimal amount of information to the inputs. Such a rule is said to follow the principle of minimal commitment. Some conditions it should comply with are studied. A cautious merging rule based on maximizing expected cardinality of the resulting belief function is proposed. It recovers the minimum operation when specialized to possibility distributions. This form of the minimal commitment principle is discussed, in particular its discriminating power and its justification when some conflict is present between the belief functions.
Independence, Conditionality and Structure of Dempster-Shafer Belief Functions
ArXiv, 2017
Several approaches of structuring (factorization, decomposition) of Dempster-Shafer joint belief functions from literature are reviewed with special emphasis on their capability to capture independence from the point of view of the claim that belief functions generalize bayes notion of probability. It is demonstrated that Zhu and Lee's {Zhu:93} logical networks and Smets' {Smets:93} directed acyclic graphs are unable to capture statistical dependence/independence of bayesian networks {Pearl:88}. On the other hand, though Shenoy and Shafer's hypergraphs can explicitly represent bayesian network factorization of bayesian belief functions, they disclaim any need for representation of independence of variables in belief functions. Cano et al. {Cano:93} reject the hypergraph representation of Shenoy and Shafer just on grounds of missing representation of variable independence, but in their frameworks some belief functions factorizable in Shenoy/Shafer framework cannot be fact...
Dempsterian-Shaferian Belief Network From Data
ArXiv, 2018
Shenoy and Shafer {Shenoy:90} demonstrated that both for Dempster-Shafer Theory and probability theory there exists a possibility to calculate efficiently marginals of joint belief distributions (by so-called local computations) provided that the joint distribution can be decomposed (factorized) into a belief network. A number of algorithms exists for decomposition of probabilistic joint belief distribution into a bayesian (belief) network from data. For example Spirtes, Glymour and Schein{Spirtes:90b} formulated a Conjecture that a direct dependence test and a head-to-head meeting test would suffice to construe bayesian network from data in such a way that Pearl's concept of d-separation {Geiger:90} applies. This paper is intended to transfer Spirtes, Glymour and Scheines {Spirtes:90b} approach onto the ground of the Dempster-Shafer Theory (DST). For this purpose, a frequentionistic interpretation of the DST developed in {Klopotek:93b} is exploited. A special notion of conditio...
Belief function combination and conflict management
Information Fusion, 2002
Within the framework of evidence theory, data fusion consists in obtaining a single belief function by the combination of several belief functions resulting from distinct information sources. The most popular rule of combination, called Dempster's rule of combination (or the orthogonal sum), has several interesting mathematical properties such as commutativity or associativity. However, combining belief functions with this operator implies normalizing the results by scaling them proportionally to the conflicting mass in order to keep some basic properties. Although this normalization seems logical, several authors have criticized it and some have proposed other solutions. In particular, Dempster's combination operator is a poor solution for the management of the conflict between the various information sources at the normalization step. Conflict management is a major problem especially during the fusion of many information sources. Indeed, the conflict increases with the number of information sources. That is why a strategy for reassigning the conflicting mass is essential. In this paper, we define a formalism to describe a family of combination operators. So, we propose to develop a generic framework in order to unify several classical rules of combination. We also propose other combination rules allowing an arbitrary or adapted assignment of the conflicting mass to subsets.
International Journal of Intelligent Systems, 2009
This paper presents an algorithm for developing models under Dempster-Shafer theory of belief functions for categorical and 'uncertain' logical relationships among binary variables. We illustrate the use of the algorithm by developing belief-function representations of the following categorical relationships: 'AND', 'OR', 'Exclusive OR (EOR)' and 'Not Exclusive OR (NEOR)', and 'AND-NEOR' and of the following uncertain relationships: 'Discounted AND', 'Conditional OR', and 'Weighted Average'. Such representations are needed to fully model and analyze a problem with a network of interrelated variables under Dempster-Shafer theory of belief functions. In addition, we compare our belief-function representation of the 'Weighted Average' relationship with the 'Weighted Average' representation developed and used by Shenoy and Shenoy 8. We find that Shenoy and Shenoy representation of the weighted average relationship is an approximation and yields significantly different values under certain conditions.
A geometric approach to conditioning belief functions
2021
Conditioning is crucial in applied science when inference involving time series is involved. Belief calculus is an effective way of handling such inference in the presence of epistemic uncertainty – unfortunately, different approaches to conditioning in the belief function framework have been proposed in the past, leaving the matter somewhat unsettled. Inspired by the geometric approach to uncertainty, in this paper we propose an approach to the conditioning of belief functions based on geometrically projecting them onto the simplex associated with the conditioning event in the space of all belief functions. We show here that such a geometric approach to conditioning often produces simple results with straightforward interpretations in terms of degrees of belief. This raises the question of whether classical approaches, such as for instance Dempster’s conditioning, can also be reduced to some form of distance minimisation in a suitable space. The study of families of combination rul...