Geometric analysis of belief space and conditional subspaces

2006

In this paper we extend the geometric approach to belief functions to the case of consistent belief functions, i.e. belief functions whose plausibility function is a possibility measure, as a step towards a unified geometric picture of a wider class of fuzzy measures. We prove that, analogously to consonant belief functions, they form a simplicial complex (a structured collection of simplices) whose maximal simplices are congruent. We also show that consistency is strictly related to the issue of combinability, a fact reflected in the similarity between the consistent complex and the complex of singular belief functions, i.e. belief functions whose core is a proper subset of their domain. Finally we draw inspiration from the approximation problem to illustrate a convex decomposition of belief functions in terms of some "consistent coordinates" on the complex which is closely related to the pignistic function, an example of how the language of discrete mathematics can help us to draw connections between belief, probability and possibility.

Proceedings of BELIEF 2010

"Conditioning is crucial in applied science when inference involving time series is involved. Belief calculus is an e ffective way of handling such inference in the presence of uncertainty, but di fferent approaches to conditioning in that framework have been proposed in the past, leaving the matter unsettled. We propose here 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. Two di fferent such simplices can be defi ned, as each belief function can be represented as either the vector of its basic probability values or the vector of its belief values. We show here that such a geometric approach to conditioning often produces simple results with straightforward interpretations in terms of degrees of belief. The question of whether classical approaches, such as for instance Dempster's conditioning, can also be reduced to some form of distance minimization remains open: the study of families of combination rules generated by (geometric) conditioning rules appears to be the natural prosecution of the presented research."

IEEE Transactions on Systems, Man, and Cybernetics, Part C, 2008

In this paper we propose a geometric approach to the theory of evidence based on convex geometric interpretations of its two key notions of belief function and Dempster's sum. On one side, we analyze the geometry of belief functions as points of a polytope in the Cartesian space called belief space, and discuss the intimate relationship between basic probability assignment and convex combination. On the other side, we study the global geometry of Dempster's rule by describing its action on those convex combinations. By proving that Dempster's sum and convex closure commute we become able to depict the geometric structure of conditional subspaces, i.e. sets of belief functions conditioned by a given function b. Natural applications of these geometric methods to classical problems like probabilistic approximation and canonical decomposition are outlined.

Advances in Soft Computing, 2000

Commutative monoids of belief states have been defined by imposing one or more of the usual axioms and employing a combination rule. Familiar operations such as normalization and the Voorbraak map are surjective homomorphisms. The latter, in particular, takes values in a space of Bayesian states. The pignistic map is not a homomorphism between these same spaces. We demonstrate an impact this may have on robust decision making for frames of cardinality at least 3. We adapt the measure zero reflection property of some maps between probability spaces to define a category of belief states having plausibility zero reflecting functions as morphisms. Our definition encapsulates a generalization of the notion of absolute continuity to the context of belief spaces. We show that the Voorbraak map is a functor valued in this category.

Proc. of ISAIM, 2008

In this paper we study the problem of conditioning a belief function (b.f.) b with respect to an event A by geometrically projecting such belief function onto the simplex associated with A in the space of all belief functions. Defining geometric conditional b.f.s by minimizing L p distances between b and the conditioning simplex in such "belief" space (rather than in the "mass" space) produces complex results with less natural interpretations in terms of degrees of belief. The question of weather classical approaches, such as Dempster's conditioning, can be themselves reduced to some form of distance minimization remains open: the generation of families of combination rules generated by (geometrical) conditioning appears to be the natural prosecution of this line of research.

In this paper we extend the geometric approach to the theory of evidence in order to include the class of necessity measures, represented on a finite domain of “frame” by consonant belief functions (b.f.s). The correspondence between chains of subsets and convex sets of b.f.s is studied and its properties analyzed, eventually yielding an elegant representation of the region of consonant belief functions in terms of the notion of “simplicial complex”. In particular we focus on the set of outer consonant approximations of a belief function, showing that for each maximal chain of subsets these approximations form a polytope. The maximal such approximation with respect to the weak inclusion relation between b.f.s is one of the vertices of this polytope, and is generated by a permutation of the elements of the frame.

2006

In this paper we extend the geometric approach to the theory of evidence in order to include other important finite fuzzy measures. In particular we describe the geometric counterparts of the class of possibility measures represented by consonant belief functions. The correspondence between chains of subsets and convex sets of consonant functions is studied and its properties analyzed, eventually yielding an elegant representation of the region of consonant belief functions in terms of the notion of simplicial complex. Exploiting the duality of the associated norms we consider the consonant approximation problem in a simple case study, compare the geometry of the solutions with that of inner and outer approximations, and formulate conjectures on their general behavior. We also propose an optimization criterion based on Dempster's rule of combination. Abstract

submitted to the IEEE Transactions on Systems, Man …, 2005

1998

It is commonly acknowledged that we need to accept and handle uncertainty when reasoning with real world data. The most profoundly studied measure of uncertainty i s the probability. H o wever, the general feeling is that probability cannot express all types of uncertainty, including vagueness and incompleteness of knowledge. The Mathematical Theory of Evidence or the Dempster-Shafer Theory (DST) 1, 12] has been intensely investigated in the past as a means of expressing incomplete knowledge. The interesting property i n t h i s c o n text is that DST formally ts into the framework of graphoidal structures 13] which implies possibilities of e cient reasoning by local computations in large multivariate belief distributions given a factorization of the belief distribution into low dimensional component conditional belief functions. But the concept of conditional belief functions is generally not usable because composition of conditional belief functions is not granted to yield joint m ultivariate belief distribution, as some values of the belief distribution may turn out to be negative 4, 13, 15]. To o vercome this problem creation of an adequate frequency model is needed. In this paper we suggest that a Dempster-Shafer distribution results from "clustering" (merging) of objects sharing common features. Upon "clustering" two (or more) objects become indistinguishable (will be counted as one) but some attributes will behave a s i f t h e y h a ve more than one value at once. The next elements of the model needed are the concept of conditional independence and that of merger conditions. It is assumed that before merger the objects move closer in such a w ay that conditional distributions of features for the objects to merge are identical. The traditional conditional independence of feature variables is assumed before merger (thereafter only the DST conditional independence holds). Furthermore it is necessary that the objects get "closer" before the merger independly for each feature variable and only those areas merge where the conditional distributions get identical in each v ariable. The paper demonstrates that within this model, the graphoidal properties hold and a su cient condition for non-negativity of the graphoidally represented belief function is presented and its validity demonstrated.