Complexes of outer consonant approximations

2011

In this paper we introduce a novel, simpler form of the polytope of inner Bayesian approximations of a belief function, or \consistent probabilities". We prove that the set of vertices of this polytope is generated by all possible permutations of elements of the domain, mirroring a similar behavior of outer consonant approximations. An intriguing connection with the behavior of maximal outer consonant approximations is highlighted, and the notion of inner (outer) approximation of a credal set in terms of lower probabilities proposed. Finally, we generalize the main result to the case of k-additive belief functions, belief functions whose focal elements have size at most k. We prove that the set of such objects dominating a given belief function is also a polytope whose vertices are generated by permutations of focal elements of size at most k.

Proc. of ISAIM, 2008

Logics in Artificial Intelligence, 2008

2007

2008

Consistent belief functions represent collections of coherent or non-contradictory pieces of evidence. As most operators used to update or elicit evidence do not preserve consistency, the use of consistent transformations cs[·] in a reasoning process to guarantee coherence can be desirable. These are in turn linked to the problem of approximating an arbitrary belief function with a consistent one. We study here the consistent approximation problem in the case in which distances are measured using classical Lp norms. We show that for each choice of the element we want them to focus on, the partial approximations determined by the L1 and L2 norms coincide, and that they can be interpreted as the classical focused consistent transformation. Global solutions do not coincide, in general, nor are they associated with the highest plausibility element. 1 The consistent approximation problem Belief functions (b.f.s) [19] are complex objects, in which different and sometimes contradictory bod...

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.

2009

In this paper we define the class of consistent belief functions as the counterparts of consistent knowledge bases in classical logic. We prove that such class can be defined univocally no matter our definition of proposition implied by a belief function. As consistency can be desirable in decision making, the problem of transforming an arbitrary belief function into a consistent one arise, and can be posed in a geometric setup. We prove that consistent belief functions live on a structured collection of simplices called simplicial complex. Eventually, we show how each belief function naturally decomposes into consistent components on such a complex, in a fashion which recalls the pignistic transform.

International Journal of Uncertainty, Fuzziness and Knowledge-based Systems, 2009

The belief structure resulting from the combination of consonant and independent marginal random sets is not, in general, consonant. Also, the complexity of such a structure grows exponentially with the number of combined random sets, making it quickly intractable for computations. In this paper, we propose a simple guaranteed consonant outer approximation of this structure. The complexity of this outer approximation does not increase with the number of marginal random sets (i.e., of dimensions), making it easier to handle in uncertainty propagation. Features and advantages of this outer approximation are then discussed, with the help of some illustrative examples.

The study of the interplay between belief and probability can be posed in a geometric framework, in which belief and plausibility functions are represented as points of simplices in a Cartesian space. Probability approximations of belief functions form two homogeneous groups, which we call "affine" and "epistemic" families. In this paper we focus on relative plausibility, belief, and uncertainty of probabilities of singletons, the "epistemic" family. They form a coherent collection of probability transformations in terms of their behavior with respect to Dempster's rule of combination. We investigate here their geometry in both the space of all pseudo belief functions and the probability simplex, and compare it with that of the affine family. We provide sufficient conditions under which probabilities of both families coincide.

2008

In this paper we introduce a novel, simpler form of the polytope of inner Bayesian approximations. We prove that the set of vertices of this polytope is generated by all possible permutations of elements of the domain, resounding a similar behavior of outer consonant approximations.

IEEE Transactions on Fuzzy Systems (under …, 2008

Consistent belief functions represent collections of coherent or non-contradictory pieces of evidence. As most operators used to update or elicit evidence do not pre-serve consistency, the use of consistent transformations cs[·] in a reasoning process to guarantee coherence ...

Information Processing and Management of Uncertainty in Knowledge-Based Systems, 2020

We introduce measures of uncertainty that are based on Depth-Bounded Logics [4] and resemble belief functions. We show that our measures can be seen as approximation of classical probability measures over classical logic, and that a variant of the PSAT [10] problem for them is solvable in polynomial time.

International Journal of Approximate Reasoning, 2012

In this paper we discuss the semantics and properties of the relative belief transform, a probability transformation of belief functions closely related to the classical plausibility transform. We discuss its rationale in both the probability-bound and Shafer's interpretations of belief functions. Even though the resulting probability (as it is the case for the plausibility transform) is not consistent with the original belief function, an interesting rationale in terms of optimal strategies in a non-cooperative game can be given in the probability-bound interpretation to both relative belief and plausibility of singletons. On the other hand, we prove that relative belief commutes with Dempster's orthogonal sum, meets a number of properties which are the duals of those met by the relative plausibility of singletons, and commutes with convex closure in a similar way to Dempster's rule. This supports the argument that relative plausibility and belief transform are indeed naturally associated with the D-S framework, and highlights a classification of probability transformations in two families, according to the operator they relate to. Finally, we point out that relative belief is only a member of a class of \relative mass" mappings, which can be interpreted as low-cost proxies for both plausibility and pignistic transforms.

2011

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. Deflning geometric conditional b.f.s by minimizing Lp 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.

2001

In this paper the geometric structure of the space S_Theta of the belief functions defined over a discrete set Theta (belief space) is analyzed. Using the Moebius inversion lemma we prove the recursive bundle structure of the belief space and show how an arbitrary belief function can be uniquely represented as a convex combination of certain elements of the fibers, giving $\mathcal{S}$ the form of a simplex. The commutativity of orthogonal sum and convex closure operator is proved and used to depict the geometric structure of conditional subspaces, i.e. sets of belief functions conditioned by a given function s. Future applications of this geometric methods to classical problems like probabilistic approximation and canonical decomposition are outlined.