Research Theme: Belief and Possibility |

The geometry of consonant belief functions and necessity measuresConsonant approximation of belief functionsConsistent belief functionsConsistent approximation of belief functions

The geometric approach to the theory of evidence can be extended to include the class of necessity measures, represented on a finite domain or '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 focussed 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.

We can solve the problem of approximating a belief measure with a necessity measure or "consonant belief function" in our geometric framework too, where consonant belief functions form a simplicial complex in the belief space (see above).

As necessity measures are strictly related to the L1 norm it makes sense to look for approximations which minimize Lp norms. Partial approximations are first sought in each simplicial component of the consonant complex, while global solutions are obtained from the set of such partial ones. The obtained Lp consonant approximations are discussed and their interpretation in terms of degrees of belief provided.

The class of consistent belief functions can be introduced as the counterpart of consistent knowledge bases in classical logic. Such class can be defined univocally no matter the way we define the notion 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. This can be posed in a geometric setup, as consistent belief functions live on a structured collection of simplices called simplicial complex. Indeed, each belief function naturally decomposes into consistent components on such a complex, in a fashion which recalls the pignistic transform.

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 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 therefore studied the consistent approximation problem in the case in which distances are measured using classical Lp norms. We show that the approximations determined by both L1 and L2 norms are unique and both coincide, for each choice of the element we want them to focus on, with classical focused consistent transformations. The Linf norm determines for each element of the frame an entire polytope of solutions whose barycenter lies on the L1/L2 approximation. Global Linf approximations are always associated with the maximal plausibility element.