5th International Symposium on Imprecise Probabilities and Their Applications, Prague, Czech Republic, 2007 On the nature of independence in the theory of evidence Fabio Cuzzolin Perception group INRIA Rhone-Alpes, Grenoble, France Fabio.Cuzzolin@inrialpes.fr Abstract Keywords. Instructions, typing, figures and tables, fonts, L A T E X. 1 Introduction The theory of evidence has born as a contribution to a mathematically rigorous description of the notion of subjective probability. Starting from some works on the ... In subjective probability, different observers (or “ex- perts”) of the same phenomenon can possess differ- ent states of knowledge of what the decision space is. Mathematically, this translates into the necessity of admitting the existence of several distinct representa- tions of the decision space at different levels of refine- ment. The collected evidence will then be available in any of those decision spaces or frames. In order for the experts to reach a consensus of the outcome of the problem it is necessary that those frames be related with each other in some mathematically precise way. This idea is summarized in the theory of evidence by the notion of family of frames. Given the evidence available in several frames of the family (correspond- ing to different persons or sensors), this can then be “moved” to a common frame or “common refinement” in which it will then be fused. Mathematically, families of frames are nothing but collections of Boolean subalgebras of the common re- finement. In this context the notion of independence of frames plays an important role. Boolean subalge- bras admit a notion of independence As Dempster’s orthogonal sum originally derives from conditional independence assumptions on the under- lying probabilities which generate belief functions through multi-valued mappings, it is not surprising to discover that combinability (in Dempster’s approach) and independence of frames (in Shafer’s formulation of the theory of evidence) are strictly intertwined. [4] As we have recently proven, families of frames possess the algebraic structure of lattice, i.e. partially ordered sets in which each pair admit inf and sup [6]. Now it is very interesting to know that independence can be introduced in a totally abstract way on the atoms A of a semimodular lattice, i.e. the elements of the lattice which “cover” the element . In fact, independence on atoms of a semimodular lattice can be formulated in three alternative ways (which we call I 1 , I 2 and I 3 and will discuss thoroughly in this pa- per), which are equivalent on A [1, 20]. It is natural to conjecture that independence of frames could be in fact a cryptomorphic form of one of those relations. However, when one tries and extend this re- lations to arbitrary elements of the lattice, they cease to be equivalent. In Section ?? we will in fact analyze in deep the existing relationships between I 1 , I 2 and I 3 in the frame lattice (in both its upper L(Θ) and lower L ∗ (Θ) semimodular form), and their relations with the independence of frames IF . It turns out that neither of them coincides with IF , even though they do possess some meaningful inter- actions. This leaves the fundamental question unan- swered: What is the algebraic nature of independence of frames? It is easy to prove that families of frames interpreted as lattice possess other different properties. In partic- ular, finite families of frames are both algebraic and geometric lattices. This is relevant to our analysis of the notion of independence in the theory of evi- dence, as geometric lattices are strictly related to the formal mathematical description of independence, a structure known as matroid. Matroid theory has been developed in the Thirties by Whitney [23] and Birkhoff [2, 3] when they noticed that several different notions of independence formu- lated in apparently disjoint fields of mathematics (lin- ear algebra, graph theory among the others) were in fact nothing but different cryptomorphic descriptions of the same concept. matroids ... Not surprisingly, the same notion of abstract inde- pendence of atoms of a semimodular lattice can be