Complexes of outer consonant approximations

In this paper we discuss the problem of approximating a belief function (b.f.) with a possibility measure of consonant belief function (co.b.f.) from a geometric point of view. We focus in particular on outer consonant approximations, i.e. co.b.f.s less committed than the original b.f. in terms of degrees of belief. We show that for each specific chain of focal elements the set of outer consonant approximation is a polytope with interesting analogies with the simplex of inner Bayesian approximations, and the entire region of outer approximations can be triangulated to form a collection of simplices or simplicial complex in analogy the whole space of consonant b.f.s. Introduction As a natural extension of the classical Bayesian formalism the theory of evidence (ToE) (Shafer 1976) is a popular approach to uncertainty description, in which probabilities are replaced by belief functions (b.f.s), which assign values between 0 and 1 to subsets of the sample space instead of single elements. Possibility theory (Dubois & Prade 1988) is based instead on a different description of uncertainty called possibility measure, a function Pos : 2 → [0, 1] on a domain Θ such that Pos( ⋃ i Ai) = supi Pos(Ai) for any family {Ai|Ai ∈ 2, i ∈ I} where I is an arbitrary set index. It is well known that possibility measures have counterparts in the theory of evidence as consonant b.f.s, i.e. belief functions whose focal elements are nested (Shafer 1976). The problem of approximating a belief function with a consonant b.f. (Dubois & Prade 1990; Joslyn & Klir 1992; Joslyn 1997; Baroni 2004) is then equivalent to approximating a belief function with a possibility. As possibilities are completely determined by their values on the singletons Pos(x), x ∈ Θ (”membership function”) they less computationally expensive, making the approximation process quite interesting in may applications. The points of contact between evidential formalism (in the transferable belief model implementation) and possibility theory has been briefly investigated by Ph. Smets in (Smets 1990), A geometric interpretation of uncertainty theory has been recently proposed by F. Cuzzolin (Cuzzolin 2004) in which Copyright c © 2008, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. several classes of uncertainty measures (among which b.f.s, possibilities) are represented as points of a Cartesian space. In particular, while belief functions form generalized triangles or ”simplices” in R , consonant b.f.s live in a structured collection of simplices called ”simplicial complexes”. In this paper we consider the problem of approximating a belief function with a possibility (Dubois & Prade 1990), focusing in particular on the class of outer consonant approximations of belief functions from such geometric point of view, showing that they live on a collection of polytopes associated with all possible maximal chains of focal elements, which in turn form a simplicial complex in analogy with the whole consonant space. More precisely, after reviewing the basic notions of evidence and possibility theory we introduce the consonant approximation problem, and in particular the notion of outer consonant approximation. We then present the idea of simplicial complex, and recall how the space of consonant belief functions forms indeed a simplicial complex. Starting from the simple binary case we will prove that the set of outer consonant approximations of a b.f. forms a collection of polytopes, in stringent analogy with the case of inner Bayesian approximations, and can indeed be triangulated in order to make it a simplicial complex. To improve the readability of the paper all major proofs are collected in an Appendix. Belief and possibility measures A basic probability assignment (b.p.a.) over a finite set (frame of discernment (Shafer 1976)) Θ is a function m : 2 → [0, 1] on its power set 2 = {A ⊆ Θ} such that m(∅) = 0, A⊆Θ m(A) = 1, and m(A) ≥ 0 ∀A ⊆ Θ. Subsets of Θ associated with non-zero values of m are called focal elements. The belief function b : 2 → [0, 1] associated with a basic probability assignment m on Θ is defined as: b(A) = ∑ B⊆A m(B). A dual mathematical representation of the evidence encoded by a belief function b is the plausibility function (pl.f.) plb : 2 → [0, 1], A 7→ plb(A) where plb(A) . = 1 − b(A) = 1 − B⊆Ac mb(B) = ∑ B∩A 6=∅mb(B) expresses the amount of evidence not against A. A probability function is simply a peculiar belief function assigning non-zero masses to singletons only (Bayesian b.f.): mb(A) = 0 |A| > 1. A b.f. is said to be consonant if its focal elements {Ei, i = 1, ..., m} are nested: E1 ⊂ E2 ⊂ ... ⊂ Em. It can be proven that (Dubois & Prade 1988; Joslyn 1991) the plausibility function plb associated with a belief function b on a domain Θ is a possibility measure iff b is consonant. Outer consonant approximations As consonant b.f.s represent possibility measures in the theory of evidence, finding the ”best” consonant approximation of a belief function is equivalent to respond the question of how to approximate a belief measure with a possibility. Now, belief functions admit the following order relation b ≤ b′ ≡ b(A) ≤ b′(A) ∀A ⊆ Θ (1) called weak inclusion. We can then define the outer consonant approximations (Dubois & Prade 1990) of a belief function b as those co.b.f.s such that co(A) ≤ b(A) ∀A ⊆ Θ (or equivalently plco(A) ≥ plb(A) ∀A). With the purpose of finding outer approximations which are minimal with respect to the weak inclusion relation (1)) Dubois and Prade have introduced a family of outer consonant approximations obtained by considering all permutations ρ of the elements {x1, ..., xn} of the frame of discernment Θ: {xρ(1), ..., xρ(n)}. A family of nested sets can be then built {S 1 = {xρ(1)}, S 2 = {xρ(1), xρ(2)}, ..., S n = {xρ(1), ..., xρ(n)}} so that a new consonant belief function co can be defined with b.p.a. mcoρ(S ρ j ) = ∑ i:min{l:Ei⊆S l }=j mb(Ei). (2) S j concentrates all the mass of the focal elements of b included in S j but not in S ρ j−1. The complex of consonant belief functions A useful tool to represent a number of uncertainty measures and pose questions like the approximation problem is provided by convex geometry. Given a frame of discernment Θ, a b.f. b : 2 → [0, 1] is completely specified by its N −1 belief values {b(A), A ⊆ Θ, A 6= ∅}, N . = 2|Θ|, and can then be represented as a point of RN−1. The belief space associated with Θ is the set of points B of RN−1 which correspond to b.f.s. Let us call bA . = b ∈ B s.t. mb(A) = 1, mb(B) = 0 ∀B 6= A (3) the unique b.f. assigning all the mass to a single subset A of Θ (A-th basis belief function). It can be proven that (Cuzzolin 2004) the belief space B is the convex closure of all the basis belief functions bA, B = Cl(bA, ∅ ( A ⊆ Θ) where Cl denotes the convex closure operator: Cl(b1, ..., bk) = {b ∈ B : b = α1b1 + · · ·+ αkbk, ∑ i αi = 1, αi ≥ 0 ∀i}. More precisely B is an N − 2-dimensional simplex, i.e. the convex closure of N − 1 (affinely independent1) points of 1 An affine combination of k points v1, ..., vk ∈ R is a sum α1v1 + · · · + αkvk whose coefficients sum to one: ∑ i αi = 1. The affine subspace generated by the points v1, ..., vk ∈ R is the set {v ∈ R : v = α1v1 + · · ·+ αkvk, ∑ i αi = 1}. If v1, ..., vk generate an affine space of dimension k they are said to be affinely independent. the Euclidean space RN−1. The faces of a simplex are all the simplices generated by a subset of its vertices. Each belief function b ∈ B can be written as a convex sum as b = ∑ ∅(A⊆Θ mb(A)bA. Similarly the set of all Bayesian b.f.s is P = Cl(bx, x ∈ Θ). Binary example. As an example consider a frame of discernment containing only two elements, Θ2 = {x, y}. Each b.f. b : 22 → [0, 1] is determined by its belief values b(x), b(y), as b(Θ) = 1 and b(∅) = 0 ∀b. We can then collect them in a vector of RN−2 = R: [b(x) = mb(x), b(y) = mb(y)]′ ∈ R. (4) Since mb(x) ≥ 0, mb(y) ≥ 0, and mb(x) + mb(y) ≤ 1 the set B2 of all the possible b.f.s on Θ2 is the triangle of Figure 1, whose vertices are the points bΘ = [0, 0]′, bx = [1, 0]′, and by = [0, 1]′. The region P2 of all Bayesian b.f.s on b =[0,0]' Θ b =[0,1]' y