Belief Functions over Infinite State Spaces

Abstract

In order to make the following considerations more transparent, let us recall the basic idea of our definition of belief function (Def. 4.2.1) in the terms of setvalued (generalized) random variables and their probabilistic numerical characteristics (generalized quantiles). Let S be a nonempty set, let S ⊂ P (P(S)) be a nonempty σ-field of systems of subsets of S, let 〈Ω, A, P〉 be a fixed abstract probability space. Let 〈E, e〉 be a measurable space over the nonempty space E of possible empirical values, let X: 〈Ω, A, P〉 → 〈E,e〉 be a random variable, let ρ: S × E →{0,1} be a compatibility relation, let U ρ,X(x) = {s∈S: ρ(s, x) = 1} for each x ∈ E. Then the value bel*ρ,X(A) is defined by $$ be{l^{*}}_{{p,X}}(A) = P(\omega \in \Omega :\O \ne {U_{{p,X}}}\left( {X(w)} \right) \subset A\} ) $$ (9.1.1) for each A ⊂ S for which this probability is defined. In other terms we can say: let Uρ,X (X(·)) be a set-valued (generalized) random variable, i.e. measurable mapping, which takes the probability space 〈Ω, A, P〉 into a measurable space 〈P(S), S〉. Then the (non-normalized) degree of belief bel*ρ,X(A) is defined by (9.1.1) for each A ⊂ S such that the probability in (9.1.1) is defined, hence, the inverse image of A is in A, in other terms, P(A) = {B: B ⊂ A} G S ⊂ P(P(S)) holds. If, moreover, {θ} e S and P({ω ∈Ω: U ρ,X (ω) = θ}) < 1 hold, the (normalized) degree of belief belρ,X (A) is defined by the conditional probability $$ be{l_{{p,X}}}(A) = P(\omega \in \Omega :{U_{{p,X}}}(\omega ) \subset A\} /\{ \omega \in \Omega :{U_{{p,X}}}(\omega ) \ne \O \} ) $$ (9.1.2)