Belvolumes

Belvolumes are a concept in uncertainty representation used to combine the strength of belief with a notion of volume in a hypothesis space. They provide a scalar measure intended to compare propositions not only by how strongly they are believed, but also by how large or significant their scope is within the domain of interest.

Formally, let Θ be a finite frame of hypotheses. A basic belief assignment m: 2^Θ → [0,1] satisfies m(∅)=0 and ∑_{A⊆Θ} m(A)=1. For any A⊆Θ, the belief function is Bel(A) = ∑_{B⊆A} m(B). Let vol(A) denote a volume factor that captures the size or geometric weight of A within Θ (for example vol(A)=|A|). The belvolume of A is defined as BV(A) = Bel(A) × vol(A). This construction yields a family of belvolumes indexed by subsets of Θ.

Properties of belvolumes include nonnegativity and a tendency to increase with the inclusion of more elements in A, depending on the chosen vol function. The exact behavior of BV(A) is driven by both the belief mass distribution m and the volume weighting vol.

Example: Suppose Θ = {h1, h2, h3}, with m({h1})=0.4, m({h2})=0.3, m({h3})=0.2, m({h1,h2})=0.1. Then Bel({h1})=0.4, Bel({h1,h2})=0.7. If vol(A)=|A|, then BV({h1})=0.4 and BV({h1,h2})=0.7×2=1.4.

Applications of belvolumes appear in decision making under uncertainty, model selection, and risk assessment, where both belief strength and the scope of propositions inform choices. They extend belief-function concepts by adding a weight for proposition size and may be adjusted to fit specific domains. See also belief function, Dempster-Shafer theory, and plausibility.