Belief functions on lattices

TL;DR

This paper generalizes belief functions from Boolean algebras to arbitrary lattices, maintaining classical properties and simplifying decomposition proofs, thus broadening their theoretical foundation.

Contribution

It extends belief functions to general lattices, preserving classical constructs and providing a simpler, more general proof of their decomposition into simple support functions.

Findings

Classical belief function concepts are valid on any lattice.

Decomposition into simple support functions is simplified and more general.

All traditional belief function operations are preserved in the lattice setting.

Abstract

We extend the notion of belief function to the case where the underlying structure is no more the Boolean lattice of subsets of some universal set, but any lattice, which we will endow with a minimal set of properties according to our needs. We show that all classical constructions and definitions (e.g., mass allocation, commonality function, plausibility functions, necessity measures with nested focal elements, possibility distributions, Dempster rule of combination, decomposition w.r.t. simple support functions, etc.) remain valid in this general setting. Moreover, our proof of decomposition of belief functions into simple support functions is much simpler and general than the original one by Shafer.