Complementary Lipschitz continuity results for the distribution of intersections or unions of independent random sets in finite discrete spaces

TL;DR

This paper establishes that the distributions of intersections and unions of independent random sets in finite spaces are Lipschitz continuous under certain metrics, with implications for conflict measurement in belief functions.

Contribution

It introduces Lipschitz continuity results for the distribution functions of random set intersections and unions, with explicit metrics and constants, within the Dempster-Shafer framework.

Findings

Lipschitz continuity with unit constant for intersections under the $L_k$ norm.

Lipschitz continuity with unit constant for unions under the $L_k$ norm.

Discussion of conflict measures using these distances in belief functions.

Abstract

We prove that intersections and unions of independent random sets in finite spaces achieve a form of Lipschitz continuity. More precisely, given the distribution of a random set $\Xi$, the function mapping any random set distribution to the distribution of its intersection (under independence assumption) with $\Xi$ is Lipschitz continuous with unit Lipschitz constant if the space of random set distributions is endowed with a metric defined as the $L_k$ norm distance between inclusion functionals also known as commonalities. Moreover, the function mapping any random set distribution to the distribution of its union (under independence assumption) with $\Xi$ is Lipschitz continuous with unit Lipschitz constant if the space of random set distributions is endowed with a metric defined as the $L_k$ norm distance between hitting functionals also known as plausibilities. Using the epistemic random set interpretation of belief functions, we also discuss the ability of these distances to yield conflict measures. All the proofs in this paper are derived in the framework of Dempster-Shafer belief functions. Let alone the discussion on conflict measures, it is straightforward to transcribe the proofs into the general (non necessarily epistemic) random set terminology.