2 research outputs found
Complementary Lipschitz continuity results for the distribution of intersections or unions of independent random sets in finite discrete spaces
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 , the function mapping any random set
distribution to the distribution of its intersection (under independence
assumption) with is Lipschitz continuous with unit Lipschitz constant if
the space of random set distributions is endowed with a metric defined as the
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 is Lipschitz continuous
with unit Lipschitz constant if the space of random set distributions is
endowed with a metric defined as the 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