1,516 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
Assessing forensic evidence by computing belief functions
We first discuss certain problems with the classical probabilistic approach
for assessing forensic evidence, in particular its inability to distinguish
between lack of belief and disbelief, and its inability to model complete
ignorance within a given population. We then discuss Shafer belief functions, a
generalization of probability distributions, which can deal with both these
objections. We use a calculus of belief functions which does not use the much
criticized Dempster rule of combination, but only the very natural
Dempster-Shafer conditioning. We then apply this calculus to some classical
forensic problems like the various island problems and the problem of parental
identification. If we impose no prior knowledge apart from assuming that the
culprit or parent belongs to a given population (something which is possible in
our setting), then our answers differ from the classical ones when uniform or
other priors are imposed. We can actually retrieve the classical answers by
imposing the relevant priors, so our setup can and should be interpreted as a
generalization of the classical methodology, allowing more flexibility. We show
how our calculus can be used to develop an analogue of Bayes' rule, with belief
functions instead of classical probabilities. We also discuss consequences of
our theory for legal practice.Comment: arXiv admin note: text overlap with arXiv:1512.01249. Accepted for
publication in Law, Probability and Ris
On the semantics of fuzzy logic
AbstractThis paper presents a formal characterization of the major concepts and constructs of fuzzy logic in terms of notions of distance, closeness, and similarity between pairs of possible worlds. The formalism is a direct extension (by recognition of multiple degrees of accessibility, conceivability, or reachability) of the najor modal logic concepts of possible and necessary truth.Given a function that maps pairs of possible worlds into a number between 0 and 1, generalizing the conventional concept of an equivalence relation, the major constructs of fuzzy logic (conditional and unconditioned possibility distributions) are defined in terms of this similarity relation using familiar concepts from the mathematical theory of metric spaces. This interpretation is different in nature and character from the typical, chance-oriented, meanings associated with probabilistic concepts, which are grounded on the mathematical notion of set measure. The similarity structure defines a topological notion of continuity in the space of possible worlds (and in that of its subsets, i.e., propositions) that allows a form of logical “extrapolation” between possible worlds.This logical extrapolation operation corresponds to the major deductive rule of fuzzy logic — the compositional rule of inference or generalized modus ponens of Zadeh — an inferential operation that generalizes its classical counterpart by virtue of its ability to be utilized when propositions representing available evidence match only approximately the antecedents of conditional propositions. The relations between the similarity-based interpretation of the role of conditional possibility distributions and the approximate inferential procedures of Baldwin are also discussed.A straightforward extension of the theory to the case where the similarity scale is symbolic rather than numeric is described. The problem of generating similarity functions from a given set of possibility distributions, with the latter interpreted as defining a number of (graded) discernibility relations and the former as the result of combining them into a joint measure of distinguishability between possible worlds, is briefly discussed
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