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Robust Bayesianism: Relation to evidence theory
 J. Advances in Information Fusion
"... We are interested in understanding the relationship between Bayesian inference and evidence theory. The concept of a set of probability distributions is central both in robust Bayesian analysis and in some versions of DempsterShafer’s evidence theory. We interpret imprecise probabilities as impreci ..."
Abstract

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We are interested in understanding the relationship between Bayesian inference and evidence theory. The concept of a set of probability distributions is central both in robust Bayesian analysis and in some versions of DempsterShafer’s evidence theory. We interpret imprecise probabilities as imprecise posteriors obtainable from imprecise likelihoods and priors, both of which are convex sets that can be considered as evidence and represented with, e.g., DSstructures. Likelihoods and prior are in Bayesian analysis combined with Laplace’s parallel composition. The natural and simple robust combination operator makes all pairwise combinations of elements from the two sets representing prior and likelihood. Our proposed combination operator is unique, and it has interesting normative and factual properties. We compare its behavior with other proposed fusion rules, and earlier efforts to reconcile Bayesian analysis and evidence theory. The behavior of the robust rule is consistent with the behavior of Fixsen/Mahler’s modified Dempster’s (MDS) rule, but not with Dempster’s rule. The Bayesian framework is liberal in allowing all significant uncertainty concepts to be modeled and taken care of and is therefore a viable, but probably not the only, unifying structure that can be economically taught and in which alternative solutions can be modeled, compared and explained. Manuscript received April 20, 2006; released for publication April
Two kadditive generalizations of the pignistic transform for imprecise decision making
"... The Transferable Belief approach to the Theory of Evidence is based on the pignistic transform which, mapping belief functions to probability distributions, allows to make “precise ” decisions on a set of disjoint hypotheses via classical utility theory. In certain scenarios, however, such as medica ..."
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The Transferable Belief approach to the Theory of Evidence is based on the pignistic transform which, mapping belief functions to probability distributions, allows to make “precise ” decisions on a set of disjoint hypotheses via classical utility theory. In certain scenarios, however, such as medical diagnosis, the need for an “imprecise ” approach to decision making arises, in which sets of possible outcomes are compared. We propose here a framework for imprecise decision derived from the TBM, in which belief functions are mapped to kadditive belief functions (i.e., belief functions whose focal elements have maximal cardinality equal to k) rather than Bayesian ones. We do so by introducing two alternative generalizations of the pignistic transform to the case of kadditive belief functions. The latter has several interesting properties: depending on which properties are deemed the most important, the two distinct generalizations arise. The proposed generalized transforms are empirically validated by applying them to imprecise decision in concrete pattern recognition problems.