Measure-theoretic approach to the inversion problem for belief functions

Abstract

The problem how to define an inverse operation to the Dempster combination rule for basic probability assignments and belief functions possesses a natural motivation and an intuitive interpretation. Or, if the Dempster rule reflects a modification of one's system of degrees of beliefs when the subject in question becomes familiar with the degrees of beliefs of another subject and accepts the arguments on which these degrees are based, the inverse operation would enable to erase the impact of this modification, and to return back to one's original degrees of beliefs, supposing that the reliability of the second subject is put into doubts. Within the algebraic framework this inversion problem was solved by Ph. Smets in [6], here we suggest an alternative solution based on the apparatus of measure theory and conserving the idea that degrees of beliefs are measures, i.e., numerically quantified sizes, of certain sets of elementary random events defined by set-valued random variables. However, probability measures, used for these purposes when defining classical belief functions, will be replaced by the so-called signed measures, which can also take values outside the unit interval of real numbers including the negative and even infinite ones.