Uniqueness of information measure in the theory of evidence

Abstract

An evidence distribution on a set X assigns non-negative weights to the subsets of X. Such weights must sum to one and the empty set is given weight 0. An information measure can be defined for such an evidence distribution. If m i are the weights assigned to subsets A i , and a i are the cardinalities of these subsets, then the function. Σ m i log a i satisfies all the usual axioms of an information measure. In this paper we show that, conversely, these axioms are sufficient to characterize uniquely the above measure. It can be thus considered as the main uncertainty function for the theory of evidence. We demonstrate that using only the properties of symmetry, additivity and subadditivity the problem of uniqueness can be reduced to finding linear functionals on the space of functions analytic at origin. We surmise that under a suitable continuity hypothesis, all such functionals can be represented as linear combinations of the coefficients of Taylor series. Our function then represents the first derivative evaluated at 0. An alternative approach is then discussed. We assume a form of branching property, suggested by the monotonicity considerations. Now the properties of symmetry, additivity and subadditivity, together with branching again offer the unique characterization of the information function. No continuity assumption whatsoever is needed and the proof is entirely elementary.