Geometric views on conflicting mass functions: From distances to angles

• Distances or metrics should not be used to measure conflict between mass functions.
• The notion of conflict is conceptually closer to that of angle than that of distance.
• The desirable properties for conflict measures are those of semi-pseudo-metrics.

Recently, several works have focused on the study of conflict among belief functions with a geometric approach, trying to elaborate on the intuition that distant belief functions are more conflicting than neighboring ones. In this article, I discuss the extent to which the mathematical properties of a metric are compliant with what can be expected from a conflict measure. As this highlights some inconsistencies, numerous other geometric or algebraic objects are investigated for their potential to give a mathematical formulation to the measurement of conflict: divergences, pseudo-sine functions, kernels and semi-pseudo-metrics. Finally, only semi-pseudo-metrics (an intermediate object the nature of which relates to both distances and angles) seem to fit. This is practically confirmed by the fact that the outer plausibility conflict is indeed a semi-pseudo-metric.