On the mathematical consistency of structural-conflict-based belief entropy

Entropy-like measures play an important role in uncertainty modeling, information fusion, and decision processes based on Evidence Theory. When such measures are used to quantify the uncertainty represented by belief functions, their numerical values should remain consistent with the informational relations induced by the underlying credal sets. In particular, monotonicity, additivity, and subadditivity are important coherence requirements: uncertainty values should behave consistently when evidence becomes more specific, when independent bodies of evidence are combined, and when joint information is projected onto marginal domains. In this paper, we study the mathematical consistency of a structural-conflict-based belief entropy proposed in the Dempster–Shafer framework. The measure combines Deng-type entropy terms with a structural conflict factor based on the overlap between focal elements. Although this construction is intuitively appealing and may provide useful practical scores for distinguishing belief functions, we show by means of explicit counterexamples that some fundamental coherence requirements are not generally satisfied. In particular, the measure may assign larger uncertainty values to more informative bodies of evidence, according to the ordering induced by credal-set inclusion. We also show that additivity and subadditivity are not guaranteed, which may affect its interpretation in information fusion and marginalization contexts. These results suggest that structural-conflict-based belief entropy should be used with caution when it is interpreted as a total uncertainty measure, especially in applications where uncertainty values are intended to support ranking, weighting, fusion, or decision procedures.