Based on Klir’s framework of uncertainty, the total uncertainty (also called ambiguity) of belief function is linear addition of discord and nonspecificity. Though uncertainty measures of belief function have been discussed widely, there is no measure that can satisfy the monotonicity and range consistency properties at the same time. In this article, we discuss uncertainty measure of belief function from the perspective of information fractal dimension. An uncertainty quantity called evenness and its measure ${\rm Eve}$ are proposed, which can represent the belief propensity degree of belief function. We first propose the measures of diversity (normalized nonspecificity) and the element evenness (normalized discord), and then fuse them to calculate ${\rm Eve}$ . The proposed method can not only measure the subnormal mass function but also interpret the different views of Klir and Smets on “Uncertainty.” In addition, we extend Klir’s framework of uncertainty based on the proposed information quantities.

In this paper, a multi-object recognition scenario is considered to extend the random finite set into random permutation set. Probabilistic information on random permutation set can be viewed as an distribution determined by three random variables. We use another emerging uncertainty representation, order-2 information granule, to realize the probabilistic information fusion on random permutation sets. First, the probabilistic information on random permutation sets is viewed as an order-2 probability distribution. Second, corresponding information fusion approach is proposed. Finally, the proposed approach is applied to random permutation sets, resolving the decision-making issue under the multi-object recognition scenario. This paper pioneers the connection of order-2 information processing logic to a multi-object recognition task and develops order-2 probability distribution and its combination rules. Compared to the traditional probabilistic information fusion approaches, the proposed approach takes into account not only the propositions’ beliefs provided by the sources, but the structural dependency among propositions as well.

As the duality and complement of the Shannon entropy, the eXtropy is proposed in this years. The eXtropy also can be explained from the perspective of negation, which can be seen as an entropy with one-step negation operation of elements. The Dempster–Shafer theory (DST), as the generalization of probability theory (PT), it can express the more uncertain information than the Bayesian distribution. It contains the uncertainty of the event probability expressed in the Bayesian distribution and the uncertainty of the Bayesian distribution. Based on above, this article proposes a new belief entropy in DST called SU entropy, which is the first belief entropy to use the elements rather than mass functions. And then, a new negation of non-specificity part in DST is proposed, which is the first time to research the negation of non-specificity alone. Finally, a new belief eXtropy was proposed. It is different from the traditional eXtropy thinking, but from the perspective of elements’ negation. After verification, it has a satisfying performance in measurement uncertainty in DST. At the end of the paper, we discuss how to understand eXtropy from the perspective of negation, and the future research direction for the new belief eXtropy.

Based on subjective possibilistic semantics, an agent’s subjective probability mass function is dominated by a qualitative Possibility Mass Function (PossMF), which can also be transformed into a unique consonant mass function. However, the existing transformation method cannot maintain the consistency of combination rules, i.e., fusing PossMFs and consonant mass functions with same information content, respectively, the results no longer maintain the reversible transformation. To address the above issue, a novel belief functions transformation is proposed, which can be interpreted based on both Smets’ canonical decomposition and Pichon’s canonical decomposition. The proposed method is validated based on consistency of combination rules, the least commitment principle, and its application in the fusion of information. In addition, based on the two canonical decompositions, we extend the transformation to possibilistic belief structure, and offer a new perspective of relationship between possibilistic information and evidential information.

Quantifying the uncertainty of information distributions containing randomness, imprecision and fuzziness is the premise of processing them. A useful information representation in the field of intelligent computing is the information granule, which optimizes data from the perspective of specificity and coverage. We introduce information granularity into evidential information and model the Basic Probability Assignment (BPA) as a weighted information granules model. Based on the proposed model, a new uncertainty measure of BPA is derived from the quality evaluation of granules. In addition, the proposed measure is extended to fuzzy evidential information distributions. When the Fuzzy BPA (FBPA) degenerates into the Probability Mass Function (ProbMF) and Possibility Mass Function (PossMF), the proposed method degenerates to Gini entropy and Yager's specificity measure, respectively. We use a refined belief structure to interpret the meaning of FBPA in the transfer belief model, and verify the validity of the proposed method by properties analysis and numerical examples. The concept of information granule is used for the first time to model focal set and beliefs. Compared with Shannon entropy-based information measures, the proposed method provides a novel perspective on the relationship between randomness, imprecision, and fuzziness in FBPA.