  This contribution reviews critically the existing entropy measures for
probabilistic hesitant fuzzy sets (PHFSs), and demonstrates that these entropy
measures fail to effectively distinguish a variety of different PHFSs in some
cases. In the sequel, we develop a new axiomatic framework of entropy measures
for probabilistic hesitant fuzzy elements (PHFEs) by considering two facets of
uncertainty associated with PHFEs which are known as fuzziness and
nonspecificity. Respect to each kind of uncertainty, a number of formulae are
derived to permit flexible selection of PHFE entropy measures. Moreover, based
on the proposed PHFE entropy measures, we introduce some entropy-based distance
measures which are used in the portion of comparative analysis.
