  Shannon entropy, a cornerstone of information theory, statistical physics and
inference methods, is uniquely identified by the Shannon-Khinchin or
Shore-Johnson axioms. Generalizations of Shannon entropy, motivated by the
study of non-extensive or non-ergodic systems, relax some of these axioms and
lead to entropy families indexed by certain `entropic' parameters. In general,
the selection of these parameters requires pre-knowledge of the system or
encounters inconsistencies. Here we introduce a simple axiom for any entropy
family: namely, that no entropic parameter can be inferred from a completely
uninformative (uniform) probability distribution. When applied to the
Uffink-Jizba-Korbel and Hanel-Thurner entropies, the axiom selects only R\'enyi
entropy as viable. It also extends consistency with the Maximum Likelihood
principle, which can then be generalized to estimate the entropic parameter
purely from data, as we confirm numerically. Remarkably, in a generalized
maximum-entropy framework the axiom implies that the maximized log-likelihood
always equals minus Shannon entropy, even if the inferred probability
distribution maximizes a generalized entropy and not Shannon's, solving a
series of problems encountered in previous approaches.
