  Different quantities that go by the name of entropy are used in variational
principles to infer probability distributions from limited data. Shore and
Johnson showed that maximizing the Boltzmann- Gibbs form of the entropy ensures
that probability distributions inferred satisfy the multiplication rule of
probability for independent events in the absence of data coupling such events.
Other types of entropies that violate the Shore and Johnson axioms, including
nonadditive entropies such as the Tsallis entropy, violate this basic
consistency requirement. Here we use the axiomatic framework of Shore and
Johnson to show how such nonadditive entropy functions generate biases in
probability distributions that are not warranted by the underlying data.
