Generalized maximum entropy: when and why you need it

The classical maximum entropy principle based on Shannon entropy is widely used to construct least-biased probability distributions from partial information. However, the Shore-Johnson axioms that single out the Shannon functional hinge on strong system independence-an assumption often violated in real-world, strongly correlated systems. When that assumption fails, enforcing the Shannon form imposes an independence the data do not support, and can systematically distort the inferred distribution. We provide a self-contained guide to when and why practitioners should abandon the Shannon form in favour of the one-parameter Uffink-Jizba-Korbel (UJK) family of generalized entropies. After reviewing the Shore-Johnson axioms from an applied perspective, we recall the most commonly used entropy functionals and locate them within the UJK family. The need for generalized entropies is illustrated with two applications, one in economics and the other in ecology. A simple mathematical model worked out in detail further shows the power of generalized maximum-entropy approaches in cases where strong system independence does not hold. We conclude with practical guidelines for choosing an entropy measure and reporting results so that analyses remain transparent and reproducible.