Maximum Entropy Principle in statistical inference: case for non-Shannonian entropies

In this letter we show that the Shore–Johnson axioms for Maximum Entropy Principle in statistical estimation theory account for a considerably wider class of entropic functional than previously thought. Apart from a formal side of the proof, we substantiate our point by analyzing the effect of weak correlations and discuss two pertinent examples: $2$-qubit quantum system and strongly interacting nuclear systems.