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In this letter we show that the Shore--Johnson axioms for Maximum EntropyPrinciple in statistical estimation theory account for a considerably widerclass of entropic functional than previously thought. Apart from a formal sideof the proof, we substantiate our point by analyzing the effect of weakcorrelations and discuss two pertinent examples: $2$-qubit quantum system andstrongly interacting nuclear systems.

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