One-parameter generalised Fisher information
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We introduce the generalised Fisher information or the one-parameter extended class of the Fisher information. This new form of the Fisher information is obtained from the intriguing connection between the standard Fisher information and the variational principle together with the non-uniqueness property of the Lagrangian. Furthermore, one could treat this one-parameter Fisher information as a generating function for obtaining what is called Fisher information hierarchy. The generalised Cramer-Rao inequality is also derived. The interesting point is about the fact that the whole Fisher information hierarchy, except for the standard Fisher information, does not follow the additive rule. This could suggest that there is an indirect connection between the Tsallis entropy and the one-parameter Fisher information. Furthermore, the whole Fisher information hierarchy is also obtained from the two-parameter Kullback-Leibler divergence.
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