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Gauge freedom of entropies on $q$-Gaussian measures

A $q$-Gaussian measure is a generalization of a Gaussian measure. This generalization is obtained by replacing the exponential function with the power function of exponent $1/(1-q)$ ($q\neq 1$). The limit case $q=1$ recovers a Gaussian measure. For $1\leq q <3$, the set of all $q$-Gaussian densities over the real line satisfies a certain regularity condition to define information geometric structures such as an entropy and a relative entropy via escort expectations. The ordinary expectation of a random variable is the integral of the random variable with respect to its law. Escort expectations admit us to replace the law to any other measures. A choice of escort expectations on the set of all $q$-Gaussian densities determines an entropy and a relative entropy. One of most important escort expectations on the set of all $q$-Gaussian densities is the $q$-escort expectation since this escort expectation determines the Tsallis entropy and the Tsallis relative entropy. The phenomenon gauge freedom of entropies is that different escort expectations determine the same entropy, but different relative entropies. In this note, we first introduce a refinement of the $q$-logarithmic function. Then we demonstrate the phenomenon on an open set of all $q$-Gaussian densities over the real line by using the refined $q$-logarithmic functions. We write down the corresponding Riemannian metric.