  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.
