Chapter 6
Gauge Freedom of Entropies
on q-Gaussian Measures
Hiroshi Matsuzoe and Asuka Takatsu
Abstract 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 = 1). The limit case q = 1 recovers a Gaussian measure. On the set of all q-Gaussian densities over the real line with 1 ≤ q < 3, escort
expectations determine information geometric structures such as an entropy and a
relative entropy. 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
by any other measures. One of the 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 chapter, 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.
Keywords Information geometry · Gauge freedom of entropies · Refined
q-logarithmic function · q-Gaussian measure
H. Matsuzoe (B)
Department of Computer Science, Nagoya Institute of Technology, Nagoya, Japan
e-mail: matsuzoe@nitech.ac.jp
A. Takatsu
Department of Mathematical Sciences, Tokyo Metropolitan University, Tokyo, Japan
e-mail: asuka@tmu.ac.jp
RIKEN Center for Advanced Intelligence Project (AIP), Tokyo, Japan
© Springer Nature Switzerland AG 2021
F. Nielsen (ed.), Progress in Information Geometry,
Signals and Communication Technology,
https://doi.org/10.1007/978-3-030-65459-7_6
127
Gauge Freedom of Entropies
on q-Gaussian Measures
Hiroshi Matsuzoe and Asuka Takatsu
Abstract 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 = 1). The limit case q = 1 recovers a Gaussian measure. On the set of all q-Gaussian densities over the real line with 1 ≤ q < 3, escort
expectations determine information geometric structures such as an entropy and a
relative entropy. 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
by any other measures. One of the 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 chapter, 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.
Keywords Information geometry · Gauge freedom of entropies · Refined
q-logarithmic function · q-Gaussian measure
H. Matsuzoe (B)
Department of Computer Science, Nagoya Institute of Technology, Nagoya, Japan
e-mail: matsuzoe@nitech.ac.jp
A. Takatsu
Department of Mathematical Sciences, Tokyo Metropolitan University, Tokyo, Japan
e-mail: asuka@tmu.ac.jp
RIKEN Center for Advanced Intelligence Project (AIP), Tokyo, Japan
© Springer Nature Switzerland AG 2021
F. Nielsen (ed.), Progress in Information Geometry,
Signals and Communication Technology,
https://doi.org/10.1007/978-3-030-65459-7_6
127