136
H. Matsuzoe and A. Takatsu
In the next section, we demonstrate gauge freedom of entropies on an open
set of q-Gaussian densities over R for 1 ≤ q < 3, which is a typical example of
manifolds consisting of probability densities. To be precise, we show that different
escort expectations determine the same entropy up to scalar multiple, but different
relative entropies, where the entropy coincides with the Boltzmann–Shannon entropy
if q = 1, and the Tsallis entropy otherwise.
6.3 Gauge Freedom of Entropies
6.3.1 q-Gaussian Measures
To define q-Gaussian measures, we extend exp q to the whole of R by
Rexp q (τ ) := max{0, 1 + (1 − q)τ }
1
1−q
for τ ∈ R,
where by convention 0
c
:= ∞ for c < 0. We recall the following improper integral.
Although it is known, we prove it for the sake of completeness.
Lemma 2 For q ∈ R and (μ, λ) ∈ R × (0, ∞), the improper integral of the function
x → Rexp q (−λ(x − μ)
2
)
on R converges if and only if q < 3. For q < 3,
3 − q
R
Rexp q (−x
2
)dx = Z q :=
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
3 − q
q − 1
B
3 − q
2(q − 1)
,
1
2
if q > 1,
√
2π
if q = 1,
3 − q
1 − q
B
2 − q
1 − q
,
1
2
if q < 1,
where B(·, ·) stands for the beta function.
Proof By the change of variables, it is enough to show the case (μ, λ) = (0, 1). We
omit the proof for the case q = 1, which is well-known.
Assume q > 1. There exist c, C, R > 0 depending on q such that
cx
2
1−q ≤ Rexp q (−x
2
) =
1 − (1 − q)x
2
1
1−q
< C x
2
1−q
for |x| > R. Since the improper integral of the function
x → x
2
1−q