140
H. Matsuzoe and A. Takatsu
Proof The corollary trivially holds for q = 1.
Assume q > 1. We observe from (6.5) that
ln q,a ( p(x; μ, σ )) = −
1
a
− ln q
1
Z q σ
+
1
(Z q σ ) 1−q (3 − q)
x − μ
σ
2
a
for (μ, σ ) ∈ R × q . This with Lemma 3 yields that
ln q,a (r ) ∈ L
1
(ν q,a;ξ ) ⇔ a <
1
2
+
1
q − 1
+ a − 1,
(6.6)
which holds for q < 3.
Following Definition 3, we define an entropy and a relative entropy on S q . Recall
the escort expectation of a function f ∈ L
1
(ν q,a;ξ ) with respect to ν q,a;ξ is defined
by
E ν q,a;ξ [ f ] =
R
f (x)dν q,a;ξ (x) =
R
f (x) exp
q,a
ln q,a
p q (x; ξ)
dx.
Definition 6 Let 1 ≤ q < 3 and a ∈ R \ {0}. Take ξ ∈ R × q and set
p = p q (·; ξ) ∈ S q .
(1) The (q, a)-cross entropy of p with respect to r ∈ S q is defined by
d q,a ( p, r ) := −E ν q,a;ξ [ln q,a (r )].
(2) The (q, a)-entropy of p is defined by
Ent q,a ( p) := d q,a ( p, p).
(3) The (q, a)-relative entropy of p with respect to r ∈ S q is defined by
D
(q,a)
( p, r ) := −d q,a ( p, p) + d q,a ( p, r ).
Remark 4 The domain of the (q, 1)-entropy can be extended from S q to the whole
of q-Gaussian densities. The (q, 1)-entropy coincides with the Boltzmann–Shannon
entropy if q = 1, and the Tsallis entropy otherwise.
Theorem 1 (Gauge freedom of entropies) Let 1 ≤ q < 3 and a ∈ R \ {0}. Then
Ent q,1 = aEnt q,a ,
D
(q,1)
= λD
(q,a)
for a = 1 and λ ∈ R.