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The proof of Theorem 1 immediately gives the following corollary.
Corollary 5 Let 1 ≤ q < 3 and a ∈ R \ {0}. Then
d q,1 = λd q,a
for a = 1 and λ ∈ R.
6.4 Refined Riemannian Metrics
Throughout of this section, we fix 1 ≤ q < 3 and a ∈ R \ {0} such that I q,a = ∅,
namely
either q = 1 or q > 1 with 1 − a <
q
q − 1
.
In this case, t q,a = 0. Set
q,a :=
σ ∈ q
1
Z q σ
< T q,a
,
S q,a :=
p q (·; ξ) ∈ S q | ξ ∈ R × q,a
.
The manifold S q,a admits information geometric structures.
6.4.1 Derivatives of (q, a)-Relative Entropy
The (q, a)-relative entropy is nondegenerate on S q,a × S q,a .
Lemma 4 For distinct p, r ∈ S q,a , D
(q,a)
( p, r ) > 0.
Proof Proposition 1 yields that exp
q,a (ln q,a ( p(x))) > 0 in x ∈ R for p ∈ S q,a . The
strict convexity of exp q,a leads to the inequality that
r (x) = exp q,a (ln q,a (r (x)))
> exp q,a (ln q,a ( p(x))) +
ln q,a (r (x)) − ln q,a ( p(x))
exp
q,a (ln q,a ( p(x)))
= p(x) + ln q,a (r (x)) exp
q,a (ln q,a ( p(x))) − ln q,a ( p(x)) exp
q,a (ln q,a ( p(x)))
for x ∈ R and p, r ∈ S q,a . Integrating this inequality on R gives
1 > 1 − d q,a ( p, r ) + d q,a ( p, p) = 1 − D
(q,a)
( p, r ).
Let us define a function ρ
(q,a) on (x, ξ 1 , ξ 2 ) ∈ R × (R × q,a )
2 by
ρ
(q,a)
(x; ξ 1 , ξ 2 ) :=
ln q,a ( p q (x; ξ 1 )) − ln q,a ( p q (x; ξ 2 ))
exp
q,a
ln q,a ( p q (x; ξ 1 ))
.