6 Gauge Freedom of Entropies on q-Gaussian Measures
135
T > sup{ p(ω) | p ∈ S, ω ∈ }
if the above supremum is finite, otherwise T := ∞.
Definition 2 Let : (0, T ) → R be a differentiable function such that
> 0 in
(0, T ). For p ∈ S, we define a measure ν ; p on as the absolutely continuous
measure with respect to m with Radon–Nikodym derivative
dν ; p
dm
(ω) =
1
( p(ω))
.
Note that is often assumed to be concave such as the logarithmic function. In the
case = log, we have that
dν ; p
dm
= p.
Therefore, the escort expectation (6.3) is nothing but the ordinary expectation, that
is
E ν ; p [ f ] =
f (ω)dν ; p (ω) =
f (ω) p(ω)dm(ω).
Definition 3 Fix a differentiable function : (0, T ) → R such that
> 0 in (0, T )
and assume that
(r ) = ◦ r ∈ L
1
(ν ; p )
for p, r ∈ S.
(6.4)
(1) For p, r ∈ S, the -cross entropy of p with respect to r is defined by
d ( p, r ) := −E ν ; p [(r )].
(2) The -entropy of p ∈ S is defined by
Ent ( p) := d ( p, p).
(3) For p, r ∈ S, the -relative entropy of p with respect to r is defined by
D
(()
( p, r ) := −d ( p, p) + d ( p, r ).
A choice of differentiable functions : (0, T ) → R such that
> 0 in (0, T )
determines an entropy and a relative entropy on S. The phenomenon gauge freedom
of entropies is that different escort expectations determine the same entropy, but different relative entropies. This is motivated by gauge freedom of Riemannian metrics
proposed by Zhang and Naudts [6] (see also [5]).