Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
3/2026
FISHER-LIKE
METRICS
ASSOCIATED
WITH
φ
-DEFORMED
NAUDTS
ENTROPIES
II
-
THE
WEIGHTED
CASE
∗
Iulia-Elena
Hirica
†
Cristina-Liliana
Pripoae
‡
Gabriel-Teodor
Pripoae
§
Vasile
Preda
Dedicated
to
the
memory
of
Professor
Mihail
Megan
DOI
10.56082/annalsarscimath.2026.3.59
Abstract
The
paper
continues
our
previous
study
on
semi-Riemannian
gener-
alized
Fisher
metrics
and
Fisher-like
metrics
[
26
],
in
the
framework
of
weighted
φ
-deformed
(Naudts)
entropy.
The
associated
scalar
curva-
tures
carry
information
related
to
the
local
structure
of
the
space
of
the
distributions
of
probability.
The
addition
of
some
related
group
log-
arithms
extends
and
refines
results
from
the
previously
quoted
paper
on
the
non-weighted
case.
Keywords:
weighted
φ
-deformed
(Naudts)
entropy,
weighted
diver-
gence,
weighted
relative
group
entropy,
weighted
generalized
Fisher
metric,
weighted
Fisher-like
metric.
∗
Accepted
for
publication
on
March
11,
2026
†
ihirica@fmi.unibuc.ro
,
Faculty
of
Mathematics
and
Computer
Science,
University
of
Bucharest,
Academiei
14,
RO-010014
Bucharest,
Romania
‡
cristinapripoae@csie.ase.ro
,
Department
of
Applied
Mathematics,
The
Bucharest
University
of
Economic
Studies,
Piata
Romana
6,
RO-010374
Bucharest,
Romania
§
gpripoae@fmi.unibuc.ro
,
Faculty
of
Mathematics
and
Computer
Science,
University
of
Bucharest,
Academiei
14,
RO-010014
Bucharest,
Romania
vasilepreda0@gmail.com
,
Gheorghe
Mihoc-Caius
Iacob
Institute
of
Mathematical
Statistics
and
Applied
Mathematics
of
Romanian
Academy,
Calea
13
Septembrie,
no.
13,
RO-050711,
Bucharest,
Romania;
Costin
C.
Kiritescu
National
Institute
of
Economic
Research
of
Romanian
Academy,
Calea
13
Septembrie,
no.
13,
RO-050711,
Bucharest,
Ro-
mania;
Faculty
of
Mathematics
and
Computer
Science,
University
of
Bucharest,
Academiei
14,
RO-010014
Bucharest,
Romania
59
On
Fisher-like
metrics
60
MSC:
53B12,
22E70,
94A17,
53B20.
1
Introduction
Modeling
stochastic
phenomena
involves
many
different
types
of
entropies
and
divergences,
ranking
from
”simpler”
to
”generalized”
ones
[
5
,
7
,
11
,
14
,
20
–
22
,
26
,
28
,
30
,
31
,
34
–
36
]
(see
also
[
4
,
6
,
9
,
25
,
28
,
29
,
38
]
for
further
details
and
references).
One
direction
uses
differential
geometric
tools
(
[
1
,
2
,
20
]),
such
as
the
Fisher
(semi-Riemannian)
metrics,
inspired
by
the
Fisher
Information
matri-
ces.
The
under-construction
dictionary
between
Statistics
and
(Riemannian)
Geometry
is
expected
to
enrich
both
domains
with
new
insights
(
[
6
,
20
,
40
]).
The
context
can
be
generalized
by
considering
weights
associated
to
en-
tropies
and
divergences
[
7
,
8
].
These
weights
allow
additional
refinements
and
new
applications,
for
both
entropies
and
divergences,
which
impact
all
derived
notions,
in
particular
the
associated
Fisher
metrics.
A
synthesis
concerning
the
weighting
extension
method
can
be
found
in
our
previous
paper
[
8
].
The
φ
-deformed
(Naudts)
entropy
is
a
newcomer
in
the
panorama
of
modern
entropies
(
[
15
,
16
]).
Its
versatility
qualifies
it
for
various
applica-
tions
in
Physics
and
Information
theory,
beyond
the
classical
entropies
like
BGS,
Kaniadakis,
Tsallis
ones.
The
Fisher
metrics
associated
with
it
and
the
related
divergences
were
intensively
studied
(
[
12
,
16
–
19
,
39
]).
For
additional
historical
notes
and
more
details
on
generalized
entropy
and
generalized
di-
vergence
topics,
including
some
related
to
the
φ
-deformed
(Naudts)
entropy,
we
refer
to
our
previous
paper
[
26
].
1.1
The
content
of
the
paper
In
Section
2
,
we
recall
the
framework
for
the
generalized
entropies
and
diver-
gences
defined
in
[
8
,
26
],
such
as
the
(classical)
φ
-deformed
(Naudts)
entropy
and
the
generalized
group
entropy
functional.
In
Section
3
,
we
recall
the
generalized
Fisher-like
metrics
associated
with
generalized
entropies
and
divergences,
defined
and
studied
in
[
6
,
26
].
The
core
of
the
paper
is
Section
4
;
it
contains
the
definition
of
the
weighted
generalized
Fisher-like
metrics,
associated
with
group
relative
en-
tropies
based
on
weighted
φ
-deformed
(Naudts)
entropies
and
divergences.
Their
coefficients
depend
on
the
PDFs,
on
the
φ
-deformed
logarithm,
on
the
weighting
function
and
(eventually)
on
a
group
logarithm
too.
A
comparison
with
the
non-weighted
case
is
pointed
out.
I.-E.
Hirica,
C.-L.
Pripoae,
G.-T.
Pripoae,
V.
Preda
61
Examples
of
the
new
metrics
are
provided
in
Section
5
,
based
on
ex-
ponential
PDFs.
We
calculate
their
scalar
curvatures
functions
and
we
do
some
geometrically
oriented
remarks.
1.2
Conventions
We
suppose,
implicitly,
that
the
integrals
are
correctly
defined
and
that
they
commute
with
their
derivatives.
All
differential
objects
are
considered
to
be
differentiable
(”smooth”).
Unless
otherwise
stated,
the
metrics
are
supposed
to
be
(only)
non-degenerated,
i.e.
semi-Riemannian
ones.
2
Preliminaries:
the
φ
-deformed
(Naudts)
entropy,
the
formal
group
logarithm
and
the
generalized
divergences
Let
X
⊂
R
m
be
a
domain
and
ρ
=
ρ
(
x
)
a
fixed
probability
density
function
(PDF)
on
X
(i.e.
ρ
(
x
)
≥
0
and
X
ρ
(
x
)
dx
=
1).
Let
ϕ
be
a
fixed
real
valued
differentiable
function
on
X
.
We
defined
(
[
26
])
a
ϕ
-dependent
generalized
(normalized)
entropy
of
X
,
by
H
[
ρ
]
=
−
X
ρ
(
x
)
ϕ
(
ρ
(
x
))
dx.
(1)
Let
F
:
[0
,
∞
)
×
[0
,
∞
)
→
R
a
smooth
function
and
σ
an
additional
fixed
PDF.
We
define
D
(
ρ,
σ
)
:=
X
F
(
ρ
(
x
)
,
σ
(
x
))
dx.
(2)
Implicitly,
we
suppose
that
D
(
ρ,
σ
)
≥
0
and
D
(
ρ,
σ
)
=
0
if
and
only
if
ρ
=
σ
.
The
number
D
(
ρ,
σ
)
is
called
the
(generalized)
divergence
between
ρ
and
σ
and
measures
to
what
extent
σ
influences
ρ
,
as
governed
by
F
(see
[
26
]).
Example
1.
With
the
previous
notations,
we
give
two
examples;
for
more
examples,
comments
and
details,
see
[
23
,
24
,
26
,
32
].
(i)
Let
φ
:
(0
,
∞
)
→
R
be
a
positive,
differentiable,
strictly-increasing
function.
The
φ
-deformed
(Naudts)
logarithm
is
defined
by
(see
[
16
,
26
])
log
N
φ
(
y
)
:=
y
1
1
φ
(
z
)
dz.
(3)
The
function
ϕ
N
φ
:=
log
N
φ
defines
the
φ
-deformed
(Naudts)
entropy.
On
Fisher-like
metrics
62
(ii)
Let
G
=
G
(
t
)
be
a
formal
group
logarithm,
which
is
a
differentiable
real
valued
function
with
some
special
algebraic
properties,
inspired
from
the
formal
series
linking
Lie
groups
to
Lie
algebras.
More
precisely,
G
(
t
)
:=
∞
i
=0
c
i
t
i
+1
i
+
1
,
where
c
0
=
1
and
c
i
∈
Q
.
Its
inverse
is
F
(
s
)
:=
∞
i
=0
γ
i
s
i
+1
i
+
1
,
where
γ
i
∈
Q
,
γ
0
=
1
,
γ
1
=
−
c
1
,
γ
2
=
3
2
c
2
1
−
c
2
and
so
on.
(We
refer
to
[
6
,
37
,
38
]
for
details
about
these
functions).
The
simplest
example
is
G
(
t
)
=
t
.
We
defined
[
26
]
the
generalized
group
entropy
functional
(GGEF)
asso-
ciated
with
(
1
),
by
S
G
(
ρ
)
:=
X
ρ
(
x
)
G
(
ϕ
◦
ρ
(
x
))
dx.
(4)
In
particular,
for
ϕ
:=
−
log
,
we
recover
the
well-known
group
entropy
func-
tional
(
[
6
,
37
])
associated
with
(
1
)
S
G
(
ρ
)
:=
X
ρ
(
x
)
G
(
logρ
(
x
)
−
1
)
dx.
(5)
Similar
GGEFs
can
be
provided
by
replacing
the
Neperian
logarithm
by
other
“generalized”
logarithms.
For
example,
in
Section
3
,
we
shall
consider
the
geometries
associated
with
the
GGEF,
based
on
φ
-deformed
(Naudts)
entropies,
by
using
the
generalized
logarithm
log
N
φ
from
(
3
).
Example
2.
(i)
Consider
the
previous
framework;
we
recall
now
the
two
main
types
of
divergence
from
[
26
].
The
generalized
(quotient)
relative
entropy
(a.k.a.
generalized
diver-
gence)
between
ρ
and
σ
is
given
by
(see
[
3
,
8
])
˜
D
(
ρ
σ
)
:=
X
ρ
(
x
)
ϕ
(
ρ
(
x
)
σ
(
x
)
)
dx.
(6)
In
this
case,
the
previous
function
F
(
z,
y
)
:=
zϕ
(
z
y
)
.
I.-E.
Hirica,
C.-L.
Pripoae,
G.-T.
Pripoae,
V.
Preda
63
The
generalized
(difference)
relative
entropy
between
ρ
and
σ
is
given
by
D
(
ρ
σ
)
:=
X
ρ
(
x
)[
ϕ
(
ρ
(
x
))
−
ϕ
(
σ
(
x
))]
dx.
(7)
In
this
case,
the
function
F
(
z,
y
)
:=
z
[
ϕ
(
z
)
−
ϕ
(
y
)]
.
(ii)
Following
the
Example
1
(vi),
possible
generalized
divergences
as
relative
group
entropies
may
use
formal
group
logarithms
G
,
ϕ
-likelihood
functions
and
quotient/
difference
operation
upon
two
PDFs.
For
example,
from
(
7
)
we
can
derive
D
G
(
ρ
σ
)
:=
X
ρ
(
x
)
·
G
ϕ
(
ρ
(
x
))
−
ϕ
(
σ
(
x
))
dx.
(iii)
Let
ρ
1
and
ρ
2
be
two
fixed
PDFs.
Consider
a
fixed
convex
differ-
entiable
function
ψ
:
R
→
R
.
Then,
the
Bregman
divergence
is
given
by
D
ψ
(
ρ
1
ρ
2
)
:=
X
{
ψ
(
ρ
1
(
x
))
−
ψ
(
ρ
2
(
x
))
−
(
ρ
1
(
x
)
−
ρ
2
(
x
))
ψ
(
ρ
2
(
x
))
}
dx.
(8)
(iv)
Let
w
:
X
→
R
a
differentiable
”weighting”
function
(usually
sup-
posed
non-negative).
The
weighted
entropies
and
divergences
functionals
[
7
,
8
]
are
obtained
from
the
formulas
of
the
non-weighted
ones,
by
multiply-
ing
the
integrand
with
w
.
We
remark
that
any
ϕ
-generalized
entropy
from
(
1
)
and
any
generalized
divergence
from
(
2
)
can
be
put
in
the
form
of
a
weighted
entropy
or
a
weighted
divergence,
by
means
of
other
functions
ϕ
and
F
,
respectively.
For
example,
the
φ
-deformed
(Naudts)
entropy
is
(see
Example
1
(i))
H
[
ρ
]
=
−
X
ρ
(
x
)
·
log
N
φ
(
ρ
(
x
))
dx
and
the
weighted
φ
-deformed
(Naudts)
entropy
is
H
w
[
ρ
]
=
−
X
w
(
x
)
·
ρ
(
x
)
·
log
N
φ
(
ρ
(
x
))
dx.
(9)
3
Preliminaries:
Fisher-like
metrics
associated
with
generalized
entropies
and
divergences
We
recall,
following
[
6
,
8
,
26
],
the
notion
of
Fisher
metric
associated
with
(generalized)
entropies
and
divergences,
defined
on
the
space
of
parameters
of
an
arbitrary
PDF.
On
Fisher-like
metrics
64
Let
Θ
be
an
open
set
of
R
n
and
ρ
:
X
×
Θ
→
R
,
ρ
=
ρ
(
x,
θ
)
be
a
family
of
PDFs,
parameterized
by
θ
∈
Θ.
Fix
ϕ
:
R
→
R
,
ϕ
=
ϕ
(
y
)
a
differentiable
(”controlling”)
function.
By
analogy
with
formula
(
1
),
we
derived
[
26
]
the
generalized
entropy
function
H
:
Θ
→
R
H
(
θ
)
=
−
X
ρ
(
x,
θ
)
·
ϕ
(
ρ
(
x,
θ
))
dx.
(10)
Generalized
divergence
functions
can
be
defined
accordingly,
by
extending
formula
(
2
)
and
the
similar
ones.
We
defined
[
26
]
the
generalized
Fisher
metrics
of
type
1
and
type
2
(de-
noted
GFM
1
and
GFM
2,
respectively),
with
components:
g
ij
(
θ
)
:=
−
X
ρ
(
x,
θ
)
∂
2
ϕ
(
ρ
(
x,
θ
))
∂θ
i
∂θ
j
dx,
i,
j
=
1
,
n
(11)
and
˜
g
ij
(
θ
)
:=
X
ρ
(
x,
θ
)
∂ϕ
(
ρ
(
x,
θ
))
∂θ
i
·
∂ϕ
(
ρ
(
x,
θ
))
∂θ
j
dx,
i,
j
=
1
,
n.
(12)
(Recall
that,
implicitly,
we
suppose
the
matrices
(
g
ij
)
i,j
=
1
,n
and
(˜
g
ij
)
i,j
=
1
,n
are
non-degenerated
and
of
constant
index
on
Θ.)
If
non-degenerated,
the
Hessian
matrix
h
ij
(
θ
)
:=
∂
2
H
(
θ
)
∂θ
i
∂θ
j
(13)
provides
another
semi-Riemannian
metric
on
Θ.
In
[
26
],
we
defined
other
two
Fisher-like
semi-Riemannian
metrics
α
and
β
on
Θ,
with
components:
α
ij
(
θ
)
:=
X
∂
2
ρ
(
x,
θ
)
∂θ
i
∂θ
j
·
ϕ
(
ρ
(
x,
θ
))
dx
(14)
and
β
ij
(
θ
)
:=
X
∂ρ
(
x,
θ
)
∂θ
i
·
∂ϕ
(
ρ
(
x,
θ
))
∂θ
j
+
∂ρ
(
x,
θ
)
∂θ
j
·
∂ϕ
(
ρ
(
x,
θ
))
∂θ
i
dx
(15)
(under
the
same
non-degeneracy
implicit
hypothesis).
I.-E.
Hirica,
C.-L.
Pripoae,
G.-T.
Pripoae,
V.
Preda
65
Remark
1.
(i)
Consider
a
weighting
function
w
on
X
×
Θ
.
There
are
several
possibilities
to
construct
weighted
Fisher-like
metrics.
For
example,
we
can
define
the
w
-weighted
counterpart
of
the
metric
(
11
):
g
ij
(
θ
)
:=
−
X
w
(
x,
θ
)
·
ρ
(
x,
θ
)
∂
2
ϕ
(
ρ
(
x,
θ
))
∂θ
i
∂θ
j
dx,
i,
j
=
1
,
n.
(16)
In
what
follows,
we
shall
use
this
weighting
convention.
(ii)
In
[
26
],
we
studied
the
previous
Fisher-like
metrics,
in
the
particular
case
of
the
φ
-deformed
(Naudts)
entropy,
i.e.
for
ϕ
:=
log
N
φ
.
In
Section
4,
we
shall
extend
this
inquiry
in
the
weighted
φ
-deformed
(Naudts)
entropies
case.
4
Fisher
metrics
associated
with
GGEFs
based
on
weighted
φ
-deformed
(Naudts)
entropies
and
di-
vergences
We
particularize
now
the
results
from
Section
3
,
in
the
case
of
the
weighted
φ
-deformed
(Naudts)
entropies.
Consider
φ
a
positive,
differentiable
and
strictly-increasing
function
as
in
Example
1
(v)
and
the
φ
-deformed
(Naudts)
logarithm
log
N
φ
defined
in
Formula
(
3
).
Let
ρ
:
X
×
Θ
→
R
,
ρ
=
ρ
(
x,
θ
)
be
a
family
of
parameterized
PDFs,
as
in
Section
3
.
The
associated
GFM
1
metric
g
and
the
GFM
2
metric
˜
g
are
obtained
as
particular
cases
from
(
11
)
and
(
12
):
g
ij
(
θ
)
:=
−
X
ρ
(
x,
θ
)
∂
2
log
N
φ
(
ρ
(
x,
θ
))
∂θ
i
∂θ
j
dx,
i,
j
=
1
,
n
(17)
and
˜
g
ij
(
θ
)
:=
X
ρ
(
x,
θ
)
∂log
N
φ
(
ρ
(
x,
θ
))
∂θ
i
·
∂log
N
φ
(
ρ
(
x,
θ
))
∂θ
j
dx,
i,
j
=
1
,
n.
(18)
We
suppose,
as
usual,
that
g
and
˜
g
are
non-degenerated
and
that
˜
g
has
a
constant
index
on
X
.
Consider,
in
addition,
a
weighting
function
w
on
X
×
Θ.
The
associated
w
-weighted
GFM
1
metric
g
w
and
the
w
-weighted
GFM
2
metric
˜
g
w
write:
g
w
ij
(
θ
)
:=
−
X
w
(
x,
θ
)
ρ
(
x,
θ
)
∂
2
log
N
φ
(
ρ
(
x,
θ
))
∂θ
i
∂θ
j
dx,
i,
j
=
1
,
n
(19)
On
Fisher-like
metrics
66
and
˜
g
w
ij
(
θ
)
:=
X
w
(
x,
θ
)
ρ
(
x,
θ
)
∂log
N
φ
(
ρ
(
x,
θ
))
∂θ
i
·
∂log
N
φ
(
ρ
(
x,
θ
))
∂θ
j
dx,
i,
j
=
1
,
n.
(20)
We
also
consider,
via
(
13
),
the
associated
Hessian
metric
h
=
h
(
θ
)
h
ij
(
θ
)
=
−
∂
2
∂θ
i
∂θ
j
X
ρ
(
x,
θ
)
·
log
N
φ
(
ρ
(
x,
θ
))
dx
,
i,
j
=
1
,
n.
(21)
The
Hessian
metric
associated
to
the
w
-weighted
entropy
function
H
w
is
h
w
ij
(
θ
)
=
−
∂
2
∂θ
i
∂θ
j
X
w
(
x,
θ
)
ρ
(
x,
θ
)
·
log
N
φ
(
ρ
(
x,
θ
))
dx
,
i,
j
=
1
,
n.
(22)
Proposition
1.
With
the
previous
notations,
for
every
i,
j
=
1
,
n
,
we
have
g
w
ij
(
θ
)
=
X
w
(
x,
θ
)
ρ
(
x,
θ
)
∂ρ
(
x,
θ
)
∂θ
i
·
∂ρ
(
x,
θ
)
∂θ
j
·
φ
−
2
(
ρ
(
x,
θ
))
·
φ
(
ρ
(
x,
θ
))
(23)
−
∂
2
ρ
(
x,
θ
)
∂θ
i
∂θ
j
·
φ
−
1
(
ρ
(
x,
θ
))
dx,
˜
g
w
ij
(
θ
)
:=
X
w
(
x,
θ
)
ρ
(
x,
θ
)
·
∂ρ
(
x,
θ
)
∂θ
i
·
∂ρ
(
x,
θ
)
∂θ
j
·
φ
−
2
(
ρ
(
x,
θ
))
dx,
(24)
and
h
w
ij
(
θ
)
=
X
w
(
x,
θ
)
ρ
(
x,
θ
)
∂ρ
(
x,
θ
)
∂θ
i
·
∂ρ
(
x,
θ
)
∂θ
j
·
φ
−
2
(
ρ
(
x,
θ
))
·
φ
(
ρ
(
x,
θ
))
(25)
−
∂
2
ρ
(
x,
θ
)
∂θ
i
∂θ
j
·
log
N
φ
(
ρ
(
x,
θ
))
−
2
∂ρ
(
x,
θ
)
∂θ
i
·
∂ρ
(
x,
θ
)
∂θ
j
·
φ
−
1
(
ρ
(
x,
θ
))
−
ρ
(
x,
θ
)
·
∂
2
ρ
(
x,
θ
)
∂θ
i
∂θ
j
·
φ
−
1
(
ρ
(
x,
θ
))
dx
−
X
∂
2
w
(
x,
θ
)
∂θ
i
∂θ
j
ρ
(
x,
θ
)
log
N
φ
(
ρ
(
x,
θ
))
+(
∂w
(
x,
θ
)
∂θ
j
∂ρ
(
x,
θ
)
∂θ
i
+
∂w
(
x,
θ
)
∂θ
i
∂ρ
(
x,
θ
)
∂θ
j
)(log
N
φ
(
ρ
(
x,
θ
))
+
ρ
(
x,
θ
)
φ
−
1
(
ρ
(
x,
θ
)))
dx.
I.-E.
Hirica,
C.-L.
Pripoae,
G.-T.
Pripoae,
V.
Preda
67
In
this
case,
α
w
and
β
w
are
given
by
α
w
ij
(
θ
)
:=
X
∂
2
(
w
(
x,
θ
)
ρ
(
x,
θ
))
∂θ
i
∂θ
j
·
log
N
φ
(
ρ
(
x,
θ
))
dx
and
β
w
ij
(
θ
)
:=
X
∂
(
w
(
x,
θ
)
ρ
(
x,
θ
))
∂θ
i
·
∂log
N
φ
(
ρ
(
x,
θ
))
∂θ
j
+
∂
(
w
(
x,
θ
)
ρ
(
x,
θ
))
∂θ
j
·
∂log
N
φ
(
ρ
(
x,
θ
))
∂θ
i
dx.
Corollary
1.
In
a
condensed
form,
we
have
the
following
relation
h
w
=
g
w
−
α
w
−
β
w
.
We
consider
now,
in
addition,
a
fixed
formal
group
logarithm
G
,
as
in
Example
1
(vi).
Let
σ
:=
ρ
(
x,
θ
0
)
be
the
associated
parameterized
PDFs
and
D
G,φ
=
D
G,φ
(
ρ
σ
)(
θ,
θ
0
)
be
the
generalized
(difference)
group
relative
entropy
(a.k.a.
the
generalized
(difference)
group
divergence),
as
particular-
ization
from
(
7
)
and
Example
2.2
(i),
(ii),
written
as
D
G,φ
(
ρ
σ
)(
θ,
θ
0
)
=
X
ρ
(
x,
θ
)
·
G
log
N
φ
(
ρ
(
x,
θ
))
−
log
N
φ
(
ρ
(
x,
θ
0
))
dx.
Let
D
w
G,φ
(
ρ
σ
)(
θ,
θ
0
)
=
X
w
(
x,
θ
)
ρ
(
x,
θ
)
·
G
log
N
φ
(
ρ
(
x,
θ
))
−
log
N
φ
(
ρ
(
x,
θ
0
))
dx.
Denote
the
generalized
group
Fisher
metric
associated
with
D
G,φ
by
ˆ
g
jk
(
θ
0
)
:=
∂
2
D
G,φ
(
ρ
σ
)(
θ,
θ
0
)
∂θ
j
∂θ
k
|
θ
=
θ
0
.
(26)
Denote
the
w
-weighted
generalized
group
Fisher
metric
associated
with
D
w
G,φ
by
ˆ
g
w
jk
(
θ
0
)
:=
∂
2
D
w
G,φ
(
ρ
σ
)(
θ,
θ
0
)
∂θ
j
∂θ
k
|
θ
=
θ
0
.
(27)
This
Hessian-type
metric
will
be
calculated
in
the
next
result.
On
Fisher-like
metrics
68
Proposition
2.
With
the
previous
notations,
we
have
the
relation
ˆ
g
w
jk
(
θ
0
)
=
G
(0)
·
X
w
(
x,
θ
0
)
∂
2
ρ
(
x,
θ
0
)
∂θ
j
∂θ
k
·
ρ
(
x,
θ
0
)
φ
(
ρ
(
x,
θ
0
))
dx
(28)
+2
X
w
(
x,
θ
0
)
φ
(
ρ
(
x,
θ
0
))
·
∂
∂θ
j
log
N
φ
(
ρ
(
x,
θ
0
))
·
∂
∂θ
k
log
N
φ
(
ρ
(
x,
θ
0
))
dx
−
X
w
(
x,
θ
0
)
ρ
(
x,
θ
0
)
·
φ
(
ρ
(
x,
θ
0
))
·
∂
∂θ
j
log
N
φ
(
ρ
(
x,
θ
0
))
·
∂
∂θ
k
log
N
φ
(
ρ
(
x,
θ
0
))
dx
+
G

(0)
·
X
w
(
x,
θ
0
)
ρ
(
x,
θ
0
)
·
∂
∂θ
j
log
N
φ
(
ρ
(
x,
θ
0
))
·
∂
∂θ
k
log
N
φ
(
ρ
(
x,
θ
0
))
dx
+
G
(0)
X
(
∂w
(
x,
θ
0
)
∂θ
k
∂ρ
(
x,
θ
0
)
∂θ
j
+
∂w
(
x,
θ
0
)
∂θ
j
∂ρ
(
x,
θ
0
)
∂θ
k
)
ρ
(
x,
θ
0
)
φ
(
ρ
(
x,
θ
0
))
dx,
which
may
be
re-written
as
depending
only
on
φ
and
ρ
,
in
ˆ
g
w
jk
(
θ
0
)
=
G
(0)
·
X
w
(
x,
θ
0
)
·
∂
2
ρ
(
x,
θ
0
)
∂θ
j
∂θ
k
·
ρ
(
x,
θ
0
)
φ
(
ρ
(
x,
θ
0
))
dx
(29)
+2
X
w
(
x,
θ
0
)
·
φ
−
1
(
ρ
(
x,
θ
0
))
·
∂
∂θ
j
ρ
(
x,
θ
0
)
·
∂
∂θ
k
ρ
(
x,
θ
0
)
dx
−
X
w
(
x,
θ
0
)
·
ρ
(
x,
θ
0
)
·
φ
(
ρ
(
x,
θ
0
))
·
φ
−
2
(
ρ
(
x,
θ
0
))
·
∂
∂θ
j
ρ
(
x,
θ
0
)
·
∂
∂θ
k
ρ
(
x,
θ
0
)
dx
+
X
∂w
∂θ
k
(
x,
θ
0
)
·
∂ρ
∂θ
j
(
x,
θ
0
)+
∂w
∂θ
j
(
x,
θ
0
)
·
∂ρ
∂θ
k
(
x,
θ
0
)
·
ρ
(
x,
θ
0
)
·
φ
−
1
(
ρ
(
x,
θ
0
))
dx
+
G

(0)
·
X
w
(
x,
θ
0
)
·
ρ
(
x,
θ
0
)
·
φ
−
2
(
ρ
(
x,
θ
0
))
·
∂
∂θ
j
ρ
(
x,
θ
0
)
·
∂
∂θ
k
ρ
(
x,
θ
0
)
dx.
The
proof
is
similar
with
the
previous
ones
and
will
be
skipped.
Suppose,
moreover,
that
G
(
t
)
=
t
.
Then,
we
have
ˆ
g
w
jk
(
θ
0
)
=
X
w
(
x,
θ
0
)
·
∂
2
ρ
(
x,
θ
0
)
∂θ
j
∂θ
k
·
ρ
(
x,
θ
0
)
φ
(
ρ
(
x,
θ
0
))
dx
(30)
+2
X
w
(
x,
θ
0
)
·
φ
−
1
(
ρ
(
x,
θ
0
))
·
∂
∂θ
j
ρ
(
x,
θ
0
)
·
∂
∂θ
k
ρ
(
x,
θ
0
)
dx
I.-E.
Hirica,
C.-L.
Pripoae,
G.-T.
Pripoae,
V.
Preda
69
−
X
w
(
x,
θ
0
)
·
ρ
(
x,
θ
0
)
·
φ
(
ρ
(
x,
θ
0
))
·
φ
−
2
(
ρ
(
x,
θ
0
))
·
∂
∂θ
j
ρ
(
x,
θ
0
)
·
∂
∂θ
k
ρ
(
x,
θ
0
)
dx
+
X
∂w
∂θ
k
(
x,
θ
0
)
·
∂ρ
∂θ
j
(
x,
θ
0
)+
∂w
∂θ
j
(
x,
θ
0
)
·
∂ρ
∂θ
k
(
x,
θ
0
)
·
ρ
(
x,
θ
0
)
·
φ
−
1
(
ρ
(
x,
θ
0
))
dx.
By
analogy,
starting
with
a
generalized
(quotient)
group
relative
en-
tropy
(a.k.a.
the
generalized
(quotient)
group
divergence)
˜
D
G,φ
=
˜
D
G,φ
(
ρ
σ
)(
θ,
θ
0
),
as
particularization
from
(
6
),
we
shall
obtain,
in
the
sequel,
other
Fisher-like
metrics,
similar
to
the
ones
in
Proposition
2
.
Denote
the
generalized
group
Fisher
metric
associated
with
˜
D
G,φ
by
g
jk
(
θ
0
)
:=
∂
2
˜
D
G,φ
(
ρ
σ
)(
θ,
θ
0
)
∂θ
j
∂θ
k
|
θ
=
θ
0
.
(31)
The
previous
considerations
relative
to
D
G,φ
,
D
w
G,φ
,
ˆ
g
,
ˆ
g
w
can
be
adapted,
by
analogy,
for
˜
D
G,φ
,
˜
D
w
G,φ
and
associated
metrics
g
,
g
w
as
well.
A
result
similar
to
Proposition
2
can
be
easily
derived
and
proved
as
in
the
non-
weighted
case
[
26
].
Remark
2.
We
constructed
a
large
family
of
associated
Riemannian
met-
rics,
in
terms
of
the
functions
φ
and
w
.
Examples
(with
specific
calculations
focusing
on
the
scalar
curvature)
will
be
given
in
the
next
section.
5
Examples
We
particularize
now
the
results
from
Section
4
,
for
the
case
when
ρ
is
an
exponential
PDF
and
m
=
1,
n
=
2.
The
deforming
function
φ
and
the
weighting
function
w
will
be
chosen
conveniently,
in
order
to
be
able
to
compute
the
integrals.
For
simplicity,
we
will
omit
(whenever
possible)
writing
the
function
variables.
Let
X
:=
R
and
ρ
:
R
×
R
×
(0
,
∞
)
→
R
be
the
exponential
(normal)
PDF
given
by
ρ
(
x
;
θ
1
,
θ
2
)
=
1
√
2
πθ
2
·
e
−
(
x
−
θ
1
)
2
2(
θ
2
)
2
.
(32)
We
denote
the
first
order
and
the
second
order
partial
derivatives
of
ρ
,
with
respect
to
the
variables
θ
1
and
θ
2
,
by
ρ
1
,
ρ
2
,
ρ
11
,
ρ
12
,
ρ
22
.
We
have
the
formulas
(
[
8
]):
On
Fisher-like
metrics
70
ρ
1
=
x
−
θ
1
(
θ
2
)
2
·
ρ,
ρ
2
=
{
(
x
−
θ
1
)
2
(
θ
2
)
3
−
1
θ
2
}
·
ρ,
ρ
11
=
{
(
x
−
θ
1
)
2
(
θ
2
)
4
−
1
(
θ
2
)
2
}
·
ρ,
ρ
12
=
{
(
x
−
θ
1
)
3
(
θ
2
)
5
−
3(
x
−
θ
1
)
(
θ
2
)
3
}
·
ρ,
ρ
22
=
{
(
x
−
θ
1
)
4
(
θ
2
)
6
−
5(
x
−
θ
1
)
2
(
θ
2
)
4
+
2
(
θ
2
)
2
}
·
ρ.
The
classical
Fisher
metric
g
0
has
the
coefficients
g
0
11
=
(
θ
2
)
−
2
,
g
0
12
=
g
0
21
=
0
and
g
0
22
=
2(
θ
2
)
−
2
(
[
8
]).
Remark
3.
Let
A
,
k
1
,
k
2
be
fixed
real
constants,
with
k
1
=
0
,
k
2
=
0
.
Then,
the
semi-Riemannian
metric
y
−
A
·
k
1
0
0
k
2
on
the
set
y
=
0
in
R
2
has
the
scalar
curvature
−
A
·
(
k
2
)
−
1
·
y
A
−
2
(
[
8
]).
In
the
sequel,
we
give
examples
of
semi-Riemannian
metrics
from
Propo-
sition
1
,
under
various
particular
hypothesis.
Consider
a
2-parameter
family
of
weighting
functions
w
=
w
a,b
:
R
×
R
×
(0
,
∞
)
→
R
,
of
the
form
w
(
x,
θ
1
,
θ
2
)
=
(
x
−
θ
1
)
2
a
·
(
θ
2
)
2
b
.
(Here,
a
>
0
and
b
are
fixed
arbitrary
parameters.)
Suppose
φ
(
t
)
:=
t
c
,
with
c
∈
(0
,
2)
a
fixed
arbitrary
parameter.
From
Formula
(
23
),
we
calculate
the
coefficients:
g
w
11
=
K
1
(
a,
c
)
·
(
θ
2
)
2
a
+2
b
+
c
−
3
,
g
w
12
=
g
w
21
=
0
,
g
w
22
=
(
√
2
π
)
c
−
2
·
(
θ
2
)
2
b
+
c
−
8
·
[(
c
−
1)
I
a
+2
+(5
−
2
c
)(
θ
2
)
2
I
a
+1
+(
c
−
2)(
θ
2
)
4
I
a
]
,
where
K
1
(
a,
c
)
=
(2
−
c
)
−
a
−
3
2
·
(
√
2
π
)
c
−
2
·
2
a
+
1
2
·
Γ(
a
+
1
2
)
·
[(
c
−
1)(2
a
+
1)
+
2
−
c
]
I.-E.
Hirica,
C.-L.
Pripoae,
G.-T.
Pripoae,
V.
Preda
71
and
I
a
=
I
a
(
c,
θ
2
)
is
given
by
I
a
=
(
θ
2
)
2
a
+1
·
(2
−
c
)
−
a
−
1
2
·
2
a
+
1
2
·
Γ(
a
+
1
2
)
.
As
the
coefficients
of
g
w
depend
on
the
gamma-integrals,
it
is
difficult
to
derive
specific
invariant
objects
from
it.
This
is
why,
in
the
sequel,
we
suppose
a
be
a
positive
integer.
The
previous
formulas
become:
g
w
11
=
K
1
(
a,
c
)
·
(
θ
2
)
2
a
+2
b
+
c
−
3
,
g
w
12
=
g
w
21
=
0
,
(33)
g
w
22
=
K
2
(
a,
c
)
·
(
θ
2
)
2
a
+2
b
+
c
−
3
,
where
K
1
(
a,
c
)
=
(2
−
c
)
−
a
−
3
2
·
(
√
2
π
)
c
−
1
·
(2
a
−
1)!!
·
[(
c
−
1)(2
a
+
1)
+
2
−
c
]
,
K
2
(
a,
c
)
=
(2
−
c
)
−
a
−
5
2
·
(
√
2
π
)
c
−
1
·
(2
a
−
1)!!
·
·
[(
c
−
1)(2
a
+
1)(2
a
+
3)
+
(
c
−
2)(2
a
+
1)(2
c
−
5)
+
(
c
−
2)
3
]
.
Denote
by
M
the
set
of
all
pairs
(
a,
c
)
∈
N
∗
×
(0
,
2),
satisfying
the
(compatible
!)
system
of
inequalities
(
c
−
1)(2
a
+
1)
+
2
−
c
>
0
(
c
−
1)(2
a
+
1)(2
a
+
3)
+
(
c
−
2)(2
a
+
1)(2
c
−
5)
+
(
c
−
2)
3
>
0
.
For
each
pair
in
M
,
g
w
given
in
(5.2)
is
a
Riemannian
metric.
In
this
case,
as
a
consequence
of
Remark
3
,
the
scalar
curvature
S
w
of
g
w
is
S
w
=
(2
−
c
)
a
+
5
2
·
(
√
2
π
)
1
−
c
·
(2
a
+
2
b
+
c
−
3)
·
(
θ
2
)
1
−
2
a
−
2
b
−
c
[(
c
−
1)(2
a
+
1)(2
a
+
3)
+
(
c
−
2)(2
a
+
1)(2
c
−
5)
+
(
c
−
2)
3
]
·
(2
a
−
1)!!
.
(34)
On
Fisher-like
metrics
72
Remark
4.
(i)
Similar
examples
of
metrics
can
be
found
also
from
˜
g
w
,
h
w
,
α
w
,
β
w
,
ˆ
g
w
and
¯
g
w
(see
[
26
]
for
the
non-weighted
case).
(ii)
We
have
sgnS
w
=
sgn
(2
a
+
2
b
+
c
−
3)
.
The
scalar
curvature
S
w
is
constant
if
and
only
if
2
a
+
2
b
+
c
=
1
;
in
this
case,
it
is
negative
(as
expected).
(iii)
The
metrics
g
w
given
in
(5.2)
are
globally
conformal
with
the
Eu-
clidean
metric
on
the
space
of
parameters
(
θ
1
,
θ
2
)
.
The
(explicit)
conformal
factors
are
independent
of
θ
1
,
even
if
the
weighting
function
w
depends
on
both
parameters
θ
1
and
θ
2
.
The
scalar
curvature
function
S
w
in
(5.3)
is
also
dependent
only
on
the
”standard
deviation”
of
the
PDF
modeled
by
θ
2
.
Statistical
applications
dependent
only
on
the
standard
deviation
may
be
found,
for
example,
in
[
10
,
13
,
27
,
33
,
41
].
The
statistical
interpretation
of
the
sectional
curvature
of
the
Fisher-like
metrics
studied
in
the
weighted
case
is
similar
to
that
in
the
non-weighted
case
(for
the
later,
see
our
comments
in
[
26
]
and
references
therein).
The
independence
from
θ
1
,
mentioned
earlier,
is
probably
related
to
the
fact
that
the
parameter
b
is
not
subject
to
additional
algebraic
restrictions,
as
are
the
parameters
a
and
c
.
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