Digital Signal Processing Reference
In-Depth Information
3.
If
∇
is torsion free then
−
q
∇
(
x
, ξ)
is a random semi-metric in M and
−
E
(
q
∇
)
is nothing else than the Fisher information of
(
M
,
P
)
. The Fisher information
is definite iff the random semi-metric q
is definite.
∇
Hint
The proof is based on direct calculations. The efficient ingredient is the
identity
P
(
x
, ξ)(
dln
(
x
, ξ)(
X
))
d
ξ
=
0
∀
X
∈ X
(
M
),
∀
x
∈
M
.
Ξ
be a locally flat manifold. The random Hessian
Geometry versus of what we just discussed may be formulated as in the statements
below.
Roughly speaking, let
(
M
,
D
)
Theorem 4
Let
(
M
,
P
)
be a statistical model for a measurable set
(Ξ, Ω)
. Set
(
, ξ)
=
lo
g(
(
, ξ))
ln
x
P
x
. If the Fisher information quadratic form
2
g(
X
,
Y
)(
x
)
=
P
(
x
, ξ)((
dln
)(
x
, ξ))
(
X
,
Y
)
d
ξ
Ξ
is definite then M carries a random Hessian structure whose random Hessian
metric is
D
2
ln
−
q
D
(
x
, ξ)(
X
,
Y
)
=−
(
x
, ξ)(
X
,
Y
)
Hint
The degeneracy of
q
D
means that there exists non zero element
v
∈
TM
such that
q
D
(
x
, ξ)(v)
=
O
∀
ξ
∈
Ξ.
Actually the last two Theorems above learn us that in every statistical model
(
M
,
P
)
(for a measurable set
(Ξ, Ω)
)
whose Fisher information is definite there is
an emedding
∇→
q
∇
of the category of locally structures
(
M
,
∇
)
in the category
of random locally flat Codazzi structures
(
M
,
q
∇
,
∇
).
The mathematical expectation
E
(
q
∇
)
is a geometrical invariant of this embedding.
4.4 Twisted Cohomology. Transverse Statistical Structures
(Ξ, Ω)
(
,
)
(
(
, ξ))
Let data
,(M,D),
M
P
and
E
q
x
be as in the precedent section. Let
∇
be a torsion free linear connection in
M
. We have already observed that the Fisher
information quadratic form
g(
x
)
=−
E
(
q
∇
(
x
, ξ))
may be dengenerate.
